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Theorem txindislem 17587
Description: Lemma for txindis 17588. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
txindislem  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )

Proof of Theorem txindislem
StepHypRef Expression
1 xp0r 4897 . . 3  |-  ( (/)  X.  (  _I  `  B
) )  =  (/)
2 fvprc 5663 . . . 4  |-  ( -.  A  e.  _V  ->  (  _I  `  A )  =  (/) )
32xpeq1d 4842 . . 3  |-  ( -.  A  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  ( (/)  X.  (  _I  `  B ) ) )
4 simpr 448 . . . . . . . 8  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  ->  B  =  (/) )
54xpeq2d 4843 . . . . . . 7  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
( A  X.  B
)  =  ( A  X.  (/) ) )
6 xp0 5232 . . . . . . 7  |-  ( A  X.  (/) )  =  (/)
75, 6syl6eq 2436 . . . . . 6  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
( A  X.  B
)  =  (/) )
87fveq2d 5673 . . . . 5  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (  _I  `  (/) ) )
9 0ex 4281 . . . . . 6  |-  (/)  e.  _V
10 fvi 5723 . . . . . 6  |-  ( (/)  e.  _V  ->  (  _I  `  (/) )  =  (/) )
119, 10ax-mp 8 . . . . 5  |-  (  _I 
`  (/) )  =  (/)
128, 11syl6eq 2436 . . . 4  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (/) )
13 dmexg 5071 . . . . . . . 8  |-  ( ( A  X.  B )  e.  _V  ->  dom  ( A  X.  B
)  e.  _V )
14 dmxp 5029 . . . . . . . . 9  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
1514eleq1d 2454 . . . . . . . 8  |-  ( B  =/=  (/)  ->  ( dom  ( A  X.  B
)  e.  _V  <->  A  e.  _V ) )
1613, 15syl5ib 211 . . . . . . 7  |-  ( B  =/=  (/)  ->  ( ( A  X.  B )  e. 
_V  ->  A  e.  _V ) )
1716con3d 127 . . . . . 6  |-  ( B  =/=  (/)  ->  ( -.  A  e.  _V  ->  -.  ( A  X.  B
)  e.  _V )
)
1817impcom 420 . . . . 5  |-  ( ( -.  A  e.  _V  /\  B  =/=  (/) )  ->  -.  ( A  X.  B
)  e.  _V )
19 fvprc 5663 . . . . 5  |-  ( -.  ( A  X.  B
)  e.  _V  ->  (  _I  `  ( A  X.  B ) )  =  (/) )
2018, 19syl 16 . . . 4  |-  ( ( -.  A  e.  _V  /\  B  =/=  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (/) )
2112, 20pm2.61dane 2629 . . 3  |-  ( -.  A  e.  _V  ->  (  _I  `  ( A  X.  B ) )  =  (/) )
221, 3, 213eqtr4a 2446 . 2  |-  ( -.  A  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  (  _I  `  ( A  X.  B
) ) )
23 xp0 5232 . . 3  |-  ( (  _I  `  A )  X.  (/) )  =  (/)
24 fvprc 5663 . . . 4  |-  ( -.  B  e.  _V  ->  (  _I  `  B )  =  (/) )
2524xpeq2d 4843 . . 3  |-  ( -.  B  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  ( (  _I 
`  A )  X.  (/) ) )
26 simpr 448 . . . . . . . 8  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  ->  A  =  (/) )
2726xpeq1d 4842 . . . . . . 7  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
( A  X.  B
)  =  ( (/)  X.  B ) )
28 xp0r 4897 . . . . . . 7  |-  ( (/)  X.  B )  =  (/)
2927, 28syl6eq 2436 . . . . . 6  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
( A  X.  B
)  =  (/) )
3029fveq2d 5673 . . . . 5  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (  _I  `  (/) ) )
3130, 11syl6eq 2436 . . . 4  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (/) )
32 rnexg 5072 . . . . . . . 8  |-  ( ( A  X.  B )  e.  _V  ->  ran  ( A  X.  B
)  e.  _V )
33 rnxp 5240 . . . . . . . . 9  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
3433eleq1d 2454 . . . . . . . 8  |-  ( A  =/=  (/)  ->  ( ran  ( A  X.  B
)  e.  _V  <->  B  e.  _V ) )
3532, 34syl5ib 211 . . . . . . 7  |-  ( A  =/=  (/)  ->  ( ( A  X.  B )  e. 
_V  ->  B  e.  _V ) )
3635con3d 127 . . . . . 6  |-  ( A  =/=  (/)  ->  ( -.  B  e.  _V  ->  -.  ( A  X.  B
)  e.  _V )
)
3736impcom 420 . . . . 5  |-  ( ( -.  B  e.  _V  /\  A  =/=  (/) )  ->  -.  ( A  X.  B
)  e.  _V )
3837, 19syl 16 . . . 4  |-  ( ( -.  B  e.  _V  /\  A  =/=  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (/) )
3931, 38pm2.61dane 2629 . . 3  |-  ( -.  B  e.  _V  ->  (  _I  `  ( A  X.  B ) )  =  (/) )
4023, 25, 393eqtr4a 2446 . 2  |-  ( -.  B  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  (  _I  `  ( A  X.  B
) ) )
41 fvi 5723 . . . 4  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
42 fvi 5723 . . . 4  |-  ( B  e.  _V  ->  (  _I  `  B )  =  B )
43 xpeq12 4838 . . . 4  |-  ( ( (  _I  `  A
)  =  A  /\  (  _I  `  B )  =  B )  -> 
( (  _I  `  A )  X.  (  _I  `  B ) )  =  ( A  X.  B ) )
4441, 42, 43syl2an 464 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  ( A  X.  B ) )
45 xpexg 4930 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  X.  B
)  e.  _V )
46 fvi 5723 . . . 4  |-  ( ( A  X.  B )  e.  _V  ->  (  _I  `  ( A  X.  B ) )  =  ( A  X.  B
) )
4745, 46syl 16 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  (  _I  `  ( A  X.  B ) )  =  ( A  X.  B ) )
4844, 47eqtr4d 2423 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B
) ) )
4922, 40, 48ecase 909 1  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900   (/)c0 3572    _I cid 4435    X. cxp 4817   dom cdm 4819   ran crn 4820   ` cfv 5395
This theorem is referenced by:  txindis  17588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fv 5403
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