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Theorem txindislem 20007
Description: Lemma for txindis 20008. (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 0xp 5070 . . 3  |-  ( (/)  X.  (  _I  `  B
) )  =  (/)
2 fvprc 5850 . . . 4  |-  ( -.  A  e.  _V  ->  (  _I  `  A )  =  (/) )
32xpeq1d 5012 . . 3  |-  ( -.  A  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  ( (/)  X.  (  _I  `  B ) ) )
4 simpr 461 . . . . . . . 8  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  ->  B  =  (/) )
54xpeq2d 5013 . . . . . . 7  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
( A  X.  B
)  =  ( A  X.  (/) ) )
6 xp0 5415 . . . . . . 7  |-  ( A  X.  (/) )  =  (/)
75, 6syl6eq 2500 . . . . . 6  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
( A  X.  B
)  =  (/) )
87fveq2d 5860 . . . . 5  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (  _I  `  (/) ) )
9 0ex 4567 . . . . . 6  |-  (/)  e.  _V
10 fvi 5915 . . . . . 6  |-  ( (/)  e.  _V  ->  (  _I  `  (/) )  =  (/) )
119, 10ax-mp 5 . . . . 5  |-  (  _I 
`  (/) )  =  (/)
128, 11syl6eq 2500 . . . 4  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (/) )
13 dmexg 6716 . . . . . . . 8  |-  ( ( A  X.  B )  e.  _V  ->  dom  ( A  X.  B
)  e.  _V )
14 dmxp 5211 . . . . . . . . 9  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
1514eleq1d 2512 . . . . . . . 8  |-  ( B  =/=  (/)  ->  ( dom  ( A  X.  B
)  e.  _V  <->  A  e.  _V ) )
1613, 15syl5ib 219 . . . . . . 7  |-  ( B  =/=  (/)  ->  ( ( A  X.  B )  e. 
_V  ->  A  e.  _V ) )
1716con3d 133 . . . . . 6  |-  ( B  =/=  (/)  ->  ( -.  A  e.  _V  ->  -.  ( A  X.  B
)  e.  _V )
)
1817impcom 430 . . . . 5  |-  ( ( -.  A  e.  _V  /\  B  =/=  (/) )  ->  -.  ( A  X.  B
)  e.  _V )
19 fvprc 5850 . . . . 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 2761 . . 3  |-  ( -.  A  e.  _V  ->  (  _I  `  ( A  X.  B ) )  =  (/) )
221, 3, 213eqtr4a 2510 . 2  |-  ( -.  A  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  (  _I  `  ( A  X.  B
) ) )
23 xp0 5415 . . 3  |-  ( (  _I  `  A )  X.  (/) )  =  (/)
24 fvprc 5850 . . . 4  |-  ( -.  B  e.  _V  ->  (  _I  `  B )  =  (/) )
2524xpeq2d 5013 . . 3  |-  ( -.  B  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  ( (  _I 
`  A )  X.  (/) ) )
26 simpr 461 . . . . . . . 8  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  ->  A  =  (/) )
2726xpeq1d 5012 . . . . . . 7  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
( A  X.  B
)  =  ( (/)  X.  B ) )
28 0xp 5070 . . . . . . 7  |-  ( (/)  X.  B )  =  (/)
2927, 28syl6eq 2500 . . . . . 6  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
( A  X.  B
)  =  (/) )
3029fveq2d 5860 . . . . 5  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (  _I  `  (/) ) )
3130, 11syl6eq 2500 . . . 4  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (/) )
32 rnexg 6717 . . . . . . . 8  |-  ( ( A  X.  B )  e.  _V  ->  ran  ( A  X.  B
)  e.  _V )
33 rnxp 5427 . . . . . . . . 9  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
3433eleq1d 2512 . . . . . . . 8  |-  ( A  =/=  (/)  ->  ( ran  ( A  X.  B
)  e.  _V  <->  B  e.  _V ) )
3532, 34syl5ib 219 . . . . . . 7  |-  ( A  =/=  (/)  ->  ( ( A  X.  B )  e. 
_V  ->  B  e.  _V ) )
3635con3d 133 . . . . . 6  |-  ( A  =/=  (/)  ->  ( -.  B  e.  _V  ->  -.  ( A  X.  B
)  e.  _V )
)
3736impcom 430 . . . . 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 2761 . . 3  |-  ( -.  B  e.  _V  ->  (  _I  `  ( A  X.  B ) )  =  (/) )
4023, 25, 393eqtr4a 2510 . 2  |-  ( -.  B  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  (  _I  `  ( A  X.  B
) ) )
41 fvi 5915 . . . 4  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
42 fvi 5915 . . . 4  |-  ( B  e.  _V  ->  (  _I  `  B )  =  B )
43 xpeq12 5008 . . . 4  |-  ( ( (  _I  `  A
)  =  A  /\  (  _I  `  B )  =  B )  -> 
( (  _I  `  A )  X.  (  _I  `  B ) )  =  ( A  X.  B ) )
4441, 42, 43syl2an 477 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  ( A  X.  B ) )
45 xpexg 6587 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  X.  B
)  e.  _V )
46 fvi 5915 . . . 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 2487 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B
) ) )
4922, 40, 48ecase 942 1  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095   (/)c0 3770    _I cid 4780    X. cxp 4987   dom cdm 4989   ran crn 4990   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fv 5586
This theorem is referenced by:  txindis  20008
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