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Theorem itg1addlem5 22650
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 22627 . . . 4  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
31, 2syl 17 . . 3  |-  ( ph  ->  ran  F  e.  Fin )
4 i1fadd.2 . . . . . 6  |-  ( ph  ->  G  e.  dom  S.1 )
5 i1frn 22627 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
64, 5syl 17 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
76adantr 467 . . . 4  |-  ( (
ph  /\  y  e.  ran  F )  ->  ran  G  e.  Fin )
8 i1ff 22626 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
91, 8syl 17 . . . . . . . . 9  |-  ( ph  ->  F : RR --> RR )
10 frn 5750 . . . . . . . . 9  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
119, 10syl 17 . . . . . . . 8  |-  ( ph  ->  ran  F  C_  RR )
1211sselda 3465 . . . . . . 7  |-  ( (
ph  /\  y  e.  ran  F )  ->  y  e.  RR )
1312adantr 467 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  y  e.  RR )
1413recnd 9671 . . . . 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 22647 . . . . . . . 8  |-  ( ph  ->  I : ( RR 
X.  RR ) --> RR )
1716ad2antrr 731 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  I : ( RR  X.  RR ) --> RR )
18 i1ff 22626 . . . . . . . . . . 11  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
194, 18syl 17 . . . . . . . . . 10  |-  ( ph  ->  G : RR --> RR )
20 frn 5750 . . . . . . . . . 10  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
2119, 20syl 17 . . . . . . . . 9  |-  ( ph  ->  ran  G  C_  RR )
2221sselda 3465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ran  G )  ->  z  e.  RR )
2322adantlr 720 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  z  e.  RR )
2417, 13, 23fovrnd 6453 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y I z )  e.  RR )
2524recnd 9671 . . . . 5  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y I z )  e.  CC )
2614, 25mulcld 9665 . . . 4  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  e.  CC )
277, 26fsumcl 13792 . . 3  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( y  x.  ( y I z ) )  e.  CC )
2823recnd 9671 . . . . 5  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  z  e.  CC )
2928, 25mulcld 9665 . . . 4  |-  ( ( ( ph  /\  y  e.  ran  F )  /\  z  e.  ran  G )  ->  ( z  x.  ( y I z ) )  e.  CC )
307, 29fsumcl 13792 . . 3  |-  ( (
ph  /\  y  e.  ran  F )  ->  sum_ z  e.  ran  G ( z  x.  ( y I z ) )  e.  CC )
313, 27, 30fsumadd 13798 . 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 22649 . . 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 9670 . . . . . 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 13762 . . . . 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 13798 . . . . 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 2464 . . . 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 13762 . . 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 2464 . 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 22633 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ y  e.  ( ran  F  \  {
0 } ) ( y  x.  ( vol `  ( `' F " { y } ) ) ) )
411, 40syl 17 . . . 4  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ y  e.  ( ran  F  \  { 0 } ) ( y  x.  ( vol `  ( `' F " { y } ) ) ) )
4219adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  G : RR --> RR )
436adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ran  G  e.  Fin )
44 inss2 3684 . . . . . . . . . 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 22628 . . . . . . . . . . . 12  |-  ( F  e.  dom  S.1  ->  ( `' F " { y } )  e.  dom  vol )
471, 46syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( `' F " { y } )  e.  dom  vol )
4847ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( `' F " { y } )  e.  dom  vol )
49 i1fima 22628 . . . . . . . . . . . 12  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
504, 49syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
5150ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( `' G " { z } )  e.  dom  vol )
52 inmbl 22487 . . . . . . . . . 10  |-  ( ( ( `' F " { y } )  e.  dom  vol  /\  ( `' G " { z } )  e.  dom  vol )  ->  ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
5348, 51, 52syl2anc 666 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
5411ssdifssd 3604 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
5554sselda 3465 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  y  e.  RR )
5655adantr 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
y  e.  RR )
5721adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ran  G  C_  RR )
5857sselda 3465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
z  e.  RR )
59 eldifsni 4124 . . . . . . . . . . . . 13  |-  ( y  e.  ( ran  F  \  { 0 } )  ->  y  =/=  0
)
6059ad2antlr 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
y  =/=  0 )
61 simpl 459 . . . . . . . . . . . . 13  |-  ( ( y  =  0  /\  z  =  0 )  ->  y  =  0 )
6261necon3ai 2653 . . . . . . . . . . . 12  |-  ( y  =/=  0  ->  -.  ( y  =  0  /\  z  =  0 ) )
6360, 62syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  -.  ( y  =  0  /\  z  =  0 ) )
641, 4, 15itg1addlem3 22648 . . . . . . . . . . 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 1264 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y I z )  =  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
6616ad2antrr 731 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  ->  I : ( RR  X.  RR ) --> RR )
6766, 56, 58fovrnd 6453 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y I z )  e.  RR )
6865, 67eqeltrrd 2512 . . . . . . . . 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 22642 . . . . . . . 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 4361 . . . . . . . . . 10  |-  U_ z  e.  ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) )  =  ( ( `' F " { y } )  i^i  U_ z  e.  ran  G ( `' G " { z } ) )
711adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  F  e.  dom  S.1 )
7271, 46syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } )  e.  dom  vol )
73 mblss 22477 . . . . . . . . . . . . 13  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( `' F " { y } ) 
C_  RR )
7472, 73syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } ) 
C_  RR )
75 iunid 4352 . . . . . . . . . . . . . . 15  |-  U_ z  e.  ran  G { z }  =  ran  G
7675imaeq2i 5183 . . . . . . . . . . . . . 14  |-  ( `' G " U_ z  e.  ran  G { z } )  =  ( `' G " ran  G
)
77 imaiun 6163 . . . . . . . . . . . . . 14  |-  ( `' G " U_ z  e.  ran  G { z } )  =  U_ z  e.  ran  G ( `' G " { z } )
78 cnvimarndm 5206 . . . . . . . . . . . . . 14  |-  ( `' G " ran  G
)  =  dom  G
7976, 77, 783eqtr3i 2460 . . . . . . . . . . . . 13  |-  U_ z  e.  ran  G ( `' G " { z } )  =  dom  G
80 fdm 5748 . . . . . . . . . . . . . 14  |-  ( G : RR --> RR  ->  dom 
G  =  RR )
8142, 80syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  dom  G  =  RR )
8279, 81syl5eq 2476 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  U_ z  e.  ran  G ( `' G " { z } )  =  RR )
8374, 82sseqtr4d 3502 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } ) 
C_  U_ z  e.  ran  G ( `' G " { z } ) )
84 df-ss 3451 . . . . . . . . . . 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 200 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( ( `' F " { y } )  i^i  U_ z  e.  ran  G ( `' G " { z } ) )  =  ( `' F " { y } ) )
8670, 85syl5req 2477 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { y } )  =  U_ z  e. 
ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )
8786fveq2d 5883 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { y } ) )  =  ( vol `  U_ z  e.  ran  G ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
8865sumeq2dv 13762 . . . . . . . 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 2474 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { y } ) )  = 
sum_ z  e.  ran  G ( y I z ) )
9089oveq2d 6319 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( y  x.  ( vol `  ( `' F " { y } ) ) )  =  ( y  x. 
sum_ z  e.  ran  G ( y I z ) ) )
9155recnd 9671 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  y  e.  CC )
9267recnd 9671 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y I z )  e.  CC )
9343, 91, 92fsummulc2 13838 . . . . . 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 2464 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  ( y  x.  ( vol `  ( `' F " { y } ) ) )  =  sum_ z  e.  ran  G ( y  x.  (
y I z ) ) )
9594sumeq2dv 13762 . . . 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 3594 . . . . 5  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  ran  F )
9756recnd 9671 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
y  e.  CC )
9897, 92mulcld 9665 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  F  \  { 0 } ) )  /\  z  e. 
ran  G )  -> 
( y  x.  (
y I z ) )  e.  CC )
9943, 98fsumcl 13792 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  {
0 } ) )  ->  sum_ z  e.  ran  G ( y  x.  (
y I z ) )  e.  CC )
100 dfin4 3714 . . . . . . . 8  |-  ( ran 
F  i^i  { 0 } )  =  ( ran  F  \  ( ran  F  \  { 0 } ) )
101 inss2 3684 . . . . . . . 8  |-  ( ran 
F  i^i  { 0 } )  C_  { 0 }
102100, 101eqsstr3i 3496 . . . . . . 7  |-  ( ran 
F  \  ( ran  F 
\  { 0 } ) )  C_  { 0 }
103102sseli 3461 . . . . . 6  |-  ( y  e.  ( ran  F  \  ( ran  F  \  { 0 } ) )  ->  y  e.  { 0 } )
104 elsni 4022 . . . . . . . . . . 11  |-  ( y  e.  { 0 }  ->  y  =  0 )
105104ad2antlr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  y  = 
0 )
106105oveq1d 6318 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  =  ( 0  x.  ( y I z ) ) )
10716ad2antrr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  I :
( RR  X.  RR )
--> RR )
108 0re 9645 . . . . . . . . . . . . 13  |-  0  e.  RR
109105, 108syl6eqel 2519 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  y  e.  RR )
11022adantlr 720 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  z  e.  RR )
111107, 109, 110fovrnd 6453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y
I z )  e.  RR )
112111recnd 9671 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y
I z )  e.  CC )
113112mul02d 9833 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( 0  x.  ( y I z ) )  =  0 )
114106, 113eqtrd 2464 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  { 0 } )  /\  z  e.  ran  G )  ->  ( y  x.  ( y I z ) )  =  0 )
115114sumeq2dv 13762 . . . . . . 7  |-  ( (
ph  /\  y  e.  { 0 } )  ->  sum_ z  e.  ran  G
( y  x.  (
y I z ) )  =  sum_ z  e.  ran  G 0 )
1166adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  { 0 } )  ->  ran  G  e.  Fin )
117116olcd 395 . . . . . . . 8  |-  ( (
ph  /\  y  e.  { 0 } )  -> 
( ran  G  C_  ( ZZ>=
`  0 )  \/ 
ran  G  e.  Fin ) )
118 sumz 13781 . . . . . . . 8  |-  ( ( ran  G  C_  ( ZZ>=
`  0 )  \/ 
ran  G  e.  Fin )  ->  sum_ z  e.  ran  G 0  =  0 )
119117, 118syl 17 . . . . . . 7  |-  ( (
ph  /\  y  e.  { 0 } )  ->  sum_ z  e.  ran  G
0  =  0 )
120115, 119eqtrd 2464 . . . . . 6  |-  ( (
ph  /\  y  e.  { 0 } )  ->  sum_ z  e.  ran  G
( y  x.  (
y I z ) )  =  0 )
121103, 120sylan2 477 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  F  \  ( ran  F  \  { 0 } ) ) )  ->  sum_ z  e.  ran  G ( y  x.  (
y I z ) )  =  0 )
12296, 99, 121, 3fsumss 13784 . . . 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 2468 . . 3  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( y  x.  ( y I z ) ) )
124 itg1val 22633 . . . . 5  |-  ( G  e.  dom  S.1  ->  ( S.1 `  G )  =  sum_ z  e.  ( ran  G  \  {
0 } ) ( z  x.  ( vol `  ( `' G " { z } ) ) ) )
1254, 124syl 17 . . . 4  |-  ( ph  ->  ( S.1 `  G
)  =  sum_ z  e.  ( ran  G  \  { 0 } ) ( z  x.  ( vol `  ( `' G " { z } ) ) ) )
1269adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  F : RR --> RR )
1273adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ran  F  e.  Fin )
128 inss1 3683 . . . . . . . . . 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 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( `' F " { y } )  e.  dom  vol )
13150ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( `' G " { z } )  e.  dom  vol )
132130, 131, 52syl2anc 666 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( ( `' F " { y } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
13311adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ran  F  C_  RR )
134133sselda 3465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
y  e.  RR )
13521ssdifssd 3604 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  G  \  { 0 } ) 
C_  RR )
136135sselda 3465 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  z  e.  RR )
137136adantr 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
z  e.  RR )
138 eldifsni 4124 . . . . . . . . . . . . 13  |-  ( z  e.  ( ran  G  \  { 0 } )  ->  z  =/=  0
)
139138ad2antlr 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
z  =/=  0 )
140 simpr 463 . . . . . . . . . . . . 13  |-  ( ( y  =  0  /\  z  =  0 )  ->  z  =  0 )
141140necon3ai 2653 . . . . . . . . . . . 12  |-  ( z  =/=  0  ->  -.  ( y  =  0  /\  z  =  0 ) )
142139, 141syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  ->  -.  ( y  =  0  /\  z  =  0 ) )
143134, 137, 142, 64syl21anc 1264 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( y I z )  =  ( vol `  ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
14416ad2antrr 731 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  ->  I : ( RR  X.  RR ) --> RR )
145144, 134, 137fovrnd 6453 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( y I z )  e.  RR )
146143, 145eqeltrrd 2512 . . . . . . . . 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 22642 . . . . . . . 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 3656 . . . . . . . . . . . . 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 4316 . . . . . . . . . . 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 4361 . . . . . . . . . . 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 2452 . . . . . . . . . 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 5205 . . . . . . . . . . . . 13  |-  ( `' G " { z } )  C_  dom  G
15419, 80syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  G  =  RR )
155154adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  dom  G  =  RR )
156153, 155syl5sseq 3513 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } ) 
C_  RR )
157 iunid 4352 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ran  F { y }  =  ran  F
158157imaeq2i 5183 . . . . . . . . . . . . . 14  |-  ( `' F " U_ y  e.  ran  F { y } )  =  ( `' F " ran  F
)
159 imaiun 6163 . . . . . . . . . . . . . 14  |-  ( `' F " U_ y  e.  ran  F { y } )  =  U_ y  e.  ran  F ( `' F " { y } )
160 cnvimarndm 5206 . . . . . . . . . . . . . 14  |-  ( `' F " ran  F
)  =  dom  F
161158, 159, 1603eqtr3i 2460 . . . . . . . . . . . . 13  |-  U_ y  e.  ran  F ( `' F " { y } )  =  dom  F
162 fdm 5748 . . . . . . . . . . . . . . 15  |-  ( F : RR --> RR  ->  dom 
F  =  RR )
1639, 162syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  F  =  RR )
164163adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  dom  F  =  RR )
165161, 164syl5eq 2476 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  U_ y  e.  ran  F ( `' F " { y } )  =  RR )
166156, 165sseqtr4d 3502 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } ) 
C_  U_ y  e.  ran  F ( `' F " { y } ) )
167 df-ss 3451 . . . . . . . . . . 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 200 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( ( `' G " { z } )  i^i  U_ y  e.  ran  F ( `' F " { y } ) )  =  ( `' G " { z } ) )
169152, 168syl5req 2477 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( `' G " { z } )  =  U_ y  e. 
ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) )
170169fveq2d 5883 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( vol `  ( `' G " { z } ) )  =  ( vol `  U_ y  e.  ran  F ( ( `' F " { y } )  i^i  ( `' G " { z } ) ) ) )
171143sumeq2dv 13762 . . . . . . . 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 2474 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( vol `  ( `' G " { z } ) )  = 
sum_ y  e.  ran  F ( y I z ) )
173172oveq2d 6319 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( z  x.  ( vol `  ( `' G " { z } ) ) )  =  ( z  x. 
sum_ y  e.  ran  F ( y I z ) ) )
174136recnd 9671 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  z  e.  CC )
175145recnd 9671 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( y I z )  e.  CC )
176127, 174, 175fsummulc2 13838 . . . . . 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 2464 . . . . 5  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  ( z  x.  ( vol `  ( `' G " { z } ) ) )  =  sum_ y  e.  ran  F ( z  x.  (
y I z ) ) )
178177sumeq2dv 13762 . . . 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 3594 . . . . . 6  |-  ( ph  ->  ( ran  G  \  { 0 } ) 
C_  ran  G )
180174adantr 467 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
z  e.  CC )
181180, 175mulcld 9665 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( ran  G  \  { 0 } ) )  /\  y  e. 
ran  F )  -> 
( z  x.  (
y I z ) )  e.  CC )
182127, 181fsumcl 13792 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  {
0 } ) )  ->  sum_ y  e.  ran  F ( z  x.  (
y I z ) )  e.  CC )
183 dfin4 3714 . . . . . . . . 9  |-  ( ran 
G  i^i  { 0 } )  =  ( ran  G  \  ( ran  G  \  { 0 } ) )
184 inss2 3684 . . . . . . . . 9  |-  ( ran 
G  i^i  { 0 } )  C_  { 0 }
185183, 184eqsstr3i 3496 . . . . . . . 8  |-  ( ran 
G  \  ( ran  G 
\  { 0 } ) )  C_  { 0 }
186185sseli 3461 . . . . . . 7  |-  ( z  e.  ( ran  G  \  ( ran  G  \  { 0 } ) )  ->  z  e.  { 0 } )
187 elsni 4022 . . . . . . . . . . . 12  |-  ( z  e.  { 0 }  ->  z  =  0 )
188187ad2antlr 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  z  = 
0 )
189188oveq1d 6318 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( z  x.  ( y I z ) )  =  ( 0  x.  ( y I z ) ) )
19016ad2antrr 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  I :
( RR  X.  RR )
--> RR )
19112adantlr 720 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  y  e.  RR )
192188, 108syl6eqel 2519 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  z  e.  RR )
193190, 191, 192fovrnd 6453 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( y
I z )  e.  RR )
194193recnd 9671 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( y
I z )  e.  CC )
195194mul02d 9833 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( 0  x.  ( y I z ) )  =  0 )
196189, 195eqtrd 2464 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  { 0 } )  /\  y  e.  ran  F )  ->  ( z  x.  ( y I z ) )  =  0 )
197196sumeq2dv 13762 . . . . . . . 8  |-  ( (
ph  /\  z  e.  { 0 } )  ->  sum_ y  e.  ran  F
( z  x.  (
y I z ) )  =  sum_ y  e.  ran  F 0 )
1983adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  { 0 } )  ->  ran  F  e.  Fin )
199198olcd 395 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  { 0 } )  -> 
( ran  F  C_  ( ZZ>=
`  0 )  \/ 
ran  F  e.  Fin ) )
200 sumz 13781 . . . . . . . . 9  |-  ( ( ran  F  C_  ( ZZ>=
`  0 )  \/ 
ran  F  e.  Fin )  ->  sum_ y  e.  ran  F 0  =  0 )
201199, 200syl 17 . . . . . . . 8  |-  ( (
ph  /\  z  e.  { 0 } )  ->  sum_ y  e.  ran  F
0  =  0 )
202197, 201eqtrd 2464 . . . . . . 7  |-  ( (
ph  /\  z  e.  { 0 } )  ->  sum_ y  e.  ran  F
( z  x.  (
y I z ) )  =  0 )
203186, 202sylan2 477 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ran  G  \  ( ran  G  \  { 0 } ) ) )  ->  sum_ y  e.  ran  F ( z  x.  (
y I z ) )  =  0 )
204179, 182, 203, 6fsumss 13784 . . . . 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 467 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  RR )
206205recnd 9671 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  z  e.  CC )
20716ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  I : ( RR  X.  RR ) --> RR )
20811adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ran  G )  ->  ran  F 
C_  RR )
209208sselda 3465 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  y  e.  RR )
210207, 209, 205fovrnd 6453 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  RR )
211210recnd 9671 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( y I z )  e.  CC )
212206, 211mulcld 9665 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ran  G )  /\  y  e.  ran  F )  ->  ( z  x.  ( y I z ) )  e.  CC )
213212anasss 652 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ran  G  /\  y  e.  ran  F ) )  ->  ( z  x.  ( y I z ) )  e.  CC )
2146, 3, 213fsumcom 13829 . . . . 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 2464 . . . 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 2468 . . 3  |-  ( ph  ->  ( S.1 `  G
)  =  sum_ y  e.  ran  F sum_ z  e.  ran  G ( z  x.  ( y I z ) ) )
217123, 216oveq12d 6321 . 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 2474 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 370    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619    \ cdif 3434    i^i cin 3436    C_ wss 3437   ifcif 3910   {csn 3997   U_ciun 4297    X. cxp 4849   `'ccnv 4850   dom cdm 4851   ran crn 4852    |` cres 4853   "cima 4854   -->wf 5595   ` cfv 5599  (class class class)co 6303    |-> cmpt2 6305    oFcof 6541   Fincfn 7575   CCcc 9539   RRcr 9540   0cc0 9541    + caddc 9544    x. cmul 9546   ZZ>=cuz 11161   sum_csu 13745   volcvol 22407   S.1citg1 22565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-disj 4393  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-q 11267  df-rp 11305  df-xadd 11412  df-ioo 11641  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-seq 12215  df-exp 12274  df-hash 12517  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-clim 13545  df-sum 13746  df-xmet 18956  df-met 18957  df-ovol 22408  df-vol 22410  df-mbf 22569  df-itg1 22570
This theorem is referenced by:  itg1add  22651
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