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Theorem itg1addlem3 19053
Description: Lemma for itg1add 19056. (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 2289 . . . . 5  |-  ( i  =  A  ->  (
i  =  0  <->  A  =  0 ) )
2 eqeq1 2289 . . . . 5  |-  ( j  =  B  ->  (
j  =  0  <->  B  =  0 ) )
31, 2bi2anan9 843 . . . 4  |-  ( ( i  =  A  /\  j  =  B )  ->  ( ( i  =  0  /\  j  =  0 )  <->  ( A  =  0  /\  B  =  0 ) ) )
4 sneq 3651 . . . . . . 7  |-  ( i  =  A  ->  { i }  =  { A } )
54imaeq2d 5012 . . . . . 6  |-  ( i  =  A  ->  ( `' F " { i } )  =  ( `' F " { A } ) )
6 sneq 3651 . . . . . . 7  |-  ( j  =  B  ->  { j }  =  { B } )
76imaeq2d 5012 . . . . . 6  |-  ( j  =  B  ->  ( `' G " { j } )  =  ( `' G " { B } ) )
85, 7ineqan12d 3372 . . . . 5  |-  ( ( i  =  A  /\  j  =  B )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  =  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) )
98fveq2d 5529 . . . 4  |-  ( ( i  =  A  /\  j  =  B )  ->  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
103, 9ifbieq2d 3585 . . 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 8832 . . . 4  |-  0  e.  _V
13 fvex 5539 . . . 4  |-  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) )  e.  _V
1412, 13ifex 3623 . . 3  |-  if ( ( A  =  0  /\  B  =  0 ) ,  0 ,  ( vol `  (
( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )  e.  _V
1510, 11, 14ovmpt2a 5978 . 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 3572 . 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 2335 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151   ifcif 3565   {csn 3640   `'ccnv 4688   dom cdm 4689   "cima 4692   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   RRcr 8736   0cc0 8737   volcvol 18823   S.1citg1 18970
This theorem is referenced by:  itg1addlem4  19054  itg1addlem5  19055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-i2m1 8805
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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