Theorem isxms2 22253

Description: Express the predicate "⟨𝑋, 𝐷⟩ is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
isms.j	⊢ 𝐽 = (TopOpen‘𝐾)
isms.x	⊢ 𝑋 = (Base‘𝐾)
isms.d	⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
isxms2	⊢ (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))

Proof of Theorem isxms2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isms.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐾)
2		isms.x	. . 3 ⊢ 𝑋 = (Base‘𝐾)
3		isms.d	. . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
4	1, 2, 3	isxms 22252	. 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
5	2, 1	istps 20738	. . . 4 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
6		df-mopn 19742	. . . . . . . . . 10 ⊢ MetOpen = (𝑥 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑥)))
7	6	dmmptss 5631	. . . . . . . . 9 ⊢ dom MetOpen ⊆ ∪ ran ∞Met
8		toponmax 20730	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
9	8	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 ∈ 𝐽)
10		simpl 473	. . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐽 = (MetOpen‘𝐷))
11	9, 10	eleqtrd 2703	. . . . . . . . . 10 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 ∈ (MetOpen‘𝐷))
12		elfvdm 6220	. . . . . . . . . 10 ⊢ (𝑋 ∈ (MetOpen‘𝐷) → 𝐷 ∈ dom MetOpen)
13	11, 12	syl 17	. . . . . . . . 9 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ dom MetOpen)
14	7, 13	sseldi 3601	. . . . . . . 8 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ ∪ ran ∞Met)
15		xmetunirn 22142	. . . . . . . 8 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
16	14, 15	sylib 208	. . . . . . 7 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
17		eqid 2622	. . . . . . . . . . . . 13 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
18	17	mopntopon 22244	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → (MetOpen‘𝐷) ∈ (TopOn‘dom dom 𝐷))
19	16, 18	syl 17	. . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (MetOpen‘𝐷) ∈ (TopOn‘dom dom 𝐷))
20	10, 19	eqeltrd 2701	. . . . . . . . . 10 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘dom dom 𝐷))
21		toponuni 20719	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘dom dom 𝐷) → dom dom 𝐷 = ∪ 𝐽)
22	20, 21	syl 17	. . . . . . . . 9 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → dom dom 𝐷 = ∪ 𝐽)
23		toponuni 20719	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
24	23	adantl 482	. . . . . . . . 9 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 = ∪ 𝐽)
25	22, 24	eqtr4d 2659	. . . . . . . 8 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → dom dom 𝐷 = 𝑋)
26	25	fveq2d 6195	. . . . . . 7 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∞Met‘dom dom 𝐷) = (∞Met‘𝑋))
27	16, 26	eleqtrd 2703	. . . . . 6 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ (∞Met‘𝑋))
28	27	ex 450	. . . . 5 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)))
29	17	mopntopon 22244	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) ∈ (TopOn‘𝑋))
30		eleq1 2689	. . . . . 6 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) ↔ (MetOpen‘𝐷) ∈ (TopOn‘𝑋)))
31	29, 30	syl5ibr 236	. . . . 5 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)))
32	28, 31	impbid 202	. . . 4 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐷 ∈ (∞Met‘𝑋)))
33	5, 32	syl5bb 272	. . 3 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐾 ∈ TopSp ↔ 𝐷 ∈ (∞Met‘𝑋)))
34	33	pm5.32ri 670	. 2 ⊢ ((𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))
35	4, 34	bitri 264	1 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∪ cuni 4436   × cxp 5112  dom cdm 5114  ran crn 5115   ↾ cres 5116  ‘cfv 5888  Basecbs 15857  distcds 15950  TopOpenctopn 16082  topGenctg 16098  ∞Metcxmt 19731  ballcbl 19733  MetOpencmopn 19736  TopOnctopon 20715  TopSpctps 20736  ∞MetSpcxme 22122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125

This theorem is referenced by:  isms2  22255  xmsxmet  22261  setsxms  22284  tmsxms  22291  imasf1oxms  22294  ressxms  22330  prdsxms  22335