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Theorem txindislem 17327
Description: Lemma for txindis 17328. (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 4768 . . 3  |-  ( (/)  X.  (  _I  `  B
) )  =  (/)
2 fvprc 5519 . . . 4  |-  ( -.  A  e.  _V  ->  (  _I  `  A )  =  (/) )
32xpeq1d 4712 . . 3  |-  ( -.  A  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  ( (/)  X.  (  _I  `  B ) ) )
4 simpr 447 . . . . . . . 8  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  ->  B  =  (/) )
54xpeq2d 4713 . . . . . . 7  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
( A  X.  B
)  =  ( A  X.  (/) ) )
6 xp0 5098 . . . . . . 7  |-  ( A  X.  (/) )  =  (/)
75, 6syl6eq 2331 . . . . . 6  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
( A  X.  B
)  =  (/) )
87fveq2d 5529 . . . . 5  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (  _I  `  (/) ) )
9 0ex 4150 . . . . . 6  |-  (/)  e.  _V
10 fvi 5579 . . . . . 6  |-  ( (/)  e.  _V  ->  (  _I  `  (/) )  =  (/) )
119, 10ax-mp 8 . . . . 5  |-  (  _I 
`  (/) )  =  (/)
128, 11syl6eq 2331 . . . 4  |-  ( ( -.  A  e.  _V  /\  B  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (/) )
13 dmexg 4939 . . . . . . . 8  |-  ( ( A  X.  B )  e.  _V  ->  dom  ( A  X.  B
)  e.  _V )
14 dmxp 4897 . . . . . . . . 9  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
1514eleq1d 2349 . . . . . . . 8  |-  ( B  =/=  (/)  ->  ( dom  ( A  X.  B
)  e.  _V  <->  A  e.  _V ) )
1613, 15syl5ib 210 . . . . . . 7  |-  ( B  =/=  (/)  ->  ( ( A  X.  B )  e. 
_V  ->  A  e.  _V ) )
1716con3d 125 . . . . . 6  |-  ( B  =/=  (/)  ->  ( -.  A  e.  _V  ->  -.  ( A  X.  B
)  e.  _V )
)
1817impcom 419 . . . . 5  |-  ( ( -.  A  e.  _V  /\  B  =/=  (/) )  ->  -.  ( A  X.  B
)  e.  _V )
19 fvprc 5519 . . . . 5  |-  ( -.  ( A  X.  B
)  e.  _V  ->  (  _I  `  ( A  X.  B ) )  =  (/) )
2018, 19syl 15 . . . 4  |-  ( ( -.  A  e.  _V  /\  B  =/=  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (/) )
2112, 20pm2.61dane 2524 . . 3  |-  ( -.  A  e.  _V  ->  (  _I  `  ( A  X.  B ) )  =  (/) )
221, 3, 213eqtr4a 2341 . 2  |-  ( -.  A  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  (  _I  `  ( A  X.  B
) ) )
23 xp0 5098 . . 3  |-  ( (  _I  `  A )  X.  (/) )  =  (/)
24 fvprc 5519 . . . 4  |-  ( -.  B  e.  _V  ->  (  _I  `  B )  =  (/) )
2524xpeq2d 4713 . . 3  |-  ( -.  B  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  ( (  _I 
`  A )  X.  (/) ) )
26 simpr 447 . . . . . . . 8  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  ->  A  =  (/) )
2726xpeq1d 4712 . . . . . . 7  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
( A  X.  B
)  =  ( (/)  X.  B ) )
28 xp0r 4768 . . . . . . 7  |-  ( (/)  X.  B )  =  (/)
2927, 28syl6eq 2331 . . . . . 6  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
( A  X.  B
)  =  (/) )
3029fveq2d 5529 . . . . 5  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (  _I  `  (/) ) )
3130, 11syl6eq 2331 . . . 4  |-  ( ( -.  B  e.  _V  /\  A  =  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (/) )
32 rnexg 4940 . . . . . . . 8  |-  ( ( A  X.  B )  e.  _V  ->  ran  ( A  X.  B
)  e.  _V )
33 rnxp 5106 . . . . . . . . 9  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
3433eleq1d 2349 . . . . . . . 8  |-  ( A  =/=  (/)  ->  ( ran  ( A  X.  B
)  e.  _V  <->  B  e.  _V ) )
3532, 34syl5ib 210 . . . . . . 7  |-  ( A  =/=  (/)  ->  ( ( A  X.  B )  e. 
_V  ->  B  e.  _V ) )
3635con3d 125 . . . . . 6  |-  ( A  =/=  (/)  ->  ( -.  B  e.  _V  ->  -.  ( A  X.  B
)  e.  _V )
)
3736impcom 419 . . . . 5  |-  ( ( -.  B  e.  _V  /\  A  =/=  (/) )  ->  -.  ( A  X.  B
)  e.  _V )
3837, 19syl 15 . . . 4  |-  ( ( -.  B  e.  _V  /\  A  =/=  (/) )  -> 
(  _I  `  ( A  X.  B ) )  =  (/) )
3931, 38pm2.61dane 2524 . . 3  |-  ( -.  B  e.  _V  ->  (  _I  `  ( A  X.  B ) )  =  (/) )
4023, 25, 393eqtr4a 2341 . 2  |-  ( -.  B  e.  _V  ->  ( (  _I  `  A
)  X.  (  _I 
`  B ) )  =  (  _I  `  ( A  X.  B
) ) )
41 fvi 5579 . . . 4  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
42 fvi 5579 . . . 4  |-  ( B  e.  _V  ->  (  _I  `  B )  =  B )
43 xpeq12 4708 . . . 4  |-  ( ( (  _I  `  A
)  =  A  /\  (  _I  `  B )  =  B )  -> 
( (  _I  `  A )  X.  (  _I  `  B ) )  =  ( A  X.  B ) )
4441, 42, 43syl2an 463 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  ( A  X.  B ) )
45 xpexg 4800 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  X.  B
)  e.  _V )
46 fvi 5579 . . . 4  |-  ( ( A  X.  B )  e.  _V  ->  (  _I  `  ( A  X.  B ) )  =  ( A  X.  B
) )
4745, 46syl 15 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  (  _I  `  ( A  X.  B ) )  =  ( A  X.  B ) )
4844, 47eqtr4d 2318 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B
) ) )
4922, 40, 48ecase 908 1  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455    _I cid 4304    X. cxp 4687   dom cdm 4689   ran crn 4690   ` cfv 5255
This theorem is referenced by:  txindis  17328
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-pw 3627  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-iota 5219  df-fun 5257  df-fv 5263
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