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Theorem itg1addlem3 22231
Description: Lemma for itg1add 22234. (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 2461 . . . . 5  |-  ( i  =  A  ->  (
i  =  0  <->  A  =  0 ) )
2 eqeq1 2461 . . . . 5  |-  ( j  =  B  ->  (
j  =  0  <->  B  =  0 ) )
31, 2bi2anan9 873 . . . 4  |-  ( ( i  =  A  /\  j  =  B )  ->  ( ( i  =  0  /\  j  =  0 )  <->  ( A  =  0  /\  B  =  0 ) ) )
4 sneq 4042 . . . . . . 7  |-  ( i  =  A  ->  { i }  =  { A } )
54imaeq2d 5347 . . . . . 6  |-  ( i  =  A  ->  ( `' F " { i } )  =  ( `' F " { A } ) )
6 sneq 4042 . . . . . . 7  |-  ( j  =  B  ->  { j }  =  { B } )
76imaeq2d 5347 . . . . . 6  |-  ( j  =  B  ->  ( `' G " { j } )  =  ( `' G " { B } ) )
85, 7ineqan12d 3698 . . . . 5  |-  ( ( i  =  A  /\  j  =  B )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  =  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) )
98fveq2d 5876 . . . 4  |-  ( ( i  =  A  /\  j  =  B )  ->  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
103, 9ifbieq2d 3969 . . 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 9607 . . . 4  |-  0  e.  _V
13 fvex 5882 . . . 4  |-  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) )  e.  _V
1412, 13ifex 4013 . . 3  |-  if ( ( A  =  0  /\  B  =  0 ) ,  0 ,  ( vol `  (
( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )  e.  _V
1510, 11, 14ovmpt2a 6432 . 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 3953 . 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 2518 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 1395    e. wcel 1819    i^i cin 3470   ifcif 3944   {csn 4032   `'ccnv 5007   dom cdm 5008   "cima 5011   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   0cc0 9509   volcvol 22001   S.1citg1 22150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-mulcl 9571  ax-i2m1 9577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301
This theorem is referenced by:  itg1addlem4  22232  itg1addlem5  22233
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