Theorem txindislem 21436

Description: Lemma for txindis 21437. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
txindislem	|- ( ( _I ` A ) X. ( _I ` B ) ) = ( _I ` ( A X. B ) )

Proof of Theorem txindislem

Step	Hyp	Ref	Expression
1		0xp 5199	. . 3 |- ( (/) X. ( _I ` B ) ) = (/)
2		fvprc 6185	. . . 4 |- ( -. A e. _V -> ( _I ` A ) = (/) )
3	2	xpeq1d 5138	. . 3 |- ( -. A e. _V -> ( ( _I ` A ) X. ( _I ` B ) ) = ( (/) X. ( _I ` B ) ) )
4		simpr 477	. . . . . . . 8 |- ( ( -. A e. _V /\ B = (/) ) -> B = (/) )
5	4	xpeq2d 5139	. . . . . . 7 |- ( ( -. A e. _V /\ B = (/) ) -> ( A X. B ) = ( A X. (/) ) )
6		xp0 5552	. . . . . . 7 |- ( A X. (/) ) = (/)
7	5, 6	syl6eq 2672	. . . . . 6 |- ( ( -. A e. _V /\ B = (/) ) -> ( A X. B ) = (/) )
8	7	fveq2d 6195	. . . . 5 |- ( ( -. A e. _V /\ B = (/) ) -> ( _I ` ( A X. B ) ) = ( _I ` (/) ) )
9		0ex 4790	. . . . . 6 |- (/) e. _V
10		fvi 6255	. . . . . 6 |- ( (/) e. _V -> ( _I ` (/) ) = (/) )
11	9, 10	ax-mp 5	. . . . 5 |- ( _I ` (/) ) = (/)
12	8, 11	syl6eq 2672	. . . 4 |- ( ( -. A e. _V /\ B = (/) ) -> ( _I ` ( A X. B ) ) = (/) )
13		dmexg 7097	. . . . . . . 8 |- ( ( A X. B ) e. _V -> dom ( A X. B ) e. _V )
14		dmxp 5344	. . . . . . . . 9 |- ( B =/= (/) -> dom ( A X. B ) = A )
15	14	eleq1d 2686	. . . . . . . 8 |- ( B =/= (/) -> ( dom ( A X. B ) e. _V <-> A e. _V ) )
16	13, 15	syl5ib 234	. . . . . . 7 |- ( B =/= (/) -> ( ( A X. B ) e. _V -> A e. _V ) )
17	16	con3d 148	. . . . . 6 |- ( B =/= (/) -> ( -. A e. _V -> -. ( A X. B ) e. _V ) )
18	17	impcom 446	. . . . 5 |- ( ( -. A e. _V /\ B =/= (/) ) -> -. ( A X. B ) e. _V )
19		fvprc 6185	. . . . 5 |- ( -. ( A X. B ) e. _V -> ( _I ` ( A X. B ) ) = (/) )
20	18, 19	syl 17	. . . 4 |- ( ( -. A e. _V /\ B =/= (/) ) -> ( _I ` ( A X. B ) ) = (/) )
21	12, 20	pm2.61dane 2881	. . 3 |- ( -. A e. _V -> ( _I ` ( A X. B ) ) = (/) )
22	1, 3, 21	3eqtr4a 2682	. 2 |- ( -. A e. _V -> ( ( _I ` A ) X. ( _I ` B ) ) = ( _I ` ( A X. B ) ) )
23		xp0 5552	. . 3 |- ( ( _I ` A ) X. (/) ) = (/)
24		fvprc 6185	. . . 4 |- ( -. B e. _V -> ( _I ` B ) = (/) )
25	24	xpeq2d 5139	. . 3 |- ( -. B e. _V -> ( ( _I ` A ) X. ( _I ` B ) ) = ( ( _I ` A ) X. (/) ) )
26		simpr 477	. . . . . . . 8 |- ( ( -. B e. _V /\ A = (/) ) -> A = (/) )
27	26	xpeq1d 5138	. . . . . . 7 |- ( ( -. B e. _V /\ A = (/) ) -> ( A X. B ) = ( (/) X. B ) )
28		0xp 5199	. . . . . . 7 |- ( (/) X. B ) = (/)
29	27, 28	syl6eq 2672	. . . . . 6 |- ( ( -. B e. _V /\ A = (/) ) -> ( A X. B ) = (/) )
30	29	fveq2d 6195	. . . . 5 |- ( ( -. B e. _V /\ A = (/) ) -> ( _I ` ( A X. B ) ) = ( _I ` (/) ) )
31	30, 11	syl6eq 2672	. . . 4 |- ( ( -. B e. _V /\ A = (/) ) -> ( _I ` ( A X. B ) ) = (/) )
32		rnexg 7098	. . . . . . . 8 |- ( ( A X. B ) e. _V -> ran ( A X. B ) e. _V )
33		rnxp 5564	. . . . . . . . 9 |- ( A =/= (/) -> ran ( A X. B ) = B )
34	33	eleq1d 2686	. . . . . . . 8 |- ( A =/= (/) -> ( ran ( A X. B ) e. _V <-> B e. _V ) )
35	32, 34	syl5ib 234	. . . . . . 7 |- ( A =/= (/) -> ( ( A X. B ) e. _V -> B e. _V ) )
36	35	con3d 148	. . . . . 6 |- ( A =/= (/) -> ( -. B e. _V -> -. ( A X. B ) e. _V ) )
37	36	impcom 446	. . . . 5 |- ( ( -. B e. _V /\ A =/= (/) ) -> -. ( A X. B ) e. _V )
38	37, 19	syl 17	. . . 4 |- ( ( -. B e. _V /\ A =/= (/) ) -> ( _I ` ( A X. B ) ) = (/) )
39	31, 38	pm2.61dane 2881	. . 3 |- ( -. B e. _V -> ( _I ` ( A X. B ) ) = (/) )
40	23, 25, 39	3eqtr4a 2682	. 2 |- ( -. B e. _V -> ( ( _I ` A ) X. ( _I ` B ) ) = ( _I ` ( A X. B ) ) )
41		fvi 6255	. . . 4 |- ( A e. _V -> ( _I ` A ) = A )
42		fvi 6255	. . . 4 |- ( B e. _V -> ( _I ` B ) = B )
43		xpeq12 5134	. . . 4 |- ( ( ( _I ` A ) = A /\ ( _I ` B ) = B ) -> ( ( _I ` A ) X. ( _I ` B ) ) = ( A X. B ) )
44	41, 42, 43	syl2an 494	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( ( _I ` A ) X. ( _I ` B ) ) = ( A X. B ) )
45		xpexg 6960	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( A X. B ) e. _V )
46		fvi 6255	. . . 4 |- ( ( A X. B ) e. _V -> ( _I ` ( A X. B ) ) = ( A X. B ) )
47	45, 46	syl 17	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( _I ` ( A X. B ) ) = ( A X. B ) )
48	44, 47	eqtr4d 2659	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( ( _I ` A ) X. ( _I ` B ) ) = ( _I ` ( A X. B ) ) )
49	22, 40, 48	ecase 983	1 |- ( ( _I ` A ) X. ( _I ` B ) ) = ( _I ` ( A X. B ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   _I cid 5023   X. cxp 5112   dom cdm 5114   ran crn 5115   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  txindis  21437