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Theorem itg1addlem3 21195
Description: Lemma for itg1add 21198. (Contributed by Mario Carneiro, 26-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 } ) ) ) ) )
Assertion
Ref Expression
itg1addlem3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A I B )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
Distinct variable groups:    i, j, A    B, i, j    i, F, j    i, G, j    ph, i, j
Allowed substitution hints:    I( i, j)

Proof of Theorem itg1addlem3
StepHypRef Expression
1 eqeq1 2449 . . . . 5  |-  ( i  =  A  ->  (
i  =  0  <->  A  =  0 ) )
2 eqeq1 2449 . . . . 5  |-  ( j  =  B  ->  (
j  =  0  <->  B  =  0 ) )
31, 2bi2anan9 868 . . . 4  |-  ( ( i  =  A  /\  j  =  B )  ->  ( ( i  =  0  /\  j  =  0 )  <->  ( A  =  0  /\  B  =  0 ) ) )
4 sneq 3906 . . . . . . 7  |-  ( i  =  A  ->  { i }  =  { A } )
54imaeq2d 5188 . . . . . 6  |-  ( i  =  A  ->  ( `' F " { i } )  =  ( `' F " { A } ) )
6 sneq 3906 . . . . . . 7  |-  ( j  =  B  ->  { j }  =  { B } )
76imaeq2d 5188 . . . . . 6  |-  ( j  =  B  ->  ( `' G " { j } )  =  ( `' G " { B } ) )
85, 7ineqan12d 3573 . . . . 5  |-  ( ( i  =  A  /\  j  =  B )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  =  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) )
98fveq2d 5714 . . . 4  |-  ( ( i  =  A  /\  j  =  B )  ->  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
103, 9ifbieq2d 3833 . . 3  |-  ( ( i  =  A  /\  j  =  B )  ->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { i } )  i^i  ( `' G " { j } ) ) ) )  =  if ( ( A  =  0  /\  B  =  0 ) ,  0 ,  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) ) )
11 itg1add.3 . . 3  |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) ) ) )
12 c0ex 9399 . . . 4  |-  0  e.  _V
13 fvex 5720 . . . 4  |-  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) )  e.  _V
1412, 13ifex 3877 . . 3  |-  if ( ( A  =  0  /\  B  =  0 ) ,  0 ,  ( vol `  (
( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )  e.  _V
1510, 11, 14ovmpt2a 6240 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A I B )  =  if ( ( A  =  0  /\  B  =  0 ) ,  0 ,  ( vol `  (
( `' F " { A } )  i^i  ( `' G " { B } ) ) ) ) )
16 iffalse 3818 . 2  |-  ( -.  ( A  =  0  /\  B  =  0 )  ->  if (
( A  =  0  /\  B  =  0 ) ,  0 ,  ( vol `  (
( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
1715, 16sylan9eq 2495 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A I B )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3346   ifcif 3810   {csn 3896   `'ccnv 4858   dom cdm 4859   "cima 4862   ` cfv 5437  (class class class)co 6110    e. cmpt2 6112   RRcr 9300   0cc0 9301   volcvol 20966   S.1citg1 21114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-mulcl 9363  ax-i2m1 9369
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115
This theorem is referenced by:  itg1addlem4  21196  itg1addlem5  21197
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