txbas - Intuitionistic Logic Explorer

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Theorem txbas 12799

Description: The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.)

**Hypothesis**
Ref	Expression
txval.1

**Assertion**
Ref	Expression
txbas	$/\$

Distinct variable groups:

Allowed substitution hints:

(

)

Proof of Theorem txbas Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txval.1	. . . . . . . 8
2		xpeq1 4612	. . . . . . . . . 10
3		xpeq2 4613	. . . . . . . . . 10
4	2, 3	cbvmpov 5913	. . . . . . . . 9
5	4	rnmpo 5943	. . . . . . . 8
6	1, 5	eqtri 2185	. . . . . . 7
7	6	abeq2i 2275	. . . . . 6
8		xpeq1 4612	. . . . . . . . . 10
9		xpeq2 4613	. . . . . . . . . 10
10	8, 9	cbvmpov 5913	. . . . . . . . 9
11	10	rnmpo 5943	. . . . . . . 8
12	1, 11	eqtri 2185	. . . . . . 7
13	12	abeq2i 2275	. . . . . 6
14	7, 13	anbi12i 456	. . . . 5 $/\$ $/\$
15		reeanv 2633	. . . . 5 $/\$ $/\$
16	14, 15	bitr4i 186	. . . 4 $/\$ $/\$
17		reeanv 2633	. . . . . 6 $/\$ $/\$
18		basis2 12587	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
19	18	exp43 370	. . . . . . . . . . . . . . 15 $/\$
20	19	imp42 352	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
21		basis2 12587	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
22	21	exp43 370	. . . . . . . . . . . . . . 15 $/\$
23	22	imp42 352	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
24		reeanv 2633	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		opelxpi 4630	. . . . . . . . . . . . . . . . . . 19 $/\$
26		xpss12 4705	. . . . . . . . . . . . . . . . . . 19 $/\$
27	25, 26	anim12i 336	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
28	27	an4s 578	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
29	28	reximi 2561	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
30	29	reximi 2561	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
31	24, 30	sylbir 134	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
32	20, 23, 31	syl2an 287	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	an4s 578	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	ralrimivva 2546	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		eleq1 2227	. . . . . . . . . . . . . 14
36	35	anbi1d 461	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	2rexbidv 2489	. . . . . . . . . . . 12 $/\$ $/\$
38	37	ralxp 4741	. . . . . . . . . . 11 $/\$ $/\$
39	34, 38	sylibr 133	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an4s 578	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	anassrs 398	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		ineq12 3313	. . . . . . . . . 10 $/\$
43		inxp 4732	. . . . . . . . . 10
44	42, 43	eqtrdi 2213	. . . . . . . . 9 $/\$
45	44	sseq2d 3167	. . . . . . . . . . . 12 $/\$
46	45	anbi2d 460	. . . . . . . . . . 11 $/\$ $/\$ $/\$
47	46	rexbidv 2465	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	1	rexeqi 2664	. . . . . . . . . . 11 $/\$ $/\$
49		1stexg 6127	. . . . . . . . . . . . . . 15
50	49	elv 2725	. . . . . . . . . . . . . 14
51		2ndexg 6128	. . . . . . . . . . . . . . 15
52	51	elv 2725	. . . . . . . . . . . . . 14
53	50, 52	xpex 4713	. . . . . . . . . . . . 13
54	53	rgenw 2519	. . . . . . . . . . . 12
55		vex 2724	. . . . . . . . . . . . . . . . 17
56		vex 2724	. . . . . . . . . . . . . . . . 17
57	55, 56	op1std 6108	. . . . . . . . . . . . . . . 16
58	55, 56	op2ndd 6109	. . . . . . . . . . . . . . . 16
59	57, 58	xpeq12d 4623	. . . . . . . . . . . . . . 15
60	59	mpompt 5925	. . . . . . . . . . . . . 14
61	60	eqcomi 2168	. . . . . . . . . . . . 13
62		eleq2 2228	. . . . . . . . . . . . . 14
63		sseq1 3160	. . . . . . . . . . . . . 14
64	62, 63	anbi12d 465	. . . . . . . . . . . . 13 $/\$ $/\$
65	61, 64	rexrnmpt 5622	. . . . . . . . . . . 12 $/\$ $/\$
66	54, 65	ax-mp 5	. . . . . . . . . . 11 $/\$ $/\$
67	59	eleq2d 2234	. . . . . . . . . . . . 13
68	59	sseq1d 3166	. . . . . . . . . . . . 13
69	67, 68	anbi12d 465	. . . . . . . . . . . 12 $/\$ $/\$
70	69	rexxp 4742	. . . . . . . . . . 11 $/\$ $/\$
71	48, 66, 70	3bitri 205	. . . . . . . . . 10 $/\$ $/\$
72	47, 71	bitrdi 195	. . . . . . . . 9 $/\$ $/\$ $/\$
73	44, 72	raleqbidv 2671	. . . . . . . 8 $/\$ $/\$ $/\$
74	41, 73	syl5ibrcom 156	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	74	rexlimdvva 2589	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	17, 75	syl5bir 152	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	rexlimdvva 2589	. . . 4 $/\$ $/\$ $/\$
78	16, 77	syl5bi 151	. . 3 $/\$ $/\$ $/\$
79	78	ralrimivv 2545	. 2 $/\$ $/\$
80	1	txbasex 12798	. . 3 $/\$
81		isbasis2g 12584	. . 3 $/\$
82	80, 81	syl 14	. 2 $/\$ $/\$
83	79, 82	mpbird 166	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1342

wcel 2135

cab 2150

wral 2442

wrex 2443

cvv 2721

cin 3110

wss 3111

cop 3573

cmpt 4037

cxp 4596

crn 4599

cfv 5182

cmpo 5838

c1st 6098

c2nd 6099

ctb 12581

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405

This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 df-fv 5190 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-bases 12582

This theorem is referenced by: txtop 12801

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