txbas - Intuitionistic Logic Explorer

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Theorem txbas 13761

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 4641	. . . . . . . . . 10
3		xpeq2 4642	. . . . . . . . . 10
4	2, 3	cbvmpov 5955	. . . . . . . . 9
5	4	rnmpo 5985	. . . . . . . 8
6	1, 5	eqtri 2198	. . . . . . 7
7	6	abeq2i 2288	. . . . . 6
8		xpeq1 4641	. . . . . . . . . 10
9		xpeq2 4642	. . . . . . . . . 10
10	8, 9	cbvmpov 5955	. . . . . . . . 9
11	10	rnmpo 5985	. . . . . . . 8
12	1, 11	eqtri 2198	. . . . . . 7
13	12	abeq2i 2288	. . . . . 6
14	7, 13	anbi12i 460	. . . . 5 $/\$ $/\$
15		reeanv 2647	. . . . 5 $/\$ $/\$
16	14, 15	bitr4i 187	. . . 4 $/\$ $/\$
17		reeanv 2647	. . . . . 6 $/\$ $/\$
18		basis2 13551	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
19	18	exp43 372	. . . . . . . . . . . . . . 15 $/\$
20	19	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
21		basis2 13551	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
22	21	exp43 372	. . . . . . . . . . . . . . 15 $/\$
23	22	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
24		reeanv 2647	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		opelxpi 4659	. . . . . . . . . . . . . . . . . . 19 $/\$
26		xpss12 4734	. . . . . . . . . . . . . . . . . . 19 $/\$
27	25, 26	anim12i 338	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
28	27	an4s 588	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
29	28	reximi 2574	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
30	29	reximi 2574	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
31	24, 30	sylbir 135	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
32	20, 23, 31	syl2an 289	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	an4s 588	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	ralrimivva 2559	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		eleq1 2240	. . . . . . . . . . . . . 14
36	35	anbi1d 465	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	2rexbidv 2502	. . . . . . . . . . . 12 $/\$ $/\$
38	37	ralxp 4771	. . . . . . . . . . 11 $/\$ $/\$
39	34, 38	sylibr 134	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an4s 588	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	anassrs 400	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		ineq12 3332	. . . . . . . . . 10 $/\$
43		inxp 4762	. . . . . . . . . 10
44	42, 43	eqtrdi 2226	. . . . . . . . 9 $/\$
45	44	sseq2d 3186	. . . . . . . . . . . 12 $/\$
46	45	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$ $/\$
47	46	rexbidv 2478	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	1	rexeqi 2678	. . . . . . . . . . 11 $/\$ $/\$
49		1stexg 6168	. . . . . . . . . . . . . . 15
50	49	elv 2742	. . . . . . . . . . . . . 14
51		2ndexg 6169	. . . . . . . . . . . . . . 15
52	51	elv 2742	. . . . . . . . . . . . . 14
53	50, 52	xpex 4742	. . . . . . . . . . . . 13
54	53	rgenw 2532	. . . . . . . . . . . 12
55		vex 2741	. . . . . . . . . . . . . . . . 17
56		vex 2741	. . . . . . . . . . . . . . . . 17
57	55, 56	op1std 6149	. . . . . . . . . . . . . . . 16
58	55, 56	op2ndd 6150	. . . . . . . . . . . . . . . 16
59	57, 58	xpeq12d 4652	. . . . . . . . . . . . . . 15
60	59	mpompt 5967	. . . . . . . . . . . . . 14
61	60	eqcomi 2181	. . . . . . . . . . . . 13
62		eleq2 2241	. . . . . . . . . . . . . 14
63		sseq1 3179	. . . . . . . . . . . . . 14
64	62, 63	anbi12d 473	. . . . . . . . . . . . 13 $/\$ $/\$
65	61, 64	rexrnmpt 5660	. . . . . . . . . . . 12 $/\$ $/\$
66	54, 65	ax-mp 5	. . . . . . . . . . 11 $/\$ $/\$
67	59	eleq2d 2247	. . . . . . . . . . . . 13
68	59	sseq1d 3185	. . . . . . . . . . . . 13
69	67, 68	anbi12d 473	. . . . . . . . . . . 12 $/\$ $/\$
70	69	rexxp 4772	. . . . . . . . . . 11 $/\$ $/\$
71	48, 66, 70	3bitri 206	. . . . . . . . . 10 $/\$ $/\$
72	47, 71	bitrdi 196	. . . . . . . . 9 $/\$ $/\$ $/\$
73	44, 72	raleqbidv 2685	. . . . . . . 8 $/\$ $/\$ $/\$
74	41, 73	syl5ibrcom 157	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	74	rexlimdvva 2602	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	17, 75	biimtrrid 153	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	rexlimdvva 2602	. . . 4 $/\$ $/\$ $/\$
78	16, 77	biimtrid 152	. . 3 $/\$ $/\$ $/\$
79	78	ralrimivv 2558	. 2 $/\$ $/\$
80	1	txbasex 13760	. . 3 $/\$
81		isbasis2g 13548	. . 3 $/\$
82	80, 81	syl 14	. 2 $/\$ $/\$
83	79, 82	mpbird 167	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148

cab 2163

wral 2455

wrex 2456

cvv 2738

cin 3129

wss 3130

cop 3596

cmpt 4065

cxp 4625

crn 4628

cfv 5217

cmpo 5877

c1st 6139

c2nd 6140

ctb 13545

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-fo 5223 df-fv 5225 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-bases 13546

This theorem is referenced by: txtop 13763

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