Theorem txindis 21437

Description: The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
txindis	|- ( { (/) , A } tX { (/) , B } ) = { (/) , ( A X. B ) }

Proof of Theorem txindis

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 3930	. . . . . . 7 |- ( -. x = (/) <-> E. y y e. x )
2		indistop 20806	. . . . . . . . . . 11 |- { (/) , A } e. Top
3		indistop 20806	. . . . . . . . . . 11 |- { (/) , B } e. Top
4		eltx 21371	. . . . . . . . . . 11 |- ( ( { (/) , A } e. Top /\ { (/) , B } e. Top ) -> ( x e. ( { (/) , A } tX { (/) , B } ) <-> A. y e. x E. z e. { (/) , A } E. w e. { (/) , B } ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) )
5	2, 3, 4	mp2an 708	. . . . . . . . . 10 |- ( x e. ( { (/) , A } tX { (/) , B } ) <-> A. y e. x E. z e. { (/) , A } E. w e. { (/) , B } ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) )
6		rsp 2929	. . . . . . . . . 10 |- ( A. y e. x E. z e. { (/) , A } E. w e. { (/) , B } ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) -> ( y e. x -> E. z e. { (/) , A } E. w e. { (/) , B } ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) )
7	5, 6	sylbi 207	. . . . . . . . 9 |- ( x e. ( { (/) , A } tX { (/) , B } ) -> ( y e. x -> E. z e. { (/) , A } E. w e. { (/) , B } ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) )
8		elssuni 4467	. . . . . . . . . . . . . 14 |- ( x e. ( { (/) , A } tX { (/) , B } ) -> x C_ U. ( { (/) , A } tX { (/) , B } ) )
9		indisuni 20807	. . . . . . . . . . . . . . 15 |- ( _I ` A ) = U. { (/) , A }
10		indisuni 20807	. . . . . . . . . . . . . . 15 |- ( _I ` B ) = U. { (/) , B }
11	2, 3, 9, 10	txunii 21396	. . . . . . . . . . . . . 14 |- ( ( _I ` A ) X. ( _I ` B ) ) = U. ( { (/) , A } tX { (/) , B } )
12	8, 11	syl6sseqr 3652	. . . . . . . . . . . . 13 |- ( x e. ( { (/) , A } tX { (/) , B } ) -> x C_ ( ( _I ` A ) X. ( _I ` B ) ) )
13	12	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( x e. ( { (/) , A } tX { (/) , B } ) /\ ( z e. { (/) , A } /\ w e. { (/) , B } ) ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> x C_ ( ( _I ` A ) X. ( _I ` B ) ) )
14		ne0i 3921	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. ( z X. w ) -> ( z X. w ) =/= (/) )
15	14	ad2antrl 764	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> ( z X. w ) =/= (/) )
16		xpnz 5553	. . . . . . . . . . . . . . . . . . 19 |- ( ( z =/= (/) /\ w =/= (/) ) <-> ( z X. w ) =/= (/) )
17	15, 16	sylibr 224	. . . . . . . . . . . . . . . . . 18 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> ( z =/= (/) /\ w =/= (/) ) )
18	17	simpld 475	. . . . . . . . . . . . . . . . 17 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> z =/= (/) )
19	18	neneqd 2799	. . . . . . . . . . . . . . . 16 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> -. z = (/) )
20		simpll 790	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> z e. { (/) , A } )
21		indislem 20804	. . . . . . . . . . . . . . . . . . 19 |- { (/) , ( _I ` A ) } = { (/) , A }
22	20, 21	syl6eleqr 2712	. . . . . . . . . . . . . . . . . 18 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> z e. { (/) , ( _I ` A ) } )
23		elpri 4197	. . . . . . . . . . . . . . . . . 18 |- ( z e. { (/) , ( _I ` A ) } -> ( z = (/) \/ z = ( _I ` A ) ) )
24	22, 23	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> ( z = (/) \/ z = ( _I ` A ) ) )
25	24	ord 392	. . . . . . . . . . . . . . . 16 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> ( -. z = (/) -> z = ( _I ` A ) ) )
26	19, 25	mpd 15	. . . . . . . . . . . . . . 15 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> z = ( _I ` A ) )
27	17	simprd 479	. . . . . . . . . . . . . . . . 17 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> w =/= (/) )
28	27	neneqd 2799	. . . . . . . . . . . . . . . 16 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> -. w = (/) )
29		simplr 792	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> w e. { (/) , B } )
30		indislem 20804	. . . . . . . . . . . . . . . . . . 19 |- { (/) , ( _I ` B ) } = { (/) , B }
31	29, 30	syl6eleqr 2712	. . . . . . . . . . . . . . . . . 18 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> w e. { (/) , ( _I ` B ) } )
32		elpri 4197	. . . . . . . . . . . . . . . . . 18 |- ( w e. { (/) , ( _I ` B ) } -> ( w = (/) \/ w = ( _I ` B ) ) )
33	31, 32	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> ( w = (/) \/ w = ( _I ` B ) ) )
34	33	ord 392	. . . . . . . . . . . . . . . 16 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> ( -. w = (/) -> w = ( _I ` B ) ) )
35	28, 34	mpd 15	. . . . . . . . . . . . . . 15 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> w = ( _I ` B ) )
36	26, 35	xpeq12d 5140	. . . . . . . . . . . . . 14 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> ( z X. w ) = ( ( _I ` A ) X. ( _I ` B ) ) )
37		simprr 796	. . . . . . . . . . . . . 14 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> ( z X. w ) C_ x )
38	36, 37	eqsstr3d 3640	. . . . . . . . . . . . 13 |- ( ( ( z e. { (/) , A } /\ w e. { (/) , B } ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> ( ( _I ` A ) X. ( _I ` B ) ) C_ x )
39	38	adantll 750	. . . . . . . . . . . 12 |- ( ( ( x e. ( { (/) , A } tX { (/) , B } ) /\ ( z e. { (/) , A } /\ w e. { (/) , B } ) ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> ( ( _I ` A ) X. ( _I ` B ) ) C_ x )
40	13, 39	eqssd 3620	. . . . . . . . . . 11 |- ( ( ( x e. ( { (/) , A } tX { (/) , B } ) /\ ( z e. { (/) , A } /\ w e. { (/) , B } ) ) /\ ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) -> x = ( ( _I ` A ) X. ( _I ` B ) ) )
41	40	ex 450	. . . . . . . . . 10 |- ( ( x e. ( { (/) , A } tX { (/) , B } ) /\ ( z e. { (/) , A } /\ w e. { (/) , B } ) ) -> ( ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) -> x = ( ( _I ` A ) X. ( _I ` B ) ) ) )
42	41	rexlimdvva 3038	. . . . . . . . 9 |- ( x e. ( { (/) , A } tX { (/) , B } ) -> ( E. z e. { (/) , A } E. w e. { (/) , B } ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) -> x = ( ( _I ` A ) X. ( _I ` B ) ) ) )
43	7, 42	syld 47	. . . . . . . 8 |- ( x e. ( { (/) , A } tX { (/) , B } ) -> ( y e. x -> x = ( ( _I ` A ) X. ( _I ` B ) ) ) )
44	43	exlimdv 1861	. . . . . . 7 |- ( x e. ( { (/) , A } tX { (/) , B } ) -> ( E. y y e. x -> x = ( ( _I ` A ) X. ( _I ` B ) ) ) )
45	1, 44	syl5bi 232	. . . . . 6 |- ( x e. ( { (/) , A } tX { (/) , B } ) -> ( -. x = (/) -> x = ( ( _I ` A ) X. ( _I ` B ) ) ) )
46	45	orrd 393	. . . . 5 |- ( x e. ( { (/) , A } tX { (/) , B } ) -> ( x = (/) \/ x = ( ( _I ` A ) X. ( _I ` B ) ) ) )
47		vex 3203	. . . . . 6 |- x e. _V
48	47	elpr 4198	. . . . 5 |- ( x e. { (/) , ( ( _I ` A ) X. ( _I ` B ) ) } <-> ( x = (/) \/ x = ( ( _I ` A ) X. ( _I ` B ) ) ) )
49	46, 48	sylibr 224	. . . 4 |- ( x e. ( { (/) , A } tX { (/) , B } ) -> x e. { (/) , ( ( _I ` A ) X. ( _I ` B ) ) } )
50	49	ssriv 3607	. . 3 |- ( { (/) , A } tX { (/) , B } ) C_ { (/) , ( ( _I ` A ) X. ( _I ` B ) ) }
51	9	toptopon 20722	. . . . . . 7 |- ( { (/) , A } e. Top <-> { (/) , A } e. (TopOn ` ( _I ` A ) ) )
52	2, 51	mpbi 220	. . . . . 6 |- { (/) , A } e. (TopOn ` ( _I ` A ) )
53	10	toptopon 20722	. . . . . . 7 |- ( { (/) , B } e. Top <-> { (/) , B } e. (TopOn ` ( _I ` B ) ) )
54	3, 53	mpbi 220	. . . . . 6 |- { (/) , B } e. (TopOn ` ( _I ` B ) )
55		txtopon 21394	. . . . . 6 |- ( ( { (/) , A } e. (TopOn ` ( _I ` A ) ) /\ { (/) , B } e. (TopOn ` ( _I ` B ) ) ) -> ( { (/) , A } tX { (/) , B } ) e. (TopOn ` ( ( _I ` A ) X. ( _I ` B ) ) ) )
56	52, 54, 55	mp2an 708	. . . . 5 |- ( { (/) , A } tX { (/) , B } ) e. (TopOn ` ( ( _I ` A ) X. ( _I ` B ) ) )
57		topgele 20734	. . . . 5 |- ( ( { (/) , A } tX { (/) , B } ) e. (TopOn ` ( ( _I ` A ) X. ( _I ` B ) ) ) -> ( { (/) , ( ( _I ` A ) X. ( _I ` B ) ) } C_ ( { (/) , A } tX { (/) , B } ) /\ ( { (/) , A } tX { (/) , B } ) C_ ~P ( ( _I ` A ) X. ( _I ` B ) ) ) )
58	56, 57	ax-mp 5	. . . 4 |- ( { (/) , ( ( _I ` A ) X. ( _I ` B ) ) } C_ ( { (/) , A } tX { (/) , B } ) /\ ( { (/) , A } tX { (/) , B } ) C_ ~P ( ( _I ` A ) X. ( _I ` B ) ) )
59	58	simpli 474	. . 3 |- { (/) , ( ( _I ` A ) X. ( _I ` B ) ) } C_ ( { (/) , A } tX { (/) , B } )
60	50, 59	eqssi 3619	. 2 |- ( { (/) , A } tX { (/) , B } ) = { (/) , ( ( _I ` A ) X. ( _I ` B ) ) }
61		txindislem 21436	. . 3 |- ( ( _I ` A ) X. ( _I ` B ) ) = ( _I ` ( A X. B ) )
62	61	preq2i 4272	. 2 |- { (/) , ( ( _I ` A ) X. ( _I ` B ) ) } = { (/) , ( _I ` ( A X. B ) ) }
63		indislem 20804	. 2 |- { (/) , ( _I ` ( A X. B ) ) } = { (/) , ( A X. B ) }
64	60, 62, 63	3eqtri 2648	1 |- ( { (/) , A } tX { (/) , B } ) = { (/) , ( A X. B ) }

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {cpr 4179   U.cuni 4436   _I cid 5023   X. cxp 5112   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   tX ctx 21363

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-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-tx 21365

This theorem is referenced by: (None)