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Theorem itg1addlem3 22656
Description: Lemma for itg1add 22659. (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 2455 . . . . 5  |-  ( i  =  A  ->  (
i  =  0  <->  A  =  0 ) )
2 eqeq1 2455 . . . . 5  |-  ( j  =  B  ->  (
j  =  0  <->  B  =  0 ) )
31, 2bi2anan9 884 . . . 4  |-  ( ( i  =  A  /\  j  =  B )  ->  ( ( i  =  0  /\  j  =  0 )  <->  ( A  =  0  /\  B  =  0 ) ) )
4 sneq 3978 . . . . . . 7  |-  ( i  =  A  ->  { i }  =  { A } )
54imaeq2d 5168 . . . . . 6  |-  ( i  =  A  ->  ( `' F " { i } )  =  ( `' F " { A } ) )
6 sneq 3978 . . . . . . 7  |-  ( j  =  B  ->  { j }  =  { B } )
76imaeq2d 5168 . . . . . 6  |-  ( j  =  B  ->  ( `' G " { j } )  =  ( `' G " { B } ) )
85, 7ineqan12d 3636 . . . . 5  |-  ( ( i  =  A  /\  j  =  B )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  =  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) )
98fveq2d 5869 . . . 4  |-  ( ( i  =  A  /\  j  =  B )  ->  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
103, 9ifbieq2d 3906 . . 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 9637 . . . 4  |-  0  e.  _V
13 fvex 5875 . . . 4  |-  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) )  e.  _V
1412, 13ifex 3949 . . 3  |-  if ( ( A  =  0  /\  B  =  0 ) ,  0 ,  ( vol `  (
( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )  e.  _V
1510, 11, 14ovmpt2a 6427 . 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 3890 . 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 2505 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 371    = wceq 1444    e. wcel 1887    i^i cin 3403   ifcif 3881   {csn 3968   `'ccnv 4833   dom cdm 4834   "cima 4837   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   RRcr 9538   0cc0 9539   volcvol 22415   S.1citg1 22573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-mulcl 9601  ax-i2m1 9607
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295
This theorem is referenced by:  itg1addlem4  22657  itg1addlem5  22658
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