txbas - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > txbas		Unicode version

Theorem txbas 15123

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 4763	. . . . . . . . . 10
3		xpeq2 4764	. . . . . . . . . 10
4	2, 3	cbvmpov 6133	. . . . . . . . 9
5	4	rnmpo 6164	. . . . . . . 8
6	1, 5	eqtri 2253	. . . . . . 7
7	6	abeq2i 2343	. . . . . 6
8		xpeq1 4763	. . . . . . . . . 10
9		xpeq2 4764	. . . . . . . . . 10
10	8, 9	cbvmpov 6133	. . . . . . . . 9
11	10	rnmpo 6164	. . . . . . . 8
12	1, 11	eqtri 2253	. . . . . . 7
13	12	abeq2i 2343	. . . . . 6
14	7, 13	anbi12i 460	. . . . 5 $/\$ $/\$
15		reeanv 2713	. . . . 5 $/\$ $/\$
16	14, 15	bitr4i 187	. . . 4 $/\$ $/\$
17		reeanv 2713	. . . . . 6 $/\$ $/\$
18		basis2 14913	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
19	18	exp43 372	. . . . . . . . . . . . . . 15 $/\$
20	19	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
21		basis2 14913	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
22	21	exp43 372	. . . . . . . . . . . . . . 15 $/\$
23	22	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
24		reeanv 2713	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		opelxpi 4781	. . . . . . . . . . . . . . . . . . 19 $/\$
26		xpss12 4857	. . . . . . . . . . . . . . . . . . 19 $/\$
27	25, 26	anim12i 338	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
28	27	an4s 592	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
29	28	reximi 2639	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
30	29	reximi 2639	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
31	24, 30	sylbir 135	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
32	20, 23, 31	syl2an 289	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	an4s 592	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	ralrimivva 2624	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		eleq1 2295	. . . . . . . . . . . . . 14
36	35	anbi1d 465	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	2rexbidv 2567	. . . . . . . . . . . 12 $/\$ $/\$
38	37	ralxp 4898	. . . . . . . . . . 11 $/\$ $/\$
39	34, 38	sylibr 134	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an4s 592	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	anassrs 400	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		ineq12 3417	. . . . . . . . . 10 $/\$
43		inxp 4889	. . . . . . . . . 10
44	42, 43	eqtrdi 2281	. . . . . . . . 9 $/\$
45	44	sseq2d 3268	. . . . . . . . . . . 12 $/\$
46	45	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$ $/\$
47	46	rexbidv 2543	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	1	rexeqi 2746	. . . . . . . . . . 11 $/\$ $/\$
49		1stexg 6361	. . . . . . . . . . . . . . 15
50	49	elv 2817	. . . . . . . . . . . . . 14
51		2ndexg 6362	. . . . . . . . . . . . . . 15
52	51	elv 2817	. . . . . . . . . . . . . 14
53	50, 52	xpex 4866	. . . . . . . . . . . . 13
54	53	rgenw 2597	. . . . . . . . . . . 12
55		vex 2816	. . . . . . . . . . . . . . . . 17
56		vex 2816	. . . . . . . . . . . . . . . . 17
57	55, 56	op1std 6342	. . . . . . . . . . . . . . . 16
58	55, 56	op2ndd 6343	. . . . . . . . . . . . . . . 16
59	57, 58	xpeq12d 4774	. . . . . . . . . . . . . . 15
60	59	mpompt 6145	. . . . . . . . . . . . . 14
61	60	eqcomi 2236	. . . . . . . . . . . . 13
62		eleq2 2296	. . . . . . . . . . . . . 14
63		sseq1 3261	. . . . . . . . . . . . . 14
64	62, 63	anbi12d 473	. . . . . . . . . . . . 13 $/\$ $/\$
65	61, 64	rexrnmpt 5820	. . . . . . . . . . . 12 $/\$ $/\$
66	54, 65	ax-mp 5	. . . . . . . . . . 11 $/\$ $/\$
67	59	eleq2d 2302	. . . . . . . . . . . . 13
68	59	sseq1d 3267	. . . . . . . . . . . . 13
69	67, 68	anbi12d 473	. . . . . . . . . . . 12 $/\$ $/\$
70	69	rexxp 4899	. . . . . . . . . . 11 $/\$ $/\$
71	48, 66, 70	3bitri 206	. . . . . . . . . 10 $/\$ $/\$
72	47, 71	bitrdi 196	. . . . . . . . 9 $/\$ $/\$ $/\$
73	44, 72	raleqbidv 2757	. . . . . . . 8 $/\$ $/\$ $/\$
74	41, 73	syl5ibrcom 157	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	74	rexlimdvva 2668	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	17, 75	biimtrrid 153	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	rexlimdvva 2668	. . . 4 $/\$ $/\$ $/\$
78	16, 77	biimtrid 152	. . 3 $/\$ $/\$ $/\$
79	78	ralrimivv 2623	. 2 $/\$ $/\$
80	1	txbasex 15122	. . 3 $/\$
81		isbasis2g 14910	. . 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 1398

wcel 2203

cab 2218

wral 2520

wrex 2521

cvv 2813

cin 3210

wss 3211

cop 3692

cmpt 4171

cxp 4747

crn 4750

cfv 5352

cmpo 6052

c1st 6332

c2nd 6333

ctb 14907

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-fo 5358 df-fv 5360 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-bases 14908

This theorem is referenced by: txtop 15125

W3C validator