txbas - Intuitionistic Logic Explorer

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Theorem txbas 12898

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 4618	. . . . . . . . . 10
3		xpeq2 4619	. . . . . . . . . 10
4	2, 3	cbvmpov 5922	. . . . . . . . 9
5	4	rnmpo 5952	. . . . . . . 8
6	1, 5	eqtri 2186	. . . . . . 7
7	6	abeq2i 2277	. . . . . 6
8		xpeq1 4618	. . . . . . . . . 10
9		xpeq2 4619	. . . . . . . . . 10
10	8, 9	cbvmpov 5922	. . . . . . . . 9
11	10	rnmpo 5952	. . . . . . . 8
12	1, 11	eqtri 2186	. . . . . . 7
13	12	abeq2i 2277	. . . . . 6
14	7, 13	anbi12i 456	. . . . 5 $/\$ $/\$
15		reeanv 2635	. . . . 5 $/\$ $/\$
16	14, 15	bitr4i 186	. . . 4 $/\$ $/\$
17		reeanv 2635	. . . . . 6 $/\$ $/\$
18		basis2 12686	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
19	18	exp43 370	. . . . . . . . . . . . . . 15 $/\$
20	19	imp42 352	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
21		basis2 12686	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
22	21	exp43 370	. . . . . . . . . . . . . . 15 $/\$
23	22	imp42 352	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
24		reeanv 2635	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		opelxpi 4636	. . . . . . . . . . . . . . . . . . 19 $/\$
26		xpss12 4711	. . . . . . . . . . . . . . . . . . 19 $/\$
27	25, 26	anim12i 336	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
28	27	an4s 578	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
29	28	reximi 2563	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
30	29	reximi 2563	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
31	24, 30	sylbir 134	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
32	20, 23, 31	syl2an 287	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	an4s 578	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	ralrimivva 2548	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		eleq1 2229	. . . . . . . . . . . . . 14
36	35	anbi1d 461	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	2rexbidv 2491	. . . . . . . . . . . 12 $/\$ $/\$
38	37	ralxp 4747	. . . . . . . . . . 11 $/\$ $/\$
39	34, 38	sylibr 133	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an4s 578	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	anassrs 398	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		ineq12 3318	. . . . . . . . . 10 $/\$
43		inxp 4738	. . . . . . . . . 10
44	42, 43	eqtrdi 2215	. . . . . . . . 9 $/\$
45	44	sseq2d 3172	. . . . . . . . . . . 12 $/\$
46	45	anbi2d 460	. . . . . . . . . . 11 $/\$ $/\$ $/\$
47	46	rexbidv 2467	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	1	rexeqi 2666	. . . . . . . . . . 11 $/\$ $/\$
49		1stexg 6135	. . . . . . . . . . . . . . 15
50	49	elv 2730	. . . . . . . . . . . . . 14
51		2ndexg 6136	. . . . . . . . . . . . . . 15
52	51	elv 2730	. . . . . . . . . . . . . 14
53	50, 52	xpex 4719	. . . . . . . . . . . . 13
54	53	rgenw 2521	. . . . . . . . . . . 12
55		vex 2729	. . . . . . . . . . . . . . . . 17
56		vex 2729	. . . . . . . . . . . . . . . . 17
57	55, 56	op1std 6116	. . . . . . . . . . . . . . . 16
58	55, 56	op2ndd 6117	. . . . . . . . . . . . . . . 16
59	57, 58	xpeq12d 4629	. . . . . . . . . . . . . . 15
60	59	mpompt 5934	. . . . . . . . . . . . . 14
61	60	eqcomi 2169	. . . . . . . . . . . . 13
62		eleq2 2230	. . . . . . . . . . . . . 14
63		sseq1 3165	. . . . . . . . . . . . . 14
64	62, 63	anbi12d 465	. . . . . . . . . . . . 13 $/\$ $/\$
65	61, 64	rexrnmpt 5628	. . . . . . . . . . . 12 $/\$ $/\$
66	54, 65	ax-mp 5	. . . . . . . . . . 11 $/\$ $/\$
67	59	eleq2d 2236	. . . . . . . . . . . . 13
68	59	sseq1d 3171	. . . . . . . . . . . . 13
69	67, 68	anbi12d 465	. . . . . . . . . . . 12 $/\$ $/\$
70	69	rexxp 4748	. . . . . . . . . . 11 $/\$ $/\$
71	48, 66, 70	3bitri 205	. . . . . . . . . 10 $/\$ $/\$
72	47, 71	bitrdi 195	. . . . . . . . 9 $/\$ $/\$ $/\$
73	44, 72	raleqbidv 2673	. . . . . . . 8 $/\$ $/\$ $/\$
74	41, 73	syl5ibrcom 156	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	74	rexlimdvva 2591	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	17, 75	syl5bir 152	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	rexlimdvva 2591	. . . 4 $/\$ $/\$ $/\$
78	16, 77	syl5bi 151	. . 3 $/\$ $/\$ $/\$
79	78	ralrimivv 2547	. 2 $/\$ $/\$
80	1	txbasex 12897	. . 3 $/\$
81		isbasis2g 12683	. . 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 1343

wcel 2136

cab 2151

wral 2444

wrex 2445

cvv 2726

cin 3115

wss 3116

cop 3579

cmpt 4043

cxp 4602

crn 4605

cfv 5188

cmpo 5844

c1st 6106

c2nd 6107

ctb 12680

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-fo 5194 df-fv 5196 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-bases 12681

This theorem is referenced by: txtop 12900

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