txbas - Intuitionistic Logic Explorer

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Theorem txbas 14426

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 4673	. . . . . . . . . 10
3		xpeq2 4674	. . . . . . . . . 10
4	2, 3	cbvmpov 5998	. . . . . . . . 9
5	4	rnmpo 6029	. . . . . . . 8
6	1, 5	eqtri 2214	. . . . . . 7
7	6	abeq2i 2304	. . . . . 6
8		xpeq1 4673	. . . . . . . . . 10
9		xpeq2 4674	. . . . . . . . . 10
10	8, 9	cbvmpov 5998	. . . . . . . . 9
11	10	rnmpo 6029	. . . . . . . 8
12	1, 11	eqtri 2214	. . . . . . 7
13	12	abeq2i 2304	. . . . . 6
14	7, 13	anbi12i 460	. . . . 5 $/\$ $/\$
15		reeanv 2664	. . . . 5 $/\$ $/\$
16	14, 15	bitr4i 187	. . . 4 $/\$ $/\$
17		reeanv 2664	. . . . . 6 $/\$ $/\$
18		basis2 14216	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
19	18	exp43 372	. . . . . . . . . . . . . . 15 $/\$
20	19	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
21		basis2 14216	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
22	21	exp43 372	. . . . . . . . . . . . . . 15 $/\$
23	22	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
24		reeanv 2664	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		opelxpi 4691	. . . . . . . . . . . . . . . . . . 19 $/\$
26		xpss12 4766	. . . . . . . . . . . . . . . . . . 19 $/\$
27	25, 26	anim12i 338	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
28	27	an4s 588	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
29	28	reximi 2591	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
30	29	reximi 2591	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
31	24, 30	sylbir 135	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
32	20, 23, 31	syl2an 289	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	an4s 588	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	ralrimivva 2576	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		eleq1 2256	. . . . . . . . . . . . . 14
36	35	anbi1d 465	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	2rexbidv 2519	. . . . . . . . . . . 12 $/\$ $/\$
38	37	ralxp 4805	. . . . . . . . . . 11 $/\$ $/\$
39	34, 38	sylibr 134	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an4s 588	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	anassrs 400	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		ineq12 3355	. . . . . . . . . 10 $/\$
43		inxp 4796	. . . . . . . . . 10
44	42, 43	eqtrdi 2242	. . . . . . . . 9 $/\$
45	44	sseq2d 3209	. . . . . . . . . . . 12 $/\$
46	45	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$ $/\$
47	46	rexbidv 2495	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	1	rexeqi 2695	. . . . . . . . . . 11 $/\$ $/\$
49		1stexg 6220	. . . . . . . . . . . . . . 15
50	49	elv 2764	. . . . . . . . . . . . . 14
51		2ndexg 6221	. . . . . . . . . . . . . . 15
52	51	elv 2764	. . . . . . . . . . . . . 14
53	50, 52	xpex 4774	. . . . . . . . . . . . 13
54	53	rgenw 2549	. . . . . . . . . . . 12
55		vex 2763	. . . . . . . . . . . . . . . . 17
56		vex 2763	. . . . . . . . . . . . . . . . 17
57	55, 56	op1std 6201	. . . . . . . . . . . . . . . 16
58	55, 56	op2ndd 6202	. . . . . . . . . . . . . . . 16
59	57, 58	xpeq12d 4684	. . . . . . . . . . . . . . 15
60	59	mpompt 6010	. . . . . . . . . . . . . 14
61	60	eqcomi 2197	. . . . . . . . . . . . 13
62		eleq2 2257	. . . . . . . . . . . . . 14
63		sseq1 3202	. . . . . . . . . . . . . 14
64	62, 63	anbi12d 473	. . . . . . . . . . . . 13 $/\$ $/\$
65	61, 64	rexrnmpt 5701	. . . . . . . . . . . 12 $/\$ $/\$
66	54, 65	ax-mp 5	. . . . . . . . . . 11 $/\$ $/\$
67	59	eleq2d 2263	. . . . . . . . . . . . 13
68	59	sseq1d 3208	. . . . . . . . . . . . 13
69	67, 68	anbi12d 473	. . . . . . . . . . . 12 $/\$ $/\$
70	69	rexxp 4806	. . . . . . . . . . 11 $/\$ $/\$
71	48, 66, 70	3bitri 206	. . . . . . . . . 10 $/\$ $/\$
72	47, 71	bitrdi 196	. . . . . . . . 9 $/\$ $/\$ $/\$
73	44, 72	raleqbidv 2706	. . . . . . . 8 $/\$ $/\$ $/\$
74	41, 73	syl5ibrcom 157	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	74	rexlimdvva 2619	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	17, 75	biimtrrid 153	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	rexlimdvva 2619	. . . 4 $/\$ $/\$ $/\$
78	16, 77	biimtrid 152	. . 3 $/\$ $/\$ $/\$
79	78	ralrimivv 2575	. 2 $/\$ $/\$
80	1	txbasex 14425	. . 3 $/\$
81		isbasis2g 14213	. . 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 2164

cab 2179

wral 2472

wrex 2473

cvv 2760

cin 3152

wss 3153

cop 3621

cmpt 4090

cxp 4657

crn 4660

cfv 5254

cmpo 5920

c1st 6191

c2nd 6192

ctb 14210

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fo 5260 df-fv 5262 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-bases 14211

This theorem is referenced by: txtop 14428

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