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Theorem itg1addlem5 19108
Description: Lemma for itg1add . (Contributed by Mario Carneiro, 27-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
itg1addlem5  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  ( ( S.1 `  F
)  +  ( S.1 `  G ) ) )
Distinct variable groups:    i, j, F    i, G, j    ph, i,
j
Allowed substitution hints:    P( i, j)    I( i, j)

Proof of Theorem itg1addlem5
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . 4  |-  ( ph  ->  F  e.  dom  S.1 )
2 i1frn 19085 . . . 4  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
31, 2syl 15 . . 3  |-  ( ph  ->  ran  F  e.  Fin )
4 i1fadd.2 . . . . . 6  |-  ( ph  ->  G  e.  dom  S.1 )
5 i1frn 19085 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
64, 5syl 15 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
76adantr 451 . . . 4  |-  ( (
ph  /\  y  e.  ran  F )  ->  ran  G  e.  Fin )
8 i1ff 19084 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
91, 8syl 15 . . . . . . . . 9  |-  ( ph  ->  F : RR --> RR )
10 frn 5433 . . . . . . . . 9  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
119, 10syl 15 . . . . . . . 8  |-  ( ph  ->  ran  F  C_  RR )
1211sselda 3214 . . . . . . 7  |-  ( (
ph  /\  y  e.  ran  F )  ->  y  e.  RR )
1312adantr 451 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  y  e.  RR )
1413recnd 8906 . . . . 5  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  y  e.  CC )
15 itg1add.3 . . . . . . . . 9  |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) ) )
161, 4, 15itg1addlem2 19105 . . . . . . . 8  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
1716ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  I : ( RR  X.  RR ) --> RR )
18 i1ff 19084 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
194, 18syl 15 . . . . . . . . . 10  |-  ( ph  ->  G : RR --> RR )
20 frn 5433 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
2119, 20syl 15 . . . . . . . . 9  |-  ( ph  ->  ran  G  C_  RR )
2221sselda 3214 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  RR )
2322adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  z  e.  RR )
24 fovrn 6032 . . . . . . 7  |-  ( ( I : ( RR 
X.  RR ) --> RR 
/\  y  e.  RR  /\  z  e.  RR )  ->  ( y I z )  e.  RR )
2517, 13, 23, 24syl3anc 1182 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y I z )  e.  RR )
2625recnd 8906 . . . . 5  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y I z )  e.  CC )
2714, 26mulcld 8900 . . . 4  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  e.  CC )
287, 27fsumcl 12253 . . 3  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( y  x.  ( y I z ) )  e.  CC )
2923recnd 8906 . . . . 5  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  z  e.  CC )
3029, 26mulcld 8900 . . . 4  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( z  x.  ( y I z ) )  e.  CC )
317, 30fsumcl 12253 . . 3  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( z  x.  ( y I z ) )  e.  CC )
323, 28, 31fsumadd 12258 . 2  |-  ( ph  -> 
sum_ y  e.  ran  F ( sum_ z  e.  ran  G ( y  x.  (
y I z ) )  +  sum_ z  e.  ran  G ( z  x.  ( y I z ) ) )  =  ( sum_ y  e.  ran  F sum_ z  e.  ran  G ( y  x.  ( y I z ) )  + 
sum_ y  e.  ran  F
sum_ z  e.  ran  G ( z  x.  (
y I z ) ) ) )
33 itg1add.4 . . . 4  |-  P  =  (  +  |`  ( ran  F  X.  ran  G
) )
341, 4, 15, 33itg1addlem4 19107 . . 3  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) ) )
3514, 29, 26adddird 8905 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( ( y  +  z )  x.  ( y I z ) )  =  ( ( y  x.  (
y I z ) )  +  ( z  x.  ( y I z ) ) ) )
3635sumeq2dv 12223 . . . . 5  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) )  = 
sum_ z  e.  ran  G ( ( y  x.  ( y I z ) )  +  ( z  x.  ( y I z ) ) ) )
377, 27, 30fsumadd 12258 . . . . 5  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( ( y  x.  ( y I z ) )  +  ( z  x.  ( y I z ) ) )  =  ( sum_ z  e.  ran  G ( y  x.  (
y I z ) )  +  sum_ z  e.  ran  G ( z  x.  ( y I z ) ) ) )
3836, 37eqtrd 2348 . . . 4  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) )  =  ( sum_ z  e.  ran  G ( y  x.  (
y I z ) )  +  sum_ z  e.  ran  G ( z  x.  ( y I z ) ) ) )
3938sumeq2dv 12223 . . 3  |-  ( ph  -> 
sum_ y  e.  ran  F
sum_ z  e.  ran  G ( ( y  +  z )  x.  (
y I z ) )  =  sum_ y  e.  ran  F ( sum_ z  e.  ran  G ( y  x.  ( y I z ) )  +  sum_ z  e.  ran  G ( z  x.  (
y I z ) ) ) )
4034, 39eqtrd 2348 . 2  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  sum_ y  e.  ran  F (
sum_ z  e.  ran  G ( y  x.  (
y I z ) )  +  sum_ z  e.  ran  G ( z  x.  ( y I z ) ) ) )
41 itg1val 19091 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ y  e.  ( ran  F  \  {
0 } ) ( y  x.  ( vol `  ( `' F " { y } ) ) ) )
421, 41syl 15 . . . 4  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ y  e.  ( ran  F  \  { 0 } ) ( y  x.  ( vol `  ( `' F " { y } ) ) ) )
4319adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  G : RR --> RR )
446adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ran  G  e.  Fin )
45 inss2 3424 . . . . . . . . . 10  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )
4645a1i 10 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } ) )
47 i1fima 19086 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  ( `' F " { y } )  e.  dom  vol )
481, 47syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( `' F " { y } )  e.  dom  vol )
4948ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( `' F " { y } )  e.  dom  vol )
50 i1fima 19086 . . . . . . . . . . . 12  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
514, 50syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
5251ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( `' G " { z } )  e.  dom  vol )
53 inmbl 18952 . . . . . . . . . 10  |-  ( ( ( `' F " { y } )  e.  dom  vol  /\  ( `' G " { z } )  e.  dom  vol )  ->  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
5449, 52, 53syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
55 difss 3337 . . . . . . . . . . . . . 14  |-  ( ran 
F  \  { 0 } )  C_  ran  F
5655, 11syl5ss 3224 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
5756sselda 3214 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  y  e.  RR )
5857adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
y  e.  RR )
5921adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ran  G  C_  RR )
6059sselda 3214 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  RR )
61 eldifsni 3784 . . . . . . . . . . . . 13  |-  ( y  e.  ( ran  F  \  { 0 } )  ->  y  =/=  0
)
6261ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
y  =/=  0 )
63 simpl 443 . . . . . . . . . . . . 13  |-  ( ( y  =  0  /\  z  =  0 )  ->  y  =  0 )
6463necon3ai 2519 . . . . . . . . . . . 12  |-  ( y  =/=  0  ->  -.  ( y  =  0  /\  z  =  0 ) )
6562, 64syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  -.  ( y  =  0  /\  z  =  0 ) )
661, 4, 15itg1addlem3 19106 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  z  e.  RR )  /\  -.  ( y  =  0  /\  z  =  0 ) )  ->  ( y I z )  =  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
6758, 60, 65, 66syl21anc 1181 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y I z )  =  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
6816ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  I : ( RR  X.  RR ) --> RR )
6968, 58, 60, 24syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y I z )  e.  RR )
7067, 69eqeltrrd 2391 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( vol `  (
( `' F " { y } )  i^i  ( `' G " { z } ) ) )  e.  RR )
7143, 44, 46, 54, 70itg1addlem1 19100 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  U_ z  e.  ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )  =  sum_ z  e.  ran  G ( vol `  (
( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
72 iunin2 4003 . . . . . . . . . 10  |-  U_ z  e.  ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  =  ( ( `' F " { y } )  i^i  U_ z  e.  ran  G ( `' G " { z } ) )
731adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  F  e.  dom  S.1 )
7473, 47syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } )  e.  dom  vol )
75 mblss 18943 . . . . . . . . . . . . 13  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( `' F " { y } ) 
C_  RR )
7674, 75syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } ) 
C_  RR )
77 iunid 3994 . . . . . . . . . . . . . . 15  |-  U_ z  e.  ran  G { z }  =  ran  G
7877imaeq2i 5047 . . . . . . . . . . . . . 14  |-  ( `' G " U_ z  e.  ran  G { z } )  =  ( `' G " ran  G
)
79 imaiun 5813 . . . . . . . . . . . . . 14  |-  ( `' G " U_ z  e.  ran  G { z } )  =  U_ z  e.  ran  G ( `' G " { z } )
80 cnvimarndm 5071 . . . . . . . . . . . . . 14  |-  ( `' G " ran  G
)  =  dom  G
8178, 79, 803eqtr3i 2344 . . . . . . . . . . . . 13  |-  U_ z  e.  ran  G ( `' G " { z } )  =  dom  G
82 fdm 5431 . . . . . . . . . . . . . 14  |-  ( G : RR --> RR  ->  dom 
G  =  RR )
8343, 82syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  dom  G  =  RR )
8481, 83syl5eq 2360 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  U_ z  e.  ran  G ( `' G " { z } )  =  RR )
8576, 84sseqtr4d 3249 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } ) 
C_  U_ z  e.  ran  G ( `' G " { z } ) )
86 df-ss 3200 . . . . . . . . . . 11  |-  ( ( `' F " { y } )  C_  U_ z  e.  ran  G ( `' G " { z } )  <->  ( ( `' F " { y } )  i^i  U_ z  e.  ran  G ( `' G " { z } ) )  =  ( `' F " { y } ) )
8785, 86sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( ( `' F " { y } )  i^i  U_ z  e.  ran  G ( `' G " { z } ) )  =  ( `' F " { y } ) )
8872, 87syl5req 2361 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } )  =  U_ z  e. 
ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )
8988fveq2d 5567 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { y } ) )  =  ( vol `  U_ z  e.  ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
9067sumeq2dv 12223 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  sum_ z  e.  ran  G ( y I z )  =  sum_ z  e.  ran  G ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
9171, 89, 903eqtr4d 2358 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { y } ) )  = 
sum_ z  e.  ran  G ( y I z ) )
9291oveq2d 5916 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( y  x.  ( vol `  ( `' F " { y } ) ) )  =  ( y  x. 
sum_ z  e.  ran  G ( y I z ) ) )
9357recnd 8906 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  y  e.  CC )
9469recnd 8906 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y I z )  e.  CC )
9544, 93, 94fsummulc2 12293 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( y  x. 
sum_ z  e.  ran  G ( y I z ) )  =  sum_ z  e.  ran  G ( y  x.  ( y I z ) ) )
9692, 95eqtrd 2348 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( y  x.  ( vol `  ( `' F " { y } ) ) )  =  sum_ z  e.  ran  G ( y  x.  (
y I z ) ) )
9796sumeq2dv 12223 . . . 4  |-  ( ph  -> 
sum_ y  e.  ( ran  F  \  {
0 } ) ( y  x.  ( vol `  ( `' F " { y } ) ) )  =  sum_ y  e.  ( ran  F 
\  { 0 } ) sum_ z  e.  ran  G ( y  x.  (
y I z ) ) )
9855a1i 10 . . . . 5  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  ran  F )
9958recnd 8906 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
y  e.  CC )
10099, 94mulcld 8900 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y  x.  (
y I z ) )  e.  CC )
10144, 100fsumcl 12253 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  sum_ z  e.  ran  G ( y  x.  (
y I z ) )  e.  CC )
102 dfin4 3443 . . . . . . . 8  |-  ( ran 
F  i^i  { 0 } )  =  ( ran  F  \  ( ran  F  \  { 0 } ) )
103 inss2 3424 . . . . . . . 8  |-  ( ran 
F  i^i  { 0 } )  C_  { 0 }
104102, 103eqsstr3i 3243 . . . . . . 7  |-  ( ran 
F  \  ( ran  F 
\  { 0 } ) )  C_  { 0 }
105104sseli 3210 . . . . . 6  |-  ( y  e.  ( ran  F  \  ( ran  F  \  { 0 } ) )  ->  y  e.  { 0 } )
106 elsni 3698 . . . . . . . . . . 11  |-  ( y  e.  { 0 }  ->  y  =  0 )
107106ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  y  = 
0 )
108107oveq1d 5915 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  =  ( 0  x.  ( y I z ) ) )
10916ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  I :
( RR  X.  RR )
--> RR )
110 0re 8883 . . . . . . . . . . . . 13  |-  0  e.  RR
111107, 110syl6eqel 2404 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  y  e.  RR )
11222adantlr 695 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  z  e.  RR )
113109, 111, 112, 24syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y
I z )  e.  RR )
114113recnd 8906 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y
I z )  e.  CC )
115114mul02d 9055 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( 0  x.  ( y I z ) )  =  0 )
116108, 115eqtrd 2348 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  =  0 )
117116sumeq2dv 12223 . . . . . . 7  |-  ( (
ph  /\  y  e.  { 0 } )  ->  sum_ z  e.  ran  G
( y  x.  (
y I z ) )  =  sum_ z  e.  ran  G 0 )
1186adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  { 0 } )  ->  ran  G  e.  Fin )
119118olcd 382 . . . . . . . 8  |-  ( (
ph  /\  y  e.  { 0 } )  -> 
( ran  G  C_  ( ZZ>=
`  0 )  \/ 
ran  G  e.  Fin ) )
120 sumz 12242 . . . . . . . 8  |-  ( ( ran  G  C_  ( ZZ>=
`  0 )  \/ 
ran  G  e.  Fin )  ->  sum_ z  e.  ran  G 0  =  0 )
121119, 120syl 15 . . . . . . 7  |-  ( (
ph  /\  y  e.  { 0 } )  ->  sum_ z  e.  ran  G
0  =  0 )
122117, 121eqtrd 2348 . . . . . 6  |-  ( (
ph  /\  y  e.  { 0 } )  ->  sum_ z  e.  ran  G
( y  x.  (
y I z ) )  =  0 )
123105, 122sylan2 460 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  ( ran  F  \  { 0 } ) ) )  ->  sum_ z  e.  ran  G ( y  x.  (
y I z ) )  =  0 )
12498, 101, 123, 3fsumss 12245 . . . 4  |-  ( ph  -> 
sum_ y  e.  ( ran  F  \  {
0 } ) sum_ z  e.  ran  G ( y  x.  ( y I z ) )  =  sum_ y  e.  ran  F
sum_ z  e.  ran  G ( y  x.  (
y I z ) ) )
12542, 97, 1243eqtrd 2352 . . 3  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( y  x.  ( y I z ) ) )
126 itg1val 19091 . . . . 5  |-  ( G  e.  dom  S.1  ->  ( S.1 `  G )  =  sum_ z  e.  ( ran  G  \  {
0 } ) ( z  x.  ( vol `  ( `' G " { z } ) ) ) )
1274, 126syl 15 . . . 4  |-  ( ph  ->  ( S.1 `  G
)  =  sum_ z  e.  ( ran  G  \  { 0 } ) ( z  x.  ( vol `  ( `' G " { z } ) ) ) )
1289adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  F : RR --> RR )
1293adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ran  F  e.  Fin )
130 inss1 3423 . . . . . . . . . 10  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { y } )
131130a1i 10 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { y } ) )
13248ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( `' F " { y } )  e.  dom  vol )
13351ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( `' G " { z } )  e.  dom  vol )
134132, 133, 53syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
13511adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ran  F  C_  RR )
136135sselda 3214 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
y  e.  RR )
137 difss 3337 . . . . . . . . . . . . . 14  |-  ( ran 
G  \  { 0 } )  C_  ran  G
138137, 21syl5ss 3224 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  G  \  { 0 } ) 
C_  RR )
139138sselda 3214 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  z  e.  RR )
140139adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
z  e.  RR )
141 eldifsni 3784 . . . . . . . . . . . . 13  |-  ( z  e.  ( ran  G  \  { 0 } )  ->  z  =/=  0
)
142141ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
z  =/=  0 )
143 simpr 447 . . . . . . . . . . . . 13  |-  ( ( y  =  0  /\  z  =  0 )  ->  z  =  0 )
144143necon3ai 2519 . . . . . . . . . . . 12  |-  ( z  =/=  0  ->  -.  ( y  =  0  /\  z  =  0 ) )
145142, 144syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  ->  -.  ( y  =  0  /\  z  =  0 ) )
146136, 140, 145, 66syl21anc 1181 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( y I z )  =  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
14716ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  ->  I : ( RR  X.  RR ) --> RR )
148147, 136, 140, 24syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( y I z )  e.  RR )
149146, 148eqeltrrd 2391 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( vol `  (
( `' F " { y } )  i^i  ( `' G " { z } ) ) )  e.  RR )
150128, 129, 131, 134, 149itg1addlem1 19100 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( vol `  U_ y  e.  ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )  =  sum_ y  e.  ran  F ( vol `  (
( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
151 incom 3395 . . . . . . . . . . . . 13  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  =  ( ( `' G " { z } )  i^i  ( `' F " { y } ) )
152151a1i 10 . . . . . . . . . . . 12  |-  ( y  e.  ran  F  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  =  ( ( `' G " { z } )  i^i  ( `' F " { y } ) ) )
153152iuneq2i 3960 . . . . . . . . . . 11  |-  U_ y  e.  ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  = 
U_ y  e.  ran  F ( ( `' G " { z } )  i^i  ( `' F " { y } ) )
154 iunin2 4003 . . . . . . . . . . 11  |-  U_ y  e.  ran  F ( ( `' G " { z } )  i^i  ( `' F " { y } ) )  =  ( ( `' G " { z } )  i^i  U_ y  e.  ran  F ( `' F " { y } ) )
155153, 154eqtri 2336 . . . . . . . . . 10  |-  U_ y  e.  ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  =  ( ( `' G " { z } )  i^i  U_ y  e.  ran  F ( `' F " { y } ) )
156 cnvimass 5070 . . . . . . . . . . . . 13  |-  ( `' G " { z } )  C_  dom  G
15719, 82syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  G  =  RR )
158157adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  dom  G  =  RR )
159156, 158syl5sseq 3260 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } ) 
C_  RR )
160 iunid 3994 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ran  F { y }  =  ran  F
161160imaeq2i 5047 . . . . . . . . . . . . . 14  |-  ( `' F " U_ y  e.  ran  F { y } )  =  ( `' F " ran  F
)
162 imaiun 5813 . . . . . . . . . . . . . 14  |-  ( `' F " U_ y  e.  ran  F { y } )  =  U_ y  e.  ran  F ( `' F " { y } )
163 cnvimarndm 5071 . . . . . . . . . . . . . 14  |-  ( `' F " ran  F
)  =  dom  F
164161, 162, 1633eqtr3i 2344 . . . . . . . . . . . . 13  |-  U_ y  e.  ran  F ( `' F " { y } )  =  dom  F
165 fdm 5431 . . . . . . . . . . . . . . 15  |-  ( F : RR --> RR  ->  dom 
F  =  RR )
1669, 165syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  F  =  RR )
167166adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  dom  F  =  RR )
168164, 167syl5eq 2360 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  U_ y  e.  ran  F ( `' F " { y } )  =  RR )
169159, 168sseqtr4d 3249 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } ) 
C_  U_ y  e.  ran  F ( `' F " { y } ) )
170 df-ss 3200 . . . . . . . . . . 11  |-  ( ( `' G " { z } )  C_  U_ y  e.  ran  F ( `' F " { y } )  <->  ( ( `' G " { z } )  i^i  U_ y  e.  ran  F ( `' F " { y } ) )  =  ( `' G " { z } ) )
171169, 170sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( ( `' G " { z } )  i^i  U_ y  e.  ran  F ( `' F " { y } ) )  =  ( `' G " { z } ) )
172155, 171syl5req 2361 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } )  =  U_ y  e. 
ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )
173172fveq2d 5567 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( vol `  ( `' G " { z } ) )  =  ( vol `  U_ y  e.  ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
174146sumeq2dv 12223 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  sum_ y  e.  ran  F ( y I z )  =  sum_ y  e.  ran  F ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
175150, 173, 1743eqtr4d 2358 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( vol `  ( `' G " { z } ) )  = 
sum_ y  e.  ran  F ( y I z ) )
176175oveq2d 5916 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( z  x.  ( vol `  ( `' G " { z } ) ) )  =  ( z  x. 
sum_ y  e.  ran  F ( y I z ) ) )
177139recnd 8906 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  z  e.  CC )
178148recnd 8906 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( y I z )  e.  CC )
179129, 177, 178fsummulc2 12293 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( z  x. 
sum_ y  e.  ran  F ( y I z ) )  =  sum_ y  e.  ran  F ( z  x.  ( y I z ) ) )
180176, 179eqtrd 2348 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( z  x.  ( vol `  ( `' G " { z } ) ) )  =  sum_ y  e.  ran  F ( z  x.  (
y I z ) ) )
181180sumeq2dv 12223 . . . 4  |-  ( ph  -> 
sum_ z  e.  ( ran  G  \  {
0 } ) ( z  x.  ( vol `  ( `' G " { z } ) ) )  =  sum_ z  e.  ( ran  G 
\  { 0 } ) sum_ y  e.  ran  F ( z  x.  (
y I z ) ) )
182137a1i 10 . . . . . 6  |-  ( ph  ->  ( ran  G  \  { 0 } ) 
C_  ran  G )
183177adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
z  e.  CC )
184183, 178mulcld 8900 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( z  x.  (
y I z ) )  e.  CC )
185129, 184fsumcl 12253 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  sum_ y  e.  ran  F ( z  x.  (
y I z ) )  e.  CC )
186 dfin4 3443 . . . . . . . . 9  |-  ( ran 
G  i^i  { 0 } )  =  ( ran  G  \  ( ran  G  \  { 0 } ) )
187 inss2 3424 . . . . . . . . 9  |-  ( ran 
G  i^i  { 0 } )  C_  { 0 }
188186, 187eqsstr3i 3243 . . . . . . . 8  |-  ( ran 
G  \  ( ran  G 
\  { 0 } ) )  C_  { 0 }
189188sseli 3210 . . . . . . 7  |-  ( z  e.  ( ran  G  \  ( ran  G  \  { 0 } ) )  ->  z  e.  { 0 } )
190 elsni 3698 . . . . . . . . . . . 12  |-  ( z  e.  { 0 }  ->  z  =  0 )
191190ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  z  = 
0 )
192191oveq1d 5915 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( z  x.  ( y I z ) )  =  ( 0  x.  ( y I z ) ) )
19316ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  I :
( RR  X.  RR )
--> RR )
19412adantlr 695 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  y  e.  RR )
195191, 110syl6eqel 2404 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  z  e.  RR )
196193, 194, 195, 24syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( y
I z )  e.  RR )
197196recnd 8906 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( y
I z )  e.  CC )
198197mul02d 9055 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( 0  x.  ( y I z ) )  =  0 )
199192, 198eqtrd 2348 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( z  x.  ( y I z ) )  =  0 )
200199sumeq2dv 12223 . . . . . . . 8  |-  ( (
ph  /\  z  e.  { 0 } )  ->  sum_ y  e.  ran  F
( z  x.  (
y I z ) )  =  sum_ y  e.  ran  F 0 )
2013adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  { 0 } )  ->  ran  F  e.  Fin )
202201olcd 382 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  { 0 } )  -> 
( ran  F  C_  ( ZZ>=
`  0 )  \/ 
ran  F  e.  Fin ) )
203 sumz 12242 . . . . . . . . 9  |-  ( ( ran  F  C_  ( ZZ>=
`  0 )  \/ 
ran  F  e.  Fin )  ->  sum_ y  e.  ran  F 0  =  0 )
204202, 203syl 15 . . . . . . . 8  |-  ( (
ph  /\  z  e.  { 0 } )  ->  sum_ y  e.  ran  F
0  =  0 )
205200, 204eqtrd 2348 . . . . . . 7  |-  ( (
ph  /\  z  e.  { 0 } )  ->  sum_ y  e.  ran  F
( z  x.  (
y I z ) )  =  0 )
206189, 205sylan2 460 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  ( ran  G  \  { 0 } ) ) )  ->  sum_ y  e.  ran  F ( z  x.  (
y I z ) )  =  0 )
207182, 185, 206, 6fsumss 12245 . . . . 5  |-  ( ph  -> 
sum_ z  e.  ( ran  G  \  {
0 } ) sum_ y  e.  ran  F ( z  x.  ( y I z ) )  =  sum_ z  e.  ran  G
sum_ y  e.  ran  F ( z  x.  (
y I z ) ) )
20822adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  RR )
209208recnd 8906 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  CC )
21016ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  I : ( RR  X.  RR ) --> RR )
21111adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  RR )
212211sselda 3214 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  RR )
213210, 212, 208, 24syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  RR )
214213recnd 8906 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  CC )
215209, 214mulcld 8900 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( z  x.  ( y I z ) )  e.  CC )
216215anasss 628 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ran  G  /\  y  e.  ran  F ) )  ->  ( z  x.  ( y I z ) )  e.  CC )
2176, 3, 216fsumcom 12285 . . . . 5  |-  ( ph  -> 
sum_ z  e.  ran  G
sum_ y  e.  ran  F ( z  x.  (
y I z ) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( z  x.  ( y I z ) ) )
218207, 217eqtrd 2348 . . . 4  |-  ( ph  -> 
sum_ z  e.  ( ran  G  \  {
0 } ) sum_ y  e.  ran  F ( z  x.  ( y I z ) )  =  sum_ y  e.  ran  F
sum_ z  e.  ran  G ( z  x.  (
y I z ) ) )
219127, 181, 2183eqtrd 2352 . . 3  |-  ( ph  ->  ( S.1 `  G
)  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( z  x.  ( y I z ) ) )
220125, 219oveq12d 5918 . 2  |-  ( ph  ->  ( ( S.1 `  F
)  +  ( S.1 `  G ) )  =  ( sum_ y  e.  ran  F
sum_ z  e.  ran  G ( y  x.  (
y I z ) )  +  sum_ y  e.  ran  F sum_ z  e.  ran  G ( z  x.  ( y I z ) ) ) )
22132, 40, 2203eqtr4d 2358 1  |-  ( ph  ->  ( S.1 `  ( F  o F  +  G
) )  =  ( ( S.1 `  F
)  +  ( S.1 `  G ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479    \ cdif 3183    i^i cin 3185    C_ wss 3186   ifcif 3599   {csn 3674   U_ciun 3942    X. cxp 4724   `'ccnv 4725   dom cdm 4726   ran crn 4727    |` cres 4728   "cima 4729   -->wf 5288   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902    o Fcof 6118   Fincfn 6906   CCcc 8780   RRcr 8781   0cc0 8782    + caddc 8785    x. cmul 8787   ZZ>=cuz 10277   sum_csu 12205   volcvol 18876   S.1citg1 19023
This theorem is referenced by:  itg1add  19109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-disj 4031  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-pm 6818  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-oi 7270  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-q 10364  df-rp 10402  df-xadd 10500  df-ioo 10707  df-ico 10709  df-icc 10710  df-fz 10830  df-fzo 10918  df-fl 10972  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-clim 12009  df-sum 12206  df-xmet 16425  df-met 16426  df-ovol 18877  df-vol 18878  df-mbf 19028  df-itg1 19029
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