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Theorem tmslem 23089

Description: Lemma for tmsbas 23090, tmsds 23091, and tmstopn 23092. (Contributed by Mario Carneiro, 2-Sep-2015.)

**Hypotheses**
Ref	Expression
tmsval.m	⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}
tmsval.k	⊢ 𝐾 = (toMetSp‘𝐷)

**Assertion**
Ref	Expression
tmslem	⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾)))

**Proof of Theorem tmslem**
Step	Hyp	Ref	Expression
1		elfvdm 6677	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2		tmsval.m	. . . . 5 ⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}
3		df-ds 16579	. . . . 5 ⊢ dist = Slot ;12
4		1nn 11636	. . . . . 6 ⊢ 1 ∈ ℕ
5		2nn0 11902	. . . . . 6 ⊢ 2 ∈ ℕ₀
6		1nn0 11901	. . . . . 6 ⊢ 1 ∈ ℕ₀
7		1lt10 12225	. . . . . 6 ⊢ 1 < ;10
8	4, 5, 6, 7	declti 12124	. . . . 5 ⊢ 1 < ;12
9		2nn 11698	. . . . . 6 ⊢ 2 ∈ ℕ
10	6, 9	decnncl 12106	. . . . 5 ⊢ ;12 ∈ ℕ
11	2, 3, 8, 10	2strbas 16595	. . . 4 ⊢ (𝑋 ∈ dom ∞Met → 𝑋 = (Base‘𝑀))
12	1, 11	syl 17	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝑀))
13		xmetf 22936	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
14		ffn 6487	. . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → 𝐷 Fn (𝑋 × 𝑋))
15		fnresdm 6438	. . . . 5 ⊢ (𝐷 Fn (𝑋 × 𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
16	13, 14, 15	3syl 18	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
17	2, 3, 8, 10	2strop 16596	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝑀))
18	17	reseq1d 5817	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
19	16, 18	eqtr3d 2835	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
20		tmsval.k	. . . 4 ⊢ 𝐾 = (toMetSp‘𝐷)
21	2, 20	tmsval 23088	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
22	12, 19, 21	setsmsbas 23082	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾))
23	12, 19, 21	setsmsds 23083	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝑀) = (dist‘𝐾))
24	17, 23	eqtrd 2833	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾))
25		prex 5298	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩} ∈ V
26	2, 25	eqeltri 2886	. . . 4 ⊢ 𝑀 ∈ V
27	26	a1i 11	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑀 ∈ V)
28	12, 19, 21, 27	setsmstopn 23085	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾))
29	22, 24, 28	3jca 1125	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {cpr 4527 ⟨cop 4531 × cxp 5517 dom cdm 5519 ↾ cres 5521 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 1c1 10527 ℝ^*cxr 10663 2c2 11680 cdc 12086 ndxcnx 16472 Basecbs 16475 distcds 16566 TopOpenctopn 16687 ∞Metcxmet 20076 MetOpencmopn 20081 toMetSpctms 22926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-tset 16576 df-ds 16579 df-rest 16688 df-topn 16689 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551 df-tms 22929

This theorem is referenced by: tmsbas 23090 tmsds 23091 tmstopn 23092

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