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Theorem metss 15485

Description: Two ways of saying that metric

generates a finer topology than metric

. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)

**Hypotheses**
Ref	Expression
metequiv.3
metequiv.4

**Assertion**
Ref	Expression
metss	$/\$

Distinct variable groups:

Proof of Theorem metss Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metequiv.3	. . . . 5
2	1	mopnval 15433	. . . 4
3	2	adantr 276	. . 3 $/\$
4		metequiv.4	. . . . 5
5	4	mopnval 15433	. . . 4
6	5	adantl 277	. . 3 $/\$
7	3, 6	sseq12d 3273	. 2 $/\$
8		blbas 15424	. . . 4
9	8	adantr 276	. . 3 $/\$
10		unirnbl 15414	. . . . 5
11	10	adantr 276	. . . 4 $/\$
12		unirnbl 15414	. . . . 5
13	12	adantl 277	. . . 4 $/\$
14	11, 13	eqtr4d 2270	. . 3 $/\$
15		tgss2 15070	. . 3 $/\$ $/\$
16	9, 14, 15	syl2anc 411	. 2 $/\$ $/\$
17	11	raleqdv 2749	. . 3 $/\$ $/\$ $/\$
18		blssex 15421	. . . . . . . 8 $/\$ $/\$
19	18	adantll 476	. . . . . . 7 $/\$ $/\$ $/\$
20	19	imbi2d 230	. . . . . 6 $/\$ $/\$ $/\$
21	20	ralbidv 2544	. . . . 5 $/\$ $/\$ $/\$
22		rpxr 10012	. . . . . . . . . . 11
23		blelrn 15411	. . . . . . . . . . 11 $/\$ $/\$
24	22, 23	syl3an3 1309	. . . . . . . . . 10 $/\$ $/\$
25		blcntr 15407	. . . . . . . . . 10 $/\$ $/\$
26		eleq2 2298	. . . . . . . . . . . . 13
27		sseq2 3266	. . . . . . . . . . . . . 14
28	27	rexbidv 2545	. . . . . . . . . . . . 13
29	26, 28	imbi12d 234	. . . . . . . . . . . 12
30	29	rspcv 2919	. . . . . . . . . . 11
31	30	com23 78	. . . . . . . . . 10
32	24, 25, 31	sylc 62	. . . . . . . . 9 $/\$ $/\$
33	32	3expa 1230	. . . . . . . 8 $/\$ $/\$
34	33	adantllr 481	. . . . . . 7 $/\$ $/\$ $/\$
35	34	ralrimdva 2624	. . . . . 6 $/\$ $/\$
36		blss 15419	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	3expb 1231	. . . . . . . . . . . 12 $/\$ $/\$
38	37	adantlr 477	. . . . . . . . . . 11 $/\$ $/\$ $/\$
39	38	adantlr 477	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
40		r19.29 2682	. . . . . . . . . . . 12 $/\$ $/\$
41		sstr 3250	. . . . . . . . . . . . . . . 16 $/\$
42	41	expcom 116	. . . . . . . . . . . . . . 15
43	42	reximdv 2645	. . . . . . . . . . . . . 14
44	43	impcom 125	. . . . . . . . . . . . 13 $/\$
45	44	rexlimivw 2658	. . . . . . . . . . . 12 $/\$
46	40, 45	syl 14	. . . . . . . . . . 11 $/\$
47	46	ex 115	. . . . . . . . . 10
48	39, 47	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
49	48	expr 375	. . . . . . . 8 $/\$ $/\$ $/\$
50	49	com23 78	. . . . . . 7 $/\$ $/\$ $/\$
51	50	ralrimdva 2624	. . . . . 6 $/\$ $/\$
52	35, 51	impbid 129	. . . . 5 $/\$ $/\$
53	21, 52	bitrd 188	. . . 4 $/\$ $/\$ $/\$
54	53	ralbidva 2540	. . 3 $/\$ $/\$
55	17, 54	bitrd 188	. 2 $/\$ $/\$
56	7, 16, 55	3bitrd 214	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1005

wceq 1398

wcel 2205

wral 2522

wrex 2523

wss 3214

cuni 3919

crn 4755

cfv 5357 (class class class)co 6058

cxr 8323

crp 10004

ctg 13551

cxmet 14810

cbl 14812

cmopn 14815

ctb 15033

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-in1 619 ax-in2 620 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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-map 6897 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-xneg 10124 df-xadd 10125 df-seqfrec 10834 df-exp 10925 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-topgen 13557 df-psmet 14817 df-xmet 14818 df-bl 14820 df-mopn 14821 df-top 14989 df-bases 15034

This theorem is referenced by: metequiv 15486 metss2 15489

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