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Theorem itg1addlem5 22273
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  oF  +  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 22250 . . . 4  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
31, 2syl 16 . . 3  |-  ( ph  ->  ran  F  e.  Fin )
4 i1fadd.2 . . . . . 6  |-  ( ph  ->  G  e.  dom  S.1 )
5 i1frn 22250 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
64, 5syl 16 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
76adantr 463 . . . 4  |-  ( (
ph  /\  y  e.  ran  F )  ->  ran  G  e.  Fin )
8 i1ff 22249 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
91, 8syl 16 . . . . . . . . 9  |-  ( ph  ->  F : RR --> RR )
10 frn 5719 . . . . . . . . 9  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
119, 10syl 16 . . . . . . . 8  |-  ( ph  ->  ran  F  C_  RR )
1211sselda 3489 . . . . . . 7  |-  ( (
ph  /\  y  e.  ran  F )  ->  y  e.  RR )
1312adantr 463 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  y  e.  RR )
1413recnd 9611 . . . . 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 22270 . . . . . . . 8  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
1716ad2antrr 723 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  I : ( RR  X.  RR ) --> RR )
18 i1ff 22249 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
194, 18syl 16 . . . . . . . . . 10  |-  ( ph  ->  G : RR --> RR )
20 frn 5719 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
2119, 20syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  G  C_  RR )
2221sselda 3489 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  RR )
2322adantlr 712 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  z  e.  RR )
2417, 13, 23fovrnd 6420 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y I z )  e.  RR )
2524recnd 9611 . . . . 5  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y I z )  e.  CC )
2614, 25mulcld 9605 . . . 4  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  e.  CC )
277, 26fsumcl 13637 . . 3  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( y  x.  ( y I z ) )  e.  CC )
2823recnd 9611 . . . . 5  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  z  e.  CC )
2928, 25mulcld 9605 . . . 4  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( z  x.  ( y I z ) )  e.  CC )
307, 29fsumcl 13637 . . 3  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( z  x.  ( y I z ) )  e.  CC )
313, 27, 30fsumadd 13643 . 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 ) ) ) )
32 itg1add.4 . . . 4  |-  P  =  (  +  |`  ( ran  F  X.  ran  G
) )
331, 4, 15, 32itg1addlem4 22272 . . 3  |-  ( ph  ->  ( S.1 `  ( F  oF  +  G
) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( ( y  +  z )  x.  ( y I z ) ) )
3414, 28, 25adddird 9610 . . . . . 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 ) ) ) )
3534sumeq2dv 13607 . . . . 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 ) ) ) )
367, 26, 29fsumadd 13643 . . . . 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 ) ) ) )
3735, 36eqtrd 2495 . . . 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 ) ) ) )
3837sumeq2dv 13607 . . 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 ) ) ) )
3933, 38eqtrd 2495 . 2  |-  ( ph  ->  ( S.1 `  ( F  oF  +  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 ) ) ) )
40 itg1val 22256 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ y  e.  ( ran  F  \  {
0 } ) ( y  x.  ( vol `  ( `' F " { y } ) ) ) )
411, 40syl 16 . . . 4  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ y  e.  ( ran  F  \  { 0 } ) ( y  x.  ( vol `  ( `' F " { y } ) ) ) )
4219adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  G : RR --> RR )
436adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ran  G  e.  Fin )
44 inss2 3705 . . . . . . . . . 10  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )
4544a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } ) )
46 i1fima 22251 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  ( `' F " { y } )  e.  dom  vol )
471, 46syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( `' F " { y } )  e.  dom  vol )
4847ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( `' F " { y } )  e.  dom  vol )
49 i1fima 22251 . . . . . . . . . . . 12  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
504, 49syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
5150ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( `' G " { z } )  e.  dom  vol )
52 inmbl 22118 . . . . . . . . . 10  |-  ( ( ( `' F " { y } )  e.  dom  vol  /\  ( `' G " { z } )  e.  dom  vol )  ->  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
5348, 51, 52syl2anc 659 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
5411ssdifssd 3628 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
5554sselda 3489 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  y  e.  RR )
5655adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
y  e.  RR )
5721adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ran  G  C_  RR )
5857sselda 3489 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  RR )
59 eldifsni 4142 . . . . . . . . . . . . 13  |-  ( y  e.  ( ran  F  \  { 0 } )  ->  y  =/=  0
)
6059ad2antlr 724 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
y  =/=  0 )
61 simpl 455 . . . . . . . . . . . . 13  |-  ( ( y  =  0  /\  z  =  0 )  ->  y  =  0 )
6261necon3ai 2682 . . . . . . . . . . . 12  |-  ( y  =/=  0  ->  -.  ( y  =  0  /\  z  =  0 ) )
6360, 62syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  -.  ( y  =  0  /\  z  =  0 ) )
641, 4, 15itg1addlem3 22271 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  z  e.  RR )  /\  -.  ( y  =  0  /\  z  =  0 ) )  ->  ( y I z )  =  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
6556, 58, 63, 64syl21anc 1225 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y I z )  =  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
6616ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  I : ( RR  X.  RR ) --> RR )
6766, 56, 58fovrnd 6420 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y I z )  e.  RR )
6865, 67eqeltrrd 2543 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( vol `  (
( `' F " { y } )  i^i  ( `' G " { z } ) ) )  e.  RR )
6942, 43, 45, 53, 68itg1addlem1 22265 . . . . . . . 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 } ) ) ) )
70 iunin2 4379 . . . . . . . . . 10  |-  U_ z  e.  ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  =  ( ( `' F " { y } )  i^i  U_ z  e.  ran  G ( `' G " { z } ) )
711adantr 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  F  e.  dom  S.1 )
7271, 46syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } )  e.  dom  vol )
73 mblss 22108 . . . . . . . . . . . . 13  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( `' F " { y } ) 
C_  RR )
7472, 73syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } ) 
C_  RR )
75 iunid 4370 . . . . . . . . . . . . . . 15  |-  U_ z  e.  ran  G { z }  =  ran  G
7675imaeq2i 5323 . . . . . . . . . . . . . 14  |-  ( `' G " U_ z  e.  ran  G { z } )  =  ( `' G " ran  G
)
77 imaiun 6132 . . . . . . . . . . . . . 14  |-  ( `' G " U_ z  e.  ran  G { z } )  =  U_ z  e.  ran  G ( `' G " { z } )
78 cnvimarndm 5346 . . . . . . . . . . . . . 14  |-  ( `' G " ran  G
)  =  dom  G
7976, 77, 783eqtr3i 2491 . . . . . . . . . . . . 13  |-  U_ z  e.  ran  G ( `' G " { z } )  =  dom  G
80 fdm 5717 . . . . . . . . . . . . . 14  |-  ( G : RR --> RR  ->  dom 
G  =  RR )
8142, 80syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  dom  G  =  RR )
8279, 81syl5eq 2507 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  U_ z  e.  ran  G ( `' G " { z } )  =  RR )
8374, 82sseqtr4d 3526 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } ) 
C_  U_ z  e.  ran  G ( `' G " { z } ) )
84 df-ss 3475 . . . . . . . . . . 11  |-  ( ( `' F " { y } )  C_  U_ z  e.  ran  G ( `' G " { z } )  <->  ( ( `' F " { y } )  i^i  U_ z  e.  ran  G ( `' G " { z } ) )  =  ( `' F " { y } ) )
8583, 84sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( ( `' F " { y } )  i^i  U_ z  e.  ran  G ( `' G " { z } ) )  =  ( `' F " { y } ) )
8670, 85syl5req 2508 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } )  =  U_ z  e. 
ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )
8786fveq2d 5852 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { y } ) )  =  ( vol `  U_ z  e.  ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
8865sumeq2dv 13607 . . . . . . . 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 } ) ) ) )
8969, 87, 883eqtr4d 2505 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { y } ) )  = 
sum_ z  e.  ran  G ( y I z ) )
9089oveq2d 6286 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( y  x.  ( vol `  ( `' F " { y } ) ) )  =  ( y  x. 
sum_ z  e.  ran  G ( y I z ) ) )
9155recnd 9611 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  y  e.  CC )
9267recnd 9611 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y I z )  e.  CC )
9343, 91, 92fsummulc2 13681 . . . . . 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 ) ) )
9490, 93eqtrd 2495 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( y  x.  ( vol `  ( `' F " { y } ) ) )  =  sum_ z  e.  ran  G ( y  x.  (
y I z ) ) )
9594sumeq2dv 13607 . . . 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 ) ) )
96 difssd 3618 . . . . 5  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  ran  F )
9756recnd 9611 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
y  e.  CC )
9897, 92mulcld 9605 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y  x.  (
y I z ) )  e.  CC )
9943, 98fsumcl 13637 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  sum_ z  e.  ran  G ( y  x.  (
y I z ) )  e.  CC )
100 dfin4 3735 . . . . . . . 8  |-  ( ran 
F  i^i  { 0 } )  =  ( ran  F  \  ( ran  F  \  { 0 } ) )
101 inss2 3705 . . . . . . . 8  |-  ( ran 
F  i^i  { 0 } )  C_  { 0 }
102100, 101eqsstr3i 3520 . . . . . . 7  |-  ( ran 
F  \  ( ran  F 
\  { 0 } ) )  C_  { 0 }
103102sseli 3485 . . . . . 6  |-  ( y  e.  ( ran  F  \  ( ran  F  \  { 0 } ) )  ->  y  e.  { 0 } )
104 elsni 4041 . . . . . . . . . . 11  |-  ( y  e.  { 0 }  ->  y  =  0 )
105104ad2antlr 724 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  y  = 
0 )
106105oveq1d 6285 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  =  ( 0  x.  ( y I z ) ) )
10716ad2antrr 723 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  I :
( RR  X.  RR )
--> RR )
108 0re 9585 . . . . . . . . . . . . 13  |-  0  e.  RR
109105, 108syl6eqel 2550 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  y  e.  RR )
11022adantlr 712 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  z  e.  RR )
111107, 109, 110fovrnd 6420 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y
I z )  e.  RR )
112111recnd 9611 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y
I z )  e.  CC )
113112mul02d 9767 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( 0  x.  ( y I z ) )  =  0 )
114106, 113eqtrd 2495 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  =  0 )
115114sumeq2dv 13607 . . . . . . 7  |-  ( (
ph  /\  y  e.  { 0 } )  ->  sum_ z  e.  ran  G
( y  x.  (
y I z ) )  =  sum_ z  e.  ran  G 0 )
1166adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  { 0 } )  ->  ran  G  e.  Fin )
117116olcd 391 . . . . . . . 8  |-  ( (
ph  /\  y  e.  { 0 } )  -> 
( ran  G  C_  ( ZZ>=
`  0 )  \/ 
ran  G  e.  Fin ) )
118 sumz 13626 . . . . . . . 8  |-  ( ( ran  G  C_  ( ZZ>=
`  0 )  \/ 
ran  G  e.  Fin )  ->  sum_ z  e.  ran  G 0  =  0 )
119117, 118syl 16 . . . . . . 7  |-  ( (
ph  /\  y  e.  { 0 } )  ->  sum_ z  e.  ran  G
0  =  0 )
120115, 119eqtrd 2495 . . . . . 6  |-  ( (
ph  /\  y  e.  { 0 } )  ->  sum_ z  e.  ran  G
( y  x.  (
y I z ) )  =  0 )
121103, 120sylan2 472 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  ( ran  F  \  { 0 } ) ) )  ->  sum_ z  e.  ran  G ( y  x.  (
y I z ) )  =  0 )
12296, 99, 121, 3fsumss 13629 . . . 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 ) ) )
12341, 95, 1223eqtrd 2499 . . 3  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( y  x.  ( y I z ) ) )
124 itg1val 22256 . . . . 5  |-  ( G  e.  dom  S.1  ->  ( S.1 `  G )  =  sum_ z  e.  ( ran  G  \  {
0 } ) ( z  x.  ( vol `  ( `' G " { z } ) ) ) )
1254, 124syl 16 . . . 4  |-  ( ph  ->  ( S.1 `  G
)  =  sum_ z  e.  ( ran  G  \  { 0 } ) ( z  x.  ( vol `  ( `' G " { z } ) ) ) )
1269adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  F : RR --> RR )
1273adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ran  F  e.  Fin )
128 inss1 3704 . . . . . . . . . 10  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { y } )
129128a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  C_  ( `' F " { y } ) )
13047ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( `' F " { y } )  e.  dom  vol )
13150ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( `' G " { z } )  e.  dom  vol )
132130, 131, 52syl2anc 659 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
13311adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ran  F  C_  RR )
134133sselda 3489 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
y  e.  RR )
13521ssdifssd 3628 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  G  \  { 0 } ) 
C_  RR )
136135sselda 3489 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  z  e.  RR )
137136adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
z  e.  RR )
138 eldifsni 4142 . . . . . . . . . . . . 13  |-  ( z  e.  ( ran  G  \  { 0 } )  ->  z  =/=  0
)
139138ad2antlr 724 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
z  =/=  0 )
140 simpr 459 . . . . . . . . . . . . 13  |-  ( ( y  =  0  /\  z  =  0 )  ->  z  =  0 )
141140necon3ai 2682 . . . . . . . . . . . 12  |-  ( z  =/=  0  ->  -.  ( y  =  0  /\  z  =  0 ) )
142139, 141syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  ->  -.  ( y  =  0  /\  z  =  0 ) )
143134, 137, 142, 64syl21anc 1225 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( y I z )  =  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
14416ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  ->  I : ( RR  X.  RR ) --> RR )
145144, 134, 137fovrnd 6420 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( y I z )  e.  RR )
146143, 145eqeltrrd 2543 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( vol `  (
( `' F " { y } )  i^i  ( `' G " { z } ) ) )  e.  RR )
147126, 127, 129, 132, 146itg1addlem1 22265 . . . . . . . 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 } ) ) ) )
148 incom 3677 . . . . . . . . . . . . 13  |-  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  =  ( ( `' G " { z } )  i^i  ( `' F " { y } ) )
149148a1i 11 . . . . . . . . . . . 12  |-  ( y  e.  ran  F  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  =  ( ( `' G " { z } )  i^i  ( `' F " { y } ) ) )
150149iuneq2i 4334 . . . . . . . . . . 11  |-  U_ y  e.  ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  = 
U_ y  e.  ran  F ( ( `' G " { z } )  i^i  ( `' F " { y } ) )
151 iunin2 4379 . . . . . . . . . . 11  |-  U_ y  e.  ran  F ( ( `' G " { z } )  i^i  ( `' F " { y } ) )  =  ( ( `' G " { z } )  i^i  U_ y  e.  ran  F ( `' F " { y } ) )
152150, 151eqtri 2483 . . . . . . . . . 10  |-  U_ y  e.  ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  =  ( ( `' G " { z } )  i^i  U_ y  e.  ran  F ( `' F " { y } ) )
153 cnvimass 5345 . . . . . . . . . . . . 13  |-  ( `' G " { z } )  C_  dom  G
15419, 80syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  G  =  RR )
155154adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  dom  G  =  RR )
156153, 155syl5sseq 3537 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } ) 
C_  RR )
157 iunid 4370 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ran  F { y }  =  ran  F
158157imaeq2i 5323 . . . . . . . . . . . . . 14  |-  ( `' F " U_ y  e.  ran  F { y } )  =  ( `' F " ran  F
)
159 imaiun 6132 . . . . . . . . . . . . . 14  |-  ( `' F " U_ y  e.  ran  F { y } )  =  U_ y  e.  ran  F ( `' F " { y } )
160 cnvimarndm 5346 . . . . . . . . . . . . . 14  |-  ( `' F " ran  F
)  =  dom  F
161158, 159, 1603eqtr3i 2491 . . . . . . . . . . . . 13  |-  U_ y  e.  ran  F ( `' F " { y } )  =  dom  F
162 fdm 5717 . . . . . . . . . . . . . . 15  |-  ( F : RR --> RR  ->  dom 
F  =  RR )
1639, 162syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  F  =  RR )
164163adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  dom  F  =  RR )
165161, 164syl5eq 2507 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  U_ y  e.  ran  F ( `' F " { y } )  =  RR )
166156, 165sseqtr4d 3526 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } ) 
C_  U_ y  e.  ran  F ( `' F " { y } ) )
167 df-ss 3475 . . . . . . . . . . 11  |-  ( ( `' G " { z } )  C_  U_ y  e.  ran  F ( `' F " { y } )  <->  ( ( `' G " { z } )  i^i  U_ y  e.  ran  F ( `' F " { y } ) )  =  ( `' G " { z } ) )
168166, 167sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( ( `' G " { z } )  i^i  U_ y  e.  ran  F ( `' F " { y } ) )  =  ( `' G " { z } ) )
169152, 168syl5req 2508 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } )  =  U_ y  e. 
ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )
170169fveq2d 5852 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( vol `  ( `' G " { z } ) )  =  ( vol `  U_ y  e.  ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
171143sumeq2dv 13607 . . . . . . . 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 } ) ) ) )
172147, 170, 1713eqtr4d 2505 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( vol `  ( `' G " { z } ) )  = 
sum_ y  e.  ran  F ( y I z ) )
173172oveq2d 6286 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( z  x.  ( vol `  ( `' G " { z } ) ) )  =  ( z  x. 
sum_ y  e.  ran  F ( y I z ) ) )
174136recnd 9611 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  z  e.  CC )
175145recnd 9611 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( y I z )  e.  CC )
176127, 174, 175fsummulc2 13681 . . . . . 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 ) ) )
177173, 176eqtrd 2495 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( z  x.  ( vol `  ( `' G " { z } ) ) )  =  sum_ y  e.  ran  F ( z  x.  (
y I z ) ) )
178177sumeq2dv 13607 . . . 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 ) ) )
179 difssd 3618 . . . . . 6  |-  ( ph  ->  ( ran  G  \  { 0 } ) 
C_  ran  G )
180174adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
z  e.  CC )
181180, 175mulcld 9605 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( z  x.  (
y I z ) )  e.  CC )
182127, 181fsumcl 13637 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  sum_ y  e.  ran  F ( z  x.  (
y I z ) )  e.  CC )
183 dfin4 3735 . . . . . . . . 9  |-  ( ran 
G  i^i  { 0 } )  =  ( ran  G  \  ( ran  G  \  { 0 } ) )
184 inss2 3705 . . . . . . . . 9  |-  ( ran 
G  i^i  { 0 } )  C_  { 0 }
185183, 184eqsstr3i 3520 . . . . . . . 8  |-  ( ran 
G  \  ( ran  G 
\  { 0 } ) )  C_  { 0 }
186185sseli 3485 . . . . . . 7  |-  ( z  e.  ( ran  G  \  ( ran  G  \  { 0 } ) )  ->  z  e.  { 0 } )
187 elsni 4041 . . . . . . . . . . . 12  |-  ( z  e.  { 0 }  ->  z  =  0 )
188187ad2antlr 724 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  z  = 
0 )
189188oveq1d 6285 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( z  x.  ( y I z ) )  =  ( 0  x.  ( y I z ) ) )
19016ad2antrr 723 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  I :
( RR  X.  RR )
--> RR )
19112adantlr 712 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  y  e.  RR )
192188, 108syl6eqel 2550 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  z  e.  RR )
193190, 191, 192fovrnd 6420 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( y
I z )  e.  RR )
194193recnd 9611 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( y
I z )  e.  CC )
195194mul02d 9767 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( 0  x.  ( y I z ) )  =  0 )
196189, 195eqtrd 2495 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( z  x.  ( y I z ) )  =  0 )
197196sumeq2dv 13607 . . . . . . . 8  |-  ( (
ph  /\  z  e.  { 0 } )  ->  sum_ y  e.  ran  F
( z  x.  (
y I z ) )  =  sum_ y  e.  ran  F 0 )
1983adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  { 0 } )  ->  ran  F  e.  Fin )
199198olcd 391 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  { 0 } )  -> 
( ran  F  C_  ( ZZ>=
`  0 )  \/ 
ran  F  e.  Fin ) )
200 sumz 13626 . . . . . . . . 9  |-  ( ( ran  F  C_  ( ZZ>=
`  0 )  \/ 
ran  F  e.  Fin )  ->  sum_ y  e.  ran  F 0  =  0 )
201199, 200syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  { 0 } )  ->  sum_ y  e.  ran  F
0  =  0 )
202197, 201eqtrd 2495 . . . . . . 7  |-  ( (
ph  /\  z  e.  { 0 } )  ->  sum_ y  e.  ran  F
( z  x.  (
y I z ) )  =  0 )
203186, 202sylan2 472 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  ( ran  G  \  { 0 } ) ) )  ->  sum_ y  e.  ran  F ( z  x.  (
y I z ) )  =  0 )
204179, 182, 203, 6fsumss 13629 . . . . 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 ) ) )
20522adantr 463 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  RR )
206205recnd 9611 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  CC )
20716ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  I : ( RR  X.  RR ) --> RR )
20811adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  RR )
209208sselda 3489 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  RR )
210207, 209, 205fovrnd 6420 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  RR )
211210recnd 9611 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  CC )
212206, 211mulcld 9605 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( z  x.  ( y I z ) )  e.  CC )
213212anasss 645 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ran  G  /\  y  e.  ran  F ) )  ->  ( z  x.  ( y I z ) )  e.  CC )
2146, 3, 213fsumcom 13672 . . . . 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 ) ) )
215204, 214eqtrd 2495 . . . 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 ) ) )
216125, 178, 2153eqtrd 2499 . . 3  |-  ( ph  ->  ( S.1 `  G
)  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( z  x.  ( y I z ) ) )
217123, 216oveq12d 6288 . 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 ) ) ) )
21831, 39, 2173eqtr4d 2505 1  |-  ( ph  ->  ( S.1 `  ( F  oF  +  G
) )  =  ( ( S.1 `  F
)  +  ( S.1 `  G ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    \ cdif 3458    i^i cin 3460    C_ wss 3461   ifcif 3929   {csn 4016   U_ciun 4315    X. cxp 4986   `'ccnv 4987   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272    oFcof 6511   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    x. cmul 9486   ZZ>=cuz 11082   sum_csu 13590   volcvol 22041   S.1citg1 22190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xadd 11322  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-xmet 18607  df-met 18608  df-ovol 22042  df-vol 22043  df-mbf 22194  df-itg1 22195
This theorem is referenced by:  itg1add  22274
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