txbas - Intuitionistic Logic Explorer

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Theorem txbas 14578

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 4678	. . . . . . . . . 10
3		xpeq2 4679	. . . . . . . . . 10
4	2, 3	cbvmpov 6006	. . . . . . . . 9
5	4	rnmpo 6037	. . . . . . . 8
6	1, 5	eqtri 2217	. . . . . . 7
7	6	abeq2i 2307	. . . . . 6
8		xpeq1 4678	. . . . . . . . . 10
9		xpeq2 4679	. . . . . . . . . 10
10	8, 9	cbvmpov 6006	. . . . . . . . 9
11	10	rnmpo 6037	. . . . . . . 8
12	1, 11	eqtri 2217	. . . . . . 7
13	12	abeq2i 2307	. . . . . 6
14	7, 13	anbi12i 460	. . . . 5 $/\$ $/\$
15		reeanv 2667	. . . . 5 $/\$ $/\$
16	14, 15	bitr4i 187	. . . 4 $/\$ $/\$
17		reeanv 2667	. . . . . 6 $/\$ $/\$
18		basis2 14368	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
19	18	exp43 372	. . . . . . . . . . . . . . 15 $/\$
20	19	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
21		basis2 14368	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
22	21	exp43 372	. . . . . . . . . . . . . . 15 $/\$
23	22	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
24		reeanv 2667	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		opelxpi 4696	. . . . . . . . . . . . . . . . . . 19 $/\$
26		xpss12 4771	. . . . . . . . . . . . . . . . . . 19 $/\$
27	25, 26	anim12i 338	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
28	27	an4s 588	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
29	28	reximi 2594	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
30	29	reximi 2594	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
31	24, 30	sylbir 135	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
32	20, 23, 31	syl2an 289	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	an4s 588	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	ralrimivva 2579	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		eleq1 2259	. . . . . . . . . . . . . 14
36	35	anbi1d 465	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	2rexbidv 2522	. . . . . . . . . . . 12 $/\$ $/\$
38	37	ralxp 4810	. . . . . . . . . . 11 $/\$ $/\$
39	34, 38	sylibr 134	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an4s 588	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	anassrs 400	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		ineq12 3360	. . . . . . . . . 10 $/\$
43		inxp 4801	. . . . . . . . . 10
44	42, 43	eqtrdi 2245	. . . . . . . . 9 $/\$
45	44	sseq2d 3214	. . . . . . . . . . . 12 $/\$
46	45	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$ $/\$
47	46	rexbidv 2498	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	1	rexeqi 2698	. . . . . . . . . . 11 $/\$ $/\$
49		1stexg 6234	. . . . . . . . . . . . . . 15
50	49	elv 2767	. . . . . . . . . . . . . 14
51		2ndexg 6235	. . . . . . . . . . . . . . 15
52	51	elv 2767	. . . . . . . . . . . . . 14
53	50, 52	xpex 4779	. . . . . . . . . . . . 13
54	53	rgenw 2552	. . . . . . . . . . . 12
55		vex 2766	. . . . . . . . . . . . . . . . 17
56		vex 2766	. . . . . . . . . . . . . . . . 17
57	55, 56	op1std 6215	. . . . . . . . . . . . . . . 16
58	55, 56	op2ndd 6216	. . . . . . . . . . . . . . . 16
59	57, 58	xpeq12d 4689	. . . . . . . . . . . . . . 15
60	59	mpompt 6018	. . . . . . . . . . . . . 14
61	60	eqcomi 2200	. . . . . . . . . . . . 13
62		eleq2 2260	. . . . . . . . . . . . . 14
63		sseq1 3207	. . . . . . . . . . . . . 14
64	62, 63	anbi12d 473	. . . . . . . . . . . . 13 $/\$ $/\$
65	61, 64	rexrnmpt 5708	. . . . . . . . . . . 12 $/\$ $/\$
66	54, 65	ax-mp 5	. . . . . . . . . . 11 $/\$ $/\$
67	59	eleq2d 2266	. . . . . . . . . . . . 13
68	59	sseq1d 3213	. . . . . . . . . . . . 13
69	67, 68	anbi12d 473	. . . . . . . . . . . 12 $/\$ $/\$
70	69	rexxp 4811	. . . . . . . . . . 11 $/\$ $/\$
71	48, 66, 70	3bitri 206	. . . . . . . . . 10 $/\$ $/\$
72	47, 71	bitrdi 196	. . . . . . . . 9 $/\$ $/\$ $/\$
73	44, 72	raleqbidv 2709	. . . . . . . 8 $/\$ $/\$ $/\$
74	41, 73	syl5ibrcom 157	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	74	rexlimdvva 2622	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	17, 75	biimtrrid 153	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	rexlimdvva 2622	. . . 4 $/\$ $/\$ $/\$
78	16, 77	biimtrid 152	. . 3 $/\$ $/\$ $/\$
79	78	ralrimivv 2578	. 2 $/\$ $/\$
80	1	txbasex 14577	. . 3 $/\$
81		isbasis2g 14365	. . 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 1364

wcel 2167

cab 2182

wral 2475

wrex 2476

cvv 2763

cin 3156

wss 3157

cop 3626

cmpt 4095

cxp 4662

crn 4665

cfv 5259

cmpo 5927

c1st 6205

c2nd 6206

ctb 14362

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-fo 5265 df-fv 5267 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-bases 14363

This theorem is referenced by: txtop 14580

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