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Theorem itg1addlem4 19054
Description: Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1  |-  ( ph  ->  F  e.  dom  S.1 )
i1fadd.2  |-  ( ph  ->  G  e.  dom  S.1 )
itg1add.3  |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) ) )
itg1add.4  |-  P  =  (  +  |`  ( ran  F  X.  ran  G
) )
Assertion
Ref Expression
itg1addlem4  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) ) )
Distinct variable groups:    i, j,
y, z    y, I    y, P, z    i, F, j, y, z    i, G, j, y, z    ph, i,
j, y, z
Allowed substitution hints:    P( i, j)    I( z, i, j)

Proof of Theorem itg1addlem4
Dummy variables  w  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . 5  |-  ( ph  ->  F  e.  dom  S.1 )
2 i1fadd.2 . . . . 5  |-  ( ph  ->  G  e.  dom  S.1 )
31, 2i1fadd 19050 . . . 4  |-  ( ph  ->  ( F  o F  +  G )  e. 
dom  S.1 )
4 i1frn 19032 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
51, 4syl 15 . . . . . . 7  |-  ( ph  ->  ran  F  e.  Fin )
6 i1frn 19032 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
72, 6syl 15 . . . . . . 7  |-  ( ph  ->  ran  G  e.  Fin )
8 xpfi 7128 . . . . . . 7  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
95, 7, 8syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
10 ax-addf 8816 . . . . . . . . . 10  |-  +  :
( CC  X.  CC )
--> CC
11 ffn 5389 . . . . . . . . . 10  |-  (  +  : ( CC  X.  CC ) --> CC  ->  +  Fn  ( CC  X.  CC ) )
1210, 11ax-mp 8 . . . . . . . . 9  |-  +  Fn  ( CC  X.  CC )
13 i1ff 19031 . . . . . . . . . . . . 13  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
141, 13syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> RR )
15 frn 5395 . . . . . . . . . . . 12  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
1614, 15syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ran  F  C_  RR )
17 ax-resscn 8794 . . . . . . . . . . 11  |-  RR  C_  CC
1816, 17syl6ss 3191 . . . . . . . . . 10  |-  ( ph  ->  ran  F  C_  CC )
19 i1ff 19031 . . . . . . . . . . . . 13  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
202, 19syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  G : RR --> RR )
21 frn 5395 . . . . . . . . . . . 12  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
2220, 21syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ran  G  C_  RR )
2322, 17syl6ss 3191 . . . . . . . . . 10  |-  ( ph  ->  ran  G  C_  CC )
24 xpss12 4792 . . . . . . . . . 10  |-  ( ( ran  F  C_  CC  /\ 
ran  G  C_  CC )  ->  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )
2518, 23, 24syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )
26 fnssres 5357 . . . . . . . . 9  |-  ( (  +  Fn  ( CC 
X.  CC )  /\  ( ran  F  X.  ran  G )  C_  ( CC  X.  CC ) )  -> 
(  +  |`  ( ran  F  X.  ran  G
) )  Fn  ( ran  F  X.  ran  G
) )
2712, 25, 26sylancr 644 . . . . . . . 8  |-  ( ph  ->  (  +  |`  ( ran  F  X.  ran  G
) )  Fn  ( ran  F  X.  ran  G
) )
28 itg1add.4 . . . . . . . . 9  |-  P  =  (  +  |`  ( ran  F  X.  ran  G
) )
2928fneq1i 5338 . . . . . . . 8  |-  ( P  Fn  ( ran  F  X.  ran  G )  <->  (  +  |`  ( ran  F  X.  ran  G ) )  Fn  ( ran  F  X.  ran  G ) )
3027, 29sylibr 203 . . . . . . 7  |-  ( ph  ->  P  Fn  ( ran 
F  X.  ran  G
) )
31 dffn4 5457 . . . . . . 7  |-  ( P  Fn  ( ran  F  X.  ran  G )  <->  P :
( ran  F  X.  ran  G ) -onto-> ran  P
)
3230, 31sylib 188 . . . . . 6  |-  ( ph  ->  P : ( ran 
F  X.  ran  G
) -onto-> ran  P )
33 fofi 7142 . . . . . 6  |-  ( ( ( ran  F  X.  ran  G )  e.  Fin  /\  P : ( ran 
F  X.  ran  G
) -onto-> ran  P )  ->  ran  P  e.  Fin )
349, 32, 33syl2anc 642 . . . . 5  |-  ( ph  ->  ran  P  e.  Fin )
35 difss 3303 . . . . 5  |-  ( ran 
P  \  { 0 } )  C_  ran  P
36 ssfi 7083 . . . . 5  |-  ( ( ran  P  e.  Fin  /\  ( ran  P  \  { 0 } ) 
C_  ran  P )  ->  ( ran  P  \  { 0 } )  e.  Fin )
3734, 35, 36sylancl 643 . . . 4  |-  ( ph  ->  ( ran  P  \  { 0 } )  e.  Fin )
38 df-ov 5861 . . . . . . . . 9  |-  ( x  +  y )  =  (  +  `  <. x ,  y >. )
39 opelxpi 4721 . . . . . . . . . . 11  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G )  ->  <. x ,  y >.  e.  ( ran  F  X.  ran  G
) )
40 ffun 5391 . . . . . . . . . . . . 13  |-  (  +  : ( CC  X.  CC ) --> CC  ->  Fun  +  )
4110, 40ax-mp 8 . . . . . . . . . . . 12  |-  Fun  +
4210fdmi 5394 . . . . . . . . . . . . 13  |-  dom  +  =  ( CC  X.  CC )
4325, 42syl6sseqr 3225 . . . . . . . . . . . 12  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  dom  +  )
44 funfvima2 5754 . . . . . . . . . . . 12  |-  ( ( Fun  +  /\  ( ran  F  X.  ran  G
)  C_  dom  +  )  ->  ( <. x ,  y >.  e.  ( ran  F  X.  ran  G )  ->  (  +  ` 
<. x ,  y >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) ) )
4541, 43, 44sylancr 644 . . . . . . . . . . 11  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ran  F  X.  ran  G )  -> 
(  +  `  <. x ,  y >. )  e.  (  +  " ( ran  F  X.  ran  G
) ) ) )
4639, 45syl5 28 . . . . . . . . . 10  |-  ( ph  ->  ( ( x  e. 
ran  F  /\  y  e.  ran  G )  -> 
(  +  `  <. x ,  y >. )  e.  (  +  " ( ran  F  X.  ran  G
) ) ) )
4746imp 418 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  (  +  `  <. x ,  y >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) )
4838, 47syl5eqel 2367 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  ( x  +  y )  e.  (  +  " ( ran 
F  X.  ran  G
) ) )
4928rneqi 4905 . . . . . . . . 9  |-  ran  P  =  ran  (  +  |`  ( ran  F  X.  ran  G
) )
50 df-ima 4702 . . . . . . . . 9  |-  (  + 
" ( ran  F  X.  ran  G ) )  =  ran  (  +  |`  ( ran  F  X.  ran  G ) )
5149, 50eqtr4i 2306 . . . . . . . 8  |-  ran  P  =  (  +  " ( ran  F  X.  ran  G
) )
5248, 51syl6eleqr 2374 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  ( x  +  y )  e.  ran  P )
53 ffn 5389 . . . . . . . . 9  |-  ( F : RR --> RR  ->  F  Fn  RR )
5414, 53syl 15 . . . . . . . 8  |-  ( ph  ->  F  Fn  RR )
55 dffn3 5396 . . . . . . . 8  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
5654, 55sylib 188 . . . . . . 7  |-  ( ph  ->  F : RR --> ran  F
)
57 ffn 5389 . . . . . . . . 9  |-  ( G : RR --> RR  ->  G  Fn  RR )
5820, 57syl 15 . . . . . . . 8  |-  ( ph  ->  G  Fn  RR )
59 dffn3 5396 . . . . . . . 8  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
6058, 59sylib 188 . . . . . . 7  |-  ( ph  ->  G : RR --> ran  G
)
61 reex 8828 . . . . . . . 8  |-  RR  e.  _V
6261a1i 10 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
63 inidm 3378 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
6452, 56, 60, 62, 62, 63off 6093 . . . . . 6  |-  ( ph  ->  ( F  o F  +  G ) : RR --> ran  P )
65 frn 5395 . . . . . 6  |-  ( ( F  o F  +  G ) : RR --> ran  P  ->  ran  ( F  o F  +  G
)  C_  ran  P )
6664, 65syl 15 . . . . 5  |-  ( ph  ->  ran  ( F  o F  +  G )  C_ 
ran  P )
67 ssdif 3311 . . . . 5  |-  ( ran  ( F  o F  +  G )  C_  ran  P  ->  ( ran  ( F  o F  +  G )  \  {
0 } )  C_  ( ran  P  \  {
0 } ) )
6866, 67syl 15 . . . 4  |-  ( ph  ->  ( ran  ( F  o F  +  G
)  \  { 0 } )  C_  ( ran  P  \  { 0 } ) )
6916sselda 3180 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ran  F )  ->  y  e.  RR )
7022sselda 3180 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  RR )
7169, 70anim12dan 810 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  ran  F  /\  z  e.  ran  G ) )  ->  ( y  e.  RR  /\  z  e.  RR ) )
72 readdcl 8820 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  z  e.  RR )  ->  ( y  +  z )  e.  RR )
7371, 72syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ran  F  /\  z  e.  ran  G ) )  ->  ( y  +  z )  e.  RR )
7473ralrimivva 2635 . . . . . . 7  |-  ( ph  ->  A. y  e.  ran  F A. z  e.  ran  G ( y  +  z )  e.  RR )
75 funimassov 5997 . . . . . . . 8  |-  ( ( Fun  +  /\  ( ran  F  X.  ran  G
)  C_  dom  +  )  ->  ( (  + 
" ( ran  F  X.  ran  G ) ) 
C_  RR  <->  A. y  e.  ran  F A. z  e.  ran  G ( y  +  z )  e.  RR ) )
7641, 43, 75sylancr 644 . . . . . . 7  |-  ( ph  ->  ( (  +  "
( ran  F  X.  ran  G ) )  C_  RR 
<-> 
A. y  e.  ran  F A. z  e.  ran  G ( y  +  z )  e.  RR ) )
7774, 76mpbird 223 . . . . . 6  |-  ( ph  ->  (  +  " ( ran  F  X.  ran  G
) )  C_  RR )
7851, 77syl5eqss 3222 . . . . 5  |-  ( ph  ->  ran  P  C_  RR )
79 ssdif 3311 . . . . 5  |-  ( ran 
P  C_  RR  ->  ( ran  P  \  {
0 } )  C_  ( RR  \  { 0 } ) )
8078, 79syl 15 . . . 4  |-  ( ph  ->  ( ran  P  \  { 0 } ) 
C_  ( RR  \  { 0 } ) )
81 itg1val2 19039 . . . 4  |-  ( ( ( F  o F  +  G )  e. 
dom  S.1  /\  ( ( ran  P  \  {
0 } )  e. 
Fin  /\  ( ran  ( F  o F  +  G )  \  {
0 } )  C_  ( ran  P  \  {
0 } )  /\  ( ran  P  \  {
0 } )  C_  ( RR  \  { 0 } ) ) )  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  o F  +  G ) " {
w } ) ) ) )
823, 37, 68, 80, 81syl13anc 1184 . . 3  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  o F  +  G ) " {
w } ) ) ) )
8320adantr 451 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  G : RR --> RR )
847adantr 451 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ran  G  e.  Fin )
85 inss2 3390 . . . . . . . . 9  |-  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )
8685a1i 10 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } ) )
87 i1fima 19033 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( `' F " { ( w  -  z ) } )  e.  dom  vol )
881, 87syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( `' F " { ( w  -  z ) } )  e.  dom  vol )
89 i1fima 19033 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
902, 89syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
91 inmbl 18899 . . . . . . . . . 10  |-  ( ( ( `' F " { ( w  -  z ) } )  e.  dom  vol  /\  ( `' G " { z } )  e.  dom  vol )  ->  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
9288, 90, 91syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
9392ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
9435, 78syl5ss 3190 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  P  \  { 0 } ) 
C_  RR )
9594sselda 3180 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  w  e.  RR )
9695adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  e.  RR )
9770adantlr 695 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  RR )
9896, 97resubcld 9211 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( w  -  z
)  e.  RR )
9996recnd 8861 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  e.  CC )
10097recnd 8861 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  CC )
10199, 100npcand 9161 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z )  +  z )  =  w )
102 eldifsni 3750 . . . . . . . . . . . . 13  |-  ( w  e.  ( ran  P  \  { 0 } )  ->  w  =/=  0
)
103102ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  w  =/=  0 )
104101, 103eqnetrd 2464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z )  +  z )  =/=  0 )
105 oveq12 5867 . . . . . . . . . . . . 13  |-  ( ( ( w  -  z
)  =  0  /\  z  =  0 )  ->  ( ( w  -  z )  +  z )  =  ( 0  +  0 ) )
106 00id 8987 . . . . . . . . . . . . 13  |-  ( 0  +  0 )  =  0
107105, 106syl6eq 2331 . . . . . . . . . . . 12  |-  ( ( ( w  -  z
)  =  0  /\  z  =  0 )  ->  ( ( w  -  z )  +  z )  =  0 )
108107necon3ai 2486 . . . . . . . . . . 11  |-  ( ( ( w  -  z
)  +  z )  =/=  0  ->  -.  ( ( w  -  z )  =  0  /\  z  =  0 ) )
109104, 108syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  -.  ( ( w  -  z )  =  0  /\  z  =  0 ) )
110 itg1add.3 . . . . . . . . . . 11  |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) ) )
1111, 2, 110itg1addlem3 19053 . . . . . . . . . 10  |-  ( ( ( ( w  -  z )  e.  RR  /\  z  e.  RR )  /\  -.  ( ( w  -  z )  =  0  /\  z  =  0 ) )  ->  ( ( w  -  z ) I z )  =  ( vol `  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
11298, 97, 109, 111syl21anc 1181 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  =  ( vol `  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
1131, 2, 110itg1addlem2 19052 . . . . . . . . . . 11  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
114113ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  I : ( RR  X.  RR ) --> RR )
115 fovrn 5990 . . . . . . . . . 10  |-  ( ( I : ( RR 
X.  RR ) --> RR 
/\  ( w  -  z )  e.  RR  /\  z  e.  RR )  ->  ( ( w  -  z ) I z )  e.  RR )
116114, 98, 97, 115syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  e.  RR )
117112, 116eqeltrrd 2358 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( vol `  (
( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
11883, 84, 86, 93, 117itg1addlem1 19047 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( vol `  U_ z  e.  ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )  =  sum_ z  e.  ran  G ( vol `  (
( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
11995recnd 8861 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  w  e.  CC )
1201, 2i1faddlem 19048 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  CC )  ->  ( `' ( F  o F  +  G ) " { w } )  =  U_ z  e. 
ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )
121119, 120syldan 456 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( `' ( F  o F  +  G ) " {
w } )  = 
U_ z  e.  ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) )
122121fveq2d 5529 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( vol `  ( `' ( F  o F  +  G ) " { w } ) )  =  ( vol `  U_ z  e.  ran  G ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
123112sumeq2dv 12176 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  sum_ z  e.  ran  G ( ( w  -  z ) I z )  =  sum_ z  e.  ran  G ( vol `  ( ( `' F " { ( w  -  z ) } )  i^i  ( `' G " { z } ) ) ) )
124118, 122, 1233eqtr4d 2325 . . . . . 6  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( vol `  ( `' ( F  o F  +  G ) " { w } ) )  =  sum_ z  e.  ran  G ( ( w  -  z ) I z ) )
125124oveq2d 5874 . . . . 5  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( w  x.  ( vol `  ( `' ( F  o F  +  G ) " { w } ) ) )  =  ( w  x.  sum_ z  e.  ran  G ( ( w  -  z ) I z ) ) )
126116recnd 8861 . . . . . 6  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( w  -  z ) I z )  e.  CC )
12784, 119, 126fsummulc2 12246 . . . . 5  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( w  x. 
sum_ z  e.  ran  G ( ( w  -  z ) I z ) )  =  sum_ z  e.  ran  G ( w  x.  ( ( w  -  z ) I z ) ) )
128125, 127eqtrd 2315 . . . 4  |-  ( (
ph  /\  w  e.  ( ran  P  \  {
0 } ) )  ->  ( w  x.  ( vol `  ( `' ( F  o F  +  G ) " { w } ) ) )  =  sum_ z  e.  ran  G ( w  x.  ( ( w  -  z ) I z ) ) )
129128sumeq2dv 12176 . . 3  |-  ( ph  -> 
sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( vol `  ( `' ( F  o F  +  G ) " { w } ) ) )  =  sum_ w  e.  ( ran  P  \  { 0 } )
sum_ z  e.  ran  G ( w  x.  (
( w  -  z
) I z ) ) )
13099, 126mulcld 8855 . . . . 5  |-  ( ( ( ph  /\  w  e.  ( ran  P  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( w  x.  (
( w  -  z
) I z ) )  e.  CC )
131130anasss 628 . . . 4  |-  ( (
ph  /\  ( w  e.  ( ran  P  \  { 0 } )  /\  z  e.  ran  G ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  e.  CC )
13237, 7, 131fsumcom 12238 . . 3  |-  ( ph  -> 
sum_ w  e.  ( ran  P  \  { 0 } ) sum_ z  e.  ran  G ( w  x.  ( ( w  -  z ) I z ) )  = 
sum_ z  e.  ran  G
sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( ( w  -  z ) I z ) ) )
13382, 129, 1323eqtrd 2319 . 2  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ z  e.  ran  G sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  (
( w  -  z
) I z ) ) )
134 oveq1 5865 . . . . . . 7  |-  ( y  =  ( w  -  z )  ->  (
y  +  z )  =  ( ( w  -  z )  +  z ) )
135 oveq1 5865 . . . . . . 7  |-  ( y  =  ( w  -  z )  ->  (
y I z )  =  ( ( w  -  z ) I z ) )
136134, 135oveq12d 5876 . . . . . 6  |-  ( y  =  ( w  -  z )  ->  (
( y  +  z )  x.  ( y I z ) )  =  ( ( ( w  -  z )  +  z )  x.  ( ( w  -  z ) I z ) ) )
13734adantr 451 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  P  e.  Fin )
13878adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  P 
C_  RR )
139138sselda 3180 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  v  e.  RR )
14070adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  z  e.  RR )
141139, 140resubcld 9211 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  ( v  -  z )  e.  RR )
142141ex 423 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  ->  ( v  -  z
)  e.  RR ) )
143139recnd 8861 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  v  e.  ran  P )  ->  v  e.  CC )
144143adantrr 697 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  v  e.  CC )
14578sselda 3180 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ran  P )  ->  y  e.  RR )
146145ad2ant2rl 729 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  y  e.  RR )
147146recnd 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  y  e.  CC )
14870recnd 8861 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  CC )
149148adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  z  e.  CC )
150 subcan2 9072 . . . . . . . . . 10  |-  ( ( v  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( v  -  z
)  =  ( y  -  z )  <->  v  =  y ) )
151144, 147, 149, 150syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( v  e.  ran  P  /\  y  e.  ran  P ) )  ->  (
( v  -  z
)  =  ( y  -  z )  <->  v  =  y ) )
152151ex 423 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
( v  e.  ran  P  /\  y  e.  ran  P )  ->  ( (
v  -  z )  =  ( y  -  z )  <->  v  =  y ) ) )
153142, 152dom2lem 6901 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR )
154 f1f1orn 5483 . . . . . . 7  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR  ->  ( v  e.  ran  P  |->  ( v  -  z ) ) : ran  P -1-1-onto-> ran  (
v  e.  ran  P  |->  ( v  -  z
) ) )
155153, 154syl 15 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P -1-1-onto-> ran  ( v  e.  ran  P 
|->  ( v  -  z
) ) )
156 oveq1 5865 . . . . . . . 8  |-  ( v  =  w  ->  (
v  -  z )  =  ( w  -  z ) )
157 eqid 2283 . . . . . . . 8  |-  ( v  e.  ran  P  |->  ( v  -  z ) )  =  ( v  e.  ran  P  |->  ( v  -  z ) )
158 ovex 5883 . . . . . . . 8  |-  ( w  -  z )  e. 
_V
159156, 157, 158fvmpt 5602 . . . . . . 7  |-  ( w  e.  ran  P  -> 
( ( v  e. 
ran  P  |->  ( v  -  z ) ) `
 w )  =  ( w  -  z
) )
160159adantl 452 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( v  e.  ran  P  |->  ( v  -  z ) ) `  w )  =  ( w  -  z ) )
161 f1f 5437 . . . . . . . . . . 11  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-1-1-> RR  ->  ( v  e.  ran  P  |->  ( v  -  z ) ) : ran  P --> RR )
162 frn 5395 . . . . . . . . . . 11  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P --> RR  ->  ran  ( v  e.  ran  P  |->  ( v  -  z ) )  C_  RR )
163153, 161, 1623syl 18 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  C_  RR )
164163sselda 3180 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  y  e.  RR )
16570adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  z  e.  RR )
166164, 165readdcld 8862 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
y  +  z )  e.  RR )
167113ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  I : ( RR  X.  RR ) --> RR )
168 fovrn 5990 . . . . . . . . 9  |-  ( ( I : ( RR 
X.  RR ) --> RR 
/\  y  e.  RR  /\  z  e.  RR )  ->  ( y I z )  e.  RR )
169167, 164, 165, 168syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
y I z )  e.  RR )
170166, 169remulcld 8863 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  e.  RR )
171170recnd 8861 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  e.  CC )
172136, 137, 155, 160, 171fsumf1o 12196 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ y  e.  ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) ( ( y  +  z )  x.  (
y I z ) )  =  sum_ w  e.  ran  P ( ( ( w  -  z
)  +  z )  x.  ( ( w  -  z ) I z ) ) )
173138sselda 3180 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  w  e.  RR )
174173recnd 8861 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  w  e.  CC )
175148adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  z  e.  CC )
176174, 175npcand 9161 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( w  -  z )  +  z )  =  w )
177176oveq1d 5873 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ran  P )  ->  ( ( ( w  -  z )  +  z )  x.  ( ( w  -  z ) I z ) )  =  ( w  x.  ( ( w  -  z ) I z ) ) )
178177sumeq2dv 12176 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ w  e.  ran  P ( ( ( w  -  z
)  +  z )  x.  ( ( w  -  z ) I z ) )  = 
sum_ w  e.  ran  P ( w  x.  (
( w  -  z
) I z ) ) )
179172, 178eqtrd 2315 . . . 4  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ y  e.  ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) ( ( y  +  z )  x.  (
y I z ) )  =  sum_ w  e.  ran  P ( w  x.  ( ( w  -  z ) I z ) ) )
180 df-ov 5861 . . . . . . . . . . 11  |-  ( y  +  z )  =  (  +  `  <. y ,  z >. )
18143ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ran  F  X.  ran  G )  C_  dom  +  )
182 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  ran  F )
183 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  ran  G )
184 opelxpi 4721 . . . . . . . . . . . . 13  |-  ( ( y  e.  ran  F  /\  z  e.  ran  G )  ->  <. y ,  z >.  e.  ( ran  F  X.  ran  G
) )
185182, 183, 184syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  <. y ,  z
>.  e.  ( ran  F  X.  ran  G ) )
186 funfvima2 5754 . . . . . . . . . . . . 13  |-  ( ( Fun  +  /\  ( ran  F  X.  ran  G
)  C_  dom  +  )  ->  ( <. y ,  z >.  e.  ( ran  F  X.  ran  G )  ->  (  +  ` 
<. y ,  z >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) ) )
18741, 186mpan 651 . . . . . . . . . . . 12  |-  ( ( ran  F  X.  ran  G )  C_  dom  +  ->  (
<. y ,  z >.  e.  ( ran  F  X.  ran  G )  ->  (  +  `  <. y ,  z
>. )  e.  (  +  " ( ran  F  X.  ran  G ) ) ) )
188181, 185, 187sylc 56 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  (  +  `  <. y ,  z >.
)  e.  (  + 
" ( ran  F  X.  ran  G ) ) )
189180, 188syl5eqel 2367 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  (  +  " ( ran 
F  X.  ran  G
) ) )
190189, 51syl6eleqr 2374 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  ran  P )
19169adantlr 695 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  RR )
192191recnd 8861 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  CC )
193148adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  CC )
194192, 193pncand 9158 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  -  z )  =  y )
195194eqcomd 2288 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  =  ( ( y  +  z )  -  z ) )
196 oveq1 5865 . . . . . . . . . . 11  |-  ( v  =  ( y  +  z )  ->  (
v  -  z )  =  ( ( y  +  z )  -  z ) )
197196eqeq2d 2294 . . . . . . . . . 10  |-  ( v  =  ( y  +  z )  ->  (
y  =  ( v  -  z )  <->  y  =  ( ( y  +  z )  -  z
) ) )
198197rspcev 2884 . . . . . . . . 9  |-  ( ( ( y  +  z )  e.  ran  P  /\  y  =  (
( y  +  z )  -  z ) )  ->  E. v  e.  ran  P  y  =  ( v  -  z
) )
199190, 195, 198syl2anc 642 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  E. v  e.  ran  P  y  =  ( v  -  z ) )
200199ralrimiva 2626 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  A. y  e.  ran  F E. v  e.  ran  P  y  =  ( v  -  z
) )
201 ssabral 3244 . . . . . . 7  |-  ( ran 
F  C_  { y  |  E. v  e.  ran  P  y  =  ( v  -  z ) }  <->  A. y  e.  ran  F E. v  e.  ran  P  y  =  ( v  -  z ) )
202200, 201sylibr 203 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  { y  |  E. v  e.  ran  P  y  =  ( v  -  z ) } )
203157rnmpt 4925 . . . . . 6  |-  ran  (
v  e.  ran  P  |->  ( v  -  z
) )  =  {
y  |  E. v  e.  ran  P  y  =  ( v  -  z
) }
204202, 203syl6sseqr 3225 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  ran  ( v  e.  ran  P  |->  ( v  -  z ) ) )
20570adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  RR )
206191, 205readdcld 8862 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y  +  z )  e.  RR )
207113ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  I : ( RR  X.  RR ) --> RR )
208207, 191, 205, 168syl3anc 1182 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  RR )
209206, 208remulcld 8863 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  RR )
210209recnd 8861 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  CC )
211 ssdif 3311 . . . . . . . 8  |-  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  C_  RR  ->  ( ran  ( v  e.  ran  P  |->  ( v  -  z ) )  \  ran  F
)  C_  ( RR  \  ran  F ) )
212163, 211syl 15 . . . . . . 7  |-  ( (
ph  /\  z  e.  ran  G )  ->  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F )  C_  ( RR  \  ran  F ) )
213212sselda 3180 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F ) )  ->  y  e.  ( RR  \  ran  F ) )
214 eldifi 3298 . . . . . . . . . . . . 13  |-  ( y  e.  ( RR  \  ran  F )  ->  y  e.  RR )
215214ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  y  e.  RR )
21670adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  z  e.  RR )
217 simprr 733 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  ( y  =  0  /\  z  =  0 ) )
2181, 2, 110itg1addlem3 19053 . . . . . . . . . . . 12  |-  ( ( ( y  e.  RR  /\  z  e.  RR )  /\  -.  ( y  =  0  /\  z  =  0 ) )  ->  ( y I z )  =  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
219215, 216, 217, 218syl21anc 1181 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y I z )  =  ( vol `  (
( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
220 inss1 3389 . . . . . . . . . . . . . . 15  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { y } )
221 eldifn 3299 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( RR  \  ran  F )  ->  -.  y  e.  ran  F )
222221ad2antrl 708 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  y  e.  ran  F )
223 vex 2791 . . . . . . . . . . . . . . . . . . . 20  |-  y  e. 
_V
224 vex 2791 . . . . . . . . . . . . . . . . . . . . 21  |-  v  e. 
_V
225224eliniseg 5042 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  _V  ->  (
v  e.  ( `' F " { y } )  <->  v F
y ) )
226223, 225ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  ( v  e.  ( `' F " { y } )  <-> 
v F y )
227224, 223brelrn 4909 . . . . . . . . . . . . . . . . . . 19  |-  ( v F y  ->  y  e.  ran  F )
228226, 227sylbi 187 . . . . . . . . . . . . . . . . . 18  |-  ( v  e.  ( `' F " { y } )  ->  y  e.  ran  F )
229222, 228nsyl 113 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  -.  v  e.  ( `' F " { y } ) )
230229pm2.21d 98 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
v  e.  ( `' F " { y } )  ->  v  e.  (/) ) )
231230ssrdv 3185 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  ( `' F " { y } )  C_  (/) )
232220, 231syl5ss 3190 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  (/) )
233 ss0 3485 . . . . . . . . . . . . . 14  |-  ( ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  (/)  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  =  (/) )
234232, 233syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( `' F " { y } )  i^i  ( `' G " { z } ) )  =  (/) )
235234fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )  =  ( vol `  (/) ) )
236 0mbl 18897 . . . . . . . . . . . . . 14  |-  (/)  e.  dom  vol
237 mblvol 18889 . . . . . . . . . . . . . 14  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol * `  (/) ) )
238236, 237ax-mp 8 . . . . . . . . . . . . 13  |-  ( vol `  (/) )  =  ( vol * `  (/) )
239 ovol0 18852 . . . . . . . . . . . . 13  |-  ( vol
* `  (/) )  =  0
240238, 239eqtri 2303 . . . . . . . . . . . 12  |-  ( vol `  (/) )  =  0
241235, 240syl6eq 2331 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )  =  0 )
242219, 241eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y I z )  =  0 )
243242oveq2d 5874 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  =  ( ( y  +  z )  x.  0 ) )
244215, 216readdcld 8862 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y  +  z )  e.  RR )
245244recnd 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
y  +  z )  e.  CC )
246245mul01d 9011 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  0 )  =  0 )
247243, 246eqtrd 2315 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  ( y  e.  ( RR  \  ran  F
)  /\  -.  (
y  =  0  /\  z  =  0 ) ) )  ->  (
( y  +  z )  x.  ( y I z ) )  =  0 )
248247expr 598 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( RR  \  ran  F ) )  ->  ( -.  (
y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 ) )
249 oveq12 5867 . . . . . . . . . 10  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y  +  z )  =  ( 0  +  0 ) )
250249, 106syl6eq 2331 . . . . . . . . 9  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y  +  z )  =  0 )
251 oveq12 5867 . . . . . . . . . 10  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y I z )  =  ( 0 I 0 ) )
252 0re 8838 . . . . . . . . . . 11  |-  0  e.  RR
253 iftrue 3571 . . . . . . . . . . . 12  |-  ( ( i  =  0  /\  j  =  0 )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  0 )
254 c0ex 8832 . . . . . . . . . . . 12  |-  0  e.  _V
255253, 110, 254ovmpt2a 5978 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  0  e.  RR )  ->  ( 0 I 0 )  =  0 )
256252, 252, 255mp2an 653 . . . . . . . . . 10  |-  ( 0 I 0 )  =  0
257251, 256syl6eq 2331 . . . . . . . . 9  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( y I z )  =  0 )
258250, 257oveq12d 5876 . . . . . . . 8  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  ( 0  x.  0 ) )
259 0cn 8831 . . . . . . . . 9  |-  0  e.  CC
260259mul01i 9002 . . . . . . . 8  |-  ( 0  x.  0 )  =  0
261258, 260syl6eq 2331 . . . . . . 7  |-  ( ( y  =  0  /\  z  =  0 )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 )
262248, 261pm2.61d2 152 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( RR  \  ran  F ) )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  0 )
263213, 262syldan 456 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ( ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  \  ran  F ) )  ->  (
( y  +  z )  x.  ( y I z ) )  =  0 )
264 f1ofo 5479 . . . . . . 7  |-  ( ( v  e.  ran  P  |->  ( v  -  z
) ) : ran  P -1-1-onto-> ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-onto->
ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) )
265155, 264syl 15 . . . . . 6  |-  ( (
ph  /\  z  e.  ran  G )  ->  (
v  e.  ran  P  |->  ( v  -  z
) ) : ran  P
-onto->
ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) )
266 fofi 7142 . . . . . 6  |-  ( ( ran  P  e.  Fin  /\  ( v  e.  ran  P 
|->  ( v  -  z
) ) : ran  P
-onto->
ran  ( v  e. 
ran  P  |->  ( v  -  z ) ) )  ->  ran  ( v  e.  ran  P  |->  ( v  -  z ) )  e.  Fin )
267137, 265, 266syl2anc 642 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  ( v  e.  ran  P 
|->  ( v  -  z
) )  e.  Fin )
268204, 210, 263, 267fsumss 12198 . . . 4  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ y  e.  ran  F ( ( y  +  z )  x.  ( y I z ) )  = 
sum_ y  e.  ran  ( v  e.  ran  P 
|->  ( v  -  z
) ) ( ( y  +  z )  x.  ( y I z ) ) )
26935a1i 10 . . . . 5  |-  ( (
ph  /\  z  e.  ran  G )  ->  ( ran  P  \  { 0 } )  C_  ran  P )
270130an32s 779 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ( ran  P 
\  { 0 } ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  e.  CC )
271 dfin4 3409 . . . . . . . 8  |-  ( ran 
P  i^i  { 0 } )  =  ( ran  P  \  ( ran  P  \  { 0 } ) )
272 inss2 3390 . . . . . . . 8  |-  ( ran 
P  i^i  { 0 } )  C_  { 0 }
273271, 272eqsstr3i 3209 . . . . . . 7  |-  ( ran 
P  \  ( ran  P 
\  { 0 } ) )  C_  { 0 }
274273sseli 3176 . . . . . 6  |-  ( w  e.  ( ran  P  \  ( ran  P  \  { 0 } ) )  ->  w  e.  { 0 } )
275 elsni 3664 . . . . . . . . 9  |-  ( w  e.  { 0 }  ->  w  =  0 )
276275adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  w  =  0 )
277276oveq1d 5873 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  ( 0  x.  ( ( w  -  z ) I z ) ) )
278113ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  I : ( RR  X.  RR ) --> RR )
279276, 252syl6eqel 2371 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  w  e.  RR )
28070adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  z  e.  RR )
281279, 280resubcld 9211 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  -  z )  e.  RR )
282278, 281, 280, 115syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
( w  -  z
) I z )  e.  RR )
283282recnd 8861 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
( w  -  z
) I z )  e.  CC )
284283mul02d 9010 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
0  x.  ( ( w  -  z ) I z ) )  =  0 )
285277, 284eqtrd 2315 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  { 0 } )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  0 )
286274, 285sylan2 460 . . . . 5  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  w  e.  ( ran  P 
\  ( ran  P  \  { 0 } ) ) )  ->  (
w  x.  ( ( w  -  z ) I z ) )  =  0 )
287269, 270, 286, 137fsumss 12198 . . . 4  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  (
( w  -  z
) I z ) )  =  sum_ w  e.  ran  P ( w  x.  ( ( w  -  z ) I z ) ) )
288179, 268, 2873eqtr4d 2325 . . 3  |-  ( (
ph  /\  z  e.  ran  G )  ->  sum_ y  e.  ran  F ( ( y  +  z )  x.  ( y I z ) )  = 
sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  ( ( w  -  z ) I z ) ) )
289288sumeq2dv 12176 . 2  |-  ( ph  -> 
sum_ z  e.  ran  G
sum_ y  e.  ran  F ( ( y  +  z )  x.  (
y I z ) )  =  sum_ z  e.  ran  G sum_ w  e.  ( ran  P  \  { 0 } ) ( w  x.  (
( w  -  z
) I z ) ) )
290210anasss 628 . . 3  |-  ( (
ph  /\  ( z  e.  ran  G  /\  y  e.  ran  F ) )  ->  ( ( y  +  z )  x.  ( y I z ) )  e.  CC )
2917, 5, 290fsumcom 12238 . 2  |-  ( ph  -> 
sum_ z  e.  ran  G
sum_ y  e.  ran  F ( ( y  +  z )  x.  (
y I z ) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) ) )
292133, 289, 2913eqtr2d 2321 1  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   {csn 3640   <.cop 3643   U_ciun 3905   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    o Fcof 6076   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    - cmin 9037   sum_csu 12158   vol
*covol 18822   volcvol 18823   S.1citg1 18970
This theorem is referenced by:  itg1addlem5  19055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976
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