txbas - Intuitionistic Logic Explorer

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Theorem txbas 13425

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 4637	. . . . . . . . . 10
3		xpeq2 4638	. . . . . . . . . 10
4	2, 3	cbvmpov 5949	. . . . . . . . 9
5	4	rnmpo 5979	. . . . . . . 8
6	1, 5	eqtri 2198	. . . . . . 7
7	6	abeq2i 2288	. . . . . 6
8		xpeq1 4637	. . . . . . . . . 10
9		xpeq2 4638	. . . . . . . . . 10
10	8, 9	cbvmpov 5949	. . . . . . . . 9
11	10	rnmpo 5979	. . . . . . . 8
12	1, 11	eqtri 2198	. . . . . . 7
13	12	abeq2i 2288	. . . . . 6
14	7, 13	anbi12i 460	. . . . 5 $/\$ $/\$
15		reeanv 2646	. . . . 5 $/\$ $/\$
16	14, 15	bitr4i 187	. . . 4 $/\$ $/\$
17		reeanv 2646	. . . . . 6 $/\$ $/\$
18		basis2 13213	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
19	18	exp43 372	. . . . . . . . . . . . . . 15 $/\$
20	19	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
21		basis2 13213	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
22	21	exp43 372	. . . . . . . . . . . . . . 15 $/\$
23	22	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
24		reeanv 2646	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		opelxpi 4655	. . . . . . . . . . . . . . . . . . 19 $/\$
26		xpss12 4730	. . . . . . . . . . . . . . . . . . 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 4766	. . . . . . . . . . 11 $/\$ $/\$
39	34, 38	sylibr 134	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an4s 588	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	anassrs 400	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		ineq12 3331	. . . . . . . . . 10 $/\$
43		inxp 4757	. . . . . . . . . 10
44	42, 43	eqtrdi 2226	. . . . . . . . 9 $/\$
45	44	sseq2d 3185	. . . . . . . . . . . 12 $/\$
46	45	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$ $/\$
47	46	rexbidv 2478	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	1	rexeqi 2677	. . . . . . . . . . 11 $/\$ $/\$
49		1stexg 6162	. . . . . . . . . . . . . . 15
50	49	elv 2741	. . . . . . . . . . . . . 14
51		2ndexg 6163	. . . . . . . . . . . . . . 15
52	51	elv 2741	. . . . . . . . . . . . . 14
53	50, 52	xpex 4738	. . . . . . . . . . . . 13
54	53	rgenw 2532	. . . . . . . . . . . 12
55		vex 2740	. . . . . . . . . . . . . . . . 17
56		vex 2740	. . . . . . . . . . . . . . . . 17
57	55, 56	op1std 6143	. . . . . . . . . . . . . . . 16
58	55, 56	op2ndd 6144	. . . . . . . . . . . . . . . 16
59	57, 58	xpeq12d 4648	. . . . . . . . . . . . . . 15
60	59	mpompt 5961	. . . . . . . . . . . . . 14
61	60	eqcomi 2181	. . . . . . . . . . . . 13
62		eleq2 2241	. . . . . . . . . . . . . 14
63		sseq1 3178	. . . . . . . . . . . . . 14
64	62, 63	anbi12d 473	. . . . . . . . . . . . 13 $/\$ $/\$
65	61, 64	rexrnmpt 5655	. . . . . . . . . . . 12 $/\$ $/\$
66	54, 65	ax-mp 5	. . . . . . . . . . 11 $/\$ $/\$
67	59	eleq2d 2247	. . . . . . . . . . . . 13
68	59	sseq1d 3184	. . . . . . . . . . . . 13
69	67, 68	anbi12d 473	. . . . . . . . . . . 12 $/\$ $/\$
70	69	rexxp 4767	. . . . . . . . . . 11 $/\$ $/\$
71	48, 66, 70	3bitri 206	. . . . . . . . . 10 $/\$ $/\$
72	47, 71	bitrdi 196	. . . . . . . . 9 $/\$ $/\$ $/\$
73	44, 72	raleqbidv 2684	. . . . . . . 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 13424	. . 3 $/\$
81		isbasis2g 13210	. . 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 2737

cin 3128

wss 3129

cop 3594

cmpt 4061

cxp 4621

crn 4624

cfv 5212

cmpo 5871

c1st 6133

c2nd 6134

ctb 13207

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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430

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 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fo 5218 df-fv 5220 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-bases 13208

This theorem is referenced by: txtop 13427

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