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Theorem itg1addlem5 19055
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 19032 . . . 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 19032 . . . . . 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 19031 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
91, 8syl 15 . . . . . . . . 9  |-  ( ph  ->  F : RR --> RR )
10 frn 5395 . . . . . . . . 9  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
119, 10syl 15 . . . . . . . 8  |-  ( ph  ->  ran  F  C_  RR )
1211sselda 3180 . . . . . . 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 8861 . . . . 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 19052 . . . . . . . 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 19031 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
194, 18syl 15 . . . . . . . . . 10  |-  ( ph  ->  G : RR --> RR )
20 frn 5395 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
2119, 20syl 15 . . . . . . . . 9  |-  ( ph  ->  ran  G  C_  RR )
2221sselda 3180 . . . . . . . 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 5990 . . . . . . 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 8861 . . . . 5  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y I z )  e.  CC )
2714, 26mulcld 8855 . . . 4  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  e.  CC )
287, 27fsumcl 12206 . . 3  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( y  x.  ( y I z ) )  e.  CC )
2923recnd 8861 . . . . 5  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  z  e.  CC )
3029, 26mulcld 8855 . . . 4  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( z  x.  ( y I z ) )  e.  CC )
317, 30fsumcl 12206 . . 3  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( z  x.  ( y I z ) )  e.  CC )
323, 28, 31fsumadd 12211 . 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 19054 . . 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 8860 . . . . . 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 12176 . . . . 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 12211 . . . . 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 2315 . . . 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 12176 . . 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 2315 . 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 19038 . . . . 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 3390 . . . . . . . . . 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 19033 . . . . . . . . . . . 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 19033 . . . . . . . . . . . 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 18899 . . . . . . . . . 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 3303 . . . . . . . . . . . . . 14  |-  ( ran 
F  \  { 0 } )  C_  ran  F
5655, 11syl5ss 3190 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
5756sselda 3180 . . . . . . . . . . . 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 3180 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  RR )
61 eldifsni 3750 . . . . . . . . . . . . 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 2486 . . . . . . . . . . . 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 19053 . . . . . . . . . . 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 2358 . . . . . . . . 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 19047 . . . . . . . 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 3966 . . . . . . . . . 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 18890 . . . . . . . . . . . . 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 3957 . . . . . . . . . . . . . . 15  |-  U_ z  e.  ran  G { z }  =  ran  G
7877imaeq2i 5010 . . . . . . . . . . . . . 14  |-  ( `' G " U_ z  e.  ran  G { z } )  =  ( `' G " ran  G
)
79 imaiun 5771 . . . . . . . . . . . . . 14  |-  ( `' G " U_ z  e.  ran  G { z } )  =  U_ z  e.  ran  G ( `' G " { z } )
80 cnvimarndm 5034 . . . . . . . . . . . . . 14  |-  ( `' G " ran  G
)  =  dom  G
8178, 79, 803eqtr3i 2311 . . . . . . . . . . . . 13  |-  U_ z  e.  ran  G ( `' G " { z } )  =  dom  G
82 fdm 5393 . . . . . . . . . . . . . 14  |-  ( G : RR --> RR  ->  dom 
G  =  RR )
8343, 82syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  dom  G  =  RR )
8481, 83syl5eq 2327 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  U_ z  e.  ran  G ( `' G " { z } )  =  RR )
8576, 84sseqtr4d 3215 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } ) 
C_  U_ z  e.  ran  G ( `' G " { z } ) )
86 df-ss 3166 . . . . . . . . . . 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 2328 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } )  =  U_ z  e. 
ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )
8988fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { y } ) )  =  ( vol `  U_ z  e.  ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
9067sumeq2dv 12176 . . . . . . . 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 2325 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { y } ) )  = 
sum_ z  e.  ran  G ( y I z ) )
9291oveq2d 5874 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( y  x.  ( vol `  ( `' F " { y } ) ) )  =  ( y  x. 
sum_ z  e.  ran  G ( y I z ) ) )
9357recnd 8861 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  y  e.  CC )
9469recnd 8861 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y I z )  e.  CC )
9544, 93, 94fsummulc2 12246 . . . . . 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 2315 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( y  x.  ( vol `  ( `' F " { y } ) ) )  =  sum_ z  e.  ran  G ( y  x.  (
y I z ) ) )
9796sumeq2dv 12176 . . . 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 8861 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
y  e.  CC )
10099, 94mulcld 8855 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y  x.  (
y I z ) )  e.  CC )
10144, 100fsumcl 12206 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  sum_ z  e.  ran  G ( y  x.  (
y I z ) )  e.  CC )
102 dfin4 3409 . . . . . . . 8  |-  ( ran 
F  i^i  { 0 } )  =  ( ran  F  \  ( ran  F  \  { 0 } ) )
103 inss2 3390 . . . . . . . 8  |-  ( ran 
F  i^i  { 0 } )  C_  { 0 }
104102, 103eqsstr3i 3209 . . . . . . 7  |-  ( ran 
F  \  ( ran  F 
\  { 0 } ) )  C_  { 0 }
105104sseli 3176 . . . . . 6  |-  ( y  e.  ( ran  F  \  ( ran  F  \  { 0 } ) )  ->  y  e.  { 0 } )
106 elsni 3664 . . . . . . . . . . 11  |-  ( y  e.  { 0 }  ->  y  =  0 )
107106ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  y  = 
0 )
108107oveq1d 5873 . . . . . . . . 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 8838 . . . . . . . . . . . . 13  |-  0  e.  RR
111107, 110syl6eqel 2371 . . . . . . . . . . . 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 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y
I z )  e.  CC )
115114mul02d 9010 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( 0  x.  ( y I z ) )  =  0 )
116108, 115eqtrd 2315 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  =  0 )
117116sumeq2dv 12176 . . . . . . 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 12195 . . . . . . . 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 2315 . . . . . 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 12198 . . . 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 2319 . . 3  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( y  x.  ( y I z ) ) )
126 itg1val 19038 . . . . 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 3389 . . . . . . . . . 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 3180 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
y  e.  RR )
137 difss 3303 . . . . . . . . . . . . . 14  |-  ( ran 
G  \  { 0 } )  C_  ran  G
138137, 21syl5ss 3190 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  G  \  { 0 } ) 
C_  RR )
139138sselda 3180 . . . . . . . . . . . 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 3750 . . . . . . . . . . . . 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 2486 . . . . . . . . . . . 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 2358 . . . . . . . . 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 19047 . . . . . . . 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 3361 . . . . . . . . . . . . 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 3923 . . . . . . . . . . 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 3966 . . . . . . . . . . 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 2303 . . . . . . . . . 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 5033 . . . . . . . . . . . . 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 3226 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } ) 
C_  RR )
160 iunid 3957 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ran  F { y }  =  ran  F
161160imaeq2i 5010 . . . . . . . . . . . . . 14  |-  ( `' F " U_ y  e.  ran  F { y } )  =  ( `' F " ran  F
)
162 imaiun 5771 . . . . . . . . . . . . . 14  |-  ( `' F " U_ y  e.  ran  F { y } )  =  U_ y  e.  ran  F ( `' F " { y } )
163 cnvimarndm 5034 . . . . . . . . . . . . . 14  |-  ( `' F " ran  F
)  =  dom  F
164161, 162, 1633eqtr3i 2311 . . . . . . . . . . . . 13  |-  U_ y  e.  ran  F ( `' F " { y } )  =  dom  F
165 fdm 5393 . . . . . . . . . . . . . . 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 2327 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  U_ y  e.  ran  F ( `' F " { y } )  =  RR )
169159, 168sseqtr4d 3215 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } ) 
C_  U_ y  e.  ran  F ( `' F " { y } ) )
170 df-ss 3166 . . . . . . . . . . 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 2328 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } )  =  U_ y  e. 
ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )
173172fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( vol `  ( `' G " { z } ) )  =  ( vol `  U_ y  e.  ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
174146sumeq2dv 12176 . . . . . . . 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 2325 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( vol `  ( `' G " { z } ) )  = 
sum_ y  e.  ran  F ( y I z ) )
176175oveq2d 5874 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( z  x.  ( vol `  ( `' G " { z } ) ) )  =  ( z  x. 
sum_ y  e.  ran  F ( y I z ) ) )
177139recnd 8861 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  z  e.  CC )
178148recnd 8861 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( y I z )  e.  CC )
179129, 177, 178fsummulc2 12246 . . . . . 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 2315 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( z  x.  ( vol `  ( `' G " { z } ) ) )  =  sum_ y  e.  ran  F ( z  x.  (
y I z ) ) )
181180sumeq2dv 12176 . . . 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 8855 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( z  x.  (
y I z ) )  e.  CC )
185129, 184fsumcl 12206 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  sum_ y  e.  ran  F ( z  x.  (
y I z ) )  e.  CC )
186 dfin4 3409 . . . . . . . . 9  |-  ( ran 
G  i^i  { 0 } )  =  ( ran  G  \  ( ran  G  \  { 0 } ) )
187 inss2 3390 . . . . . . . . 9  |-  ( ran 
G  i^i  { 0 } )  C_  { 0 }
188186, 187eqsstr3i 3209 . . . . . . . 8  |-  ( ran 
G  \  ( ran  G 
\  { 0 } ) )  C_  { 0 }
189188sseli 3176 . . . . . . 7  |-  ( z  e.  ( ran  G  \  ( ran  G  \  { 0 } ) )  ->  z  e.  { 0 } )
190 elsni 3664 . . . . . . . . . . . 12  |-  ( z  e.  { 0 }  ->  z  =  0 )
191190ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  z  = 
0 )
192191oveq1d 5873 . . . . . . . . . 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 2371 . . . . . . . . . . . . 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 8861 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( y
I z )  e.  CC )
198197mul02d 9010 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( 0  x.  ( y I z ) )  =  0 )
199192, 198eqtrd 2315 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( z  x.  ( y I z ) )  =  0 )
200199sumeq2dv 12176 . . . . . . . 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 12195 . . . . . . . . 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 2315 . . . . . . 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 12198 . . . . 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 8861 . . . . . . . 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 3180 . . . . . . . . . 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 8861 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  CC )
215209, 214mulcld 8855 . . . . . . 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 12238 . . . . 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 2315 . . . 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 2319 . . 3  |-  ( ph  ->  ( S.1 `  G
)  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( z  x.  ( y I z ) ) )
220125, 219oveq12d 5876 . 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 2325 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 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    i^i cin 3151    C_ wss 3152   ifcif 3565   {csn 3640   U_ciun 3905    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   -->wf 5251   ` 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   ZZ>=cuz 10230   sum_csu 12158   volcvol 18823   S.1citg1 18970
This theorem is referenced by:  itg1add  19056
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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