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Theorem tmslem 24449

Description: Lemma for tmsbas 24450, tmsds 24451, and tmstopn 24452. (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 6876	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2		tmsval.m	. . . . 5 ⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}
3		basendxltdsndx 17353	. . . . 5 ⊢ (Base‘ndx) < (dist‘ndx)
4		dsndxnn 17352	. . . . 5 ⊢ (dist‘ndx) ∈ ℕ
5	2, 3, 4	2strbas 17200	. . . 4 ⊢ (𝑋 ∈ dom ∞Met → 𝑋 = (Base‘𝑀))
6	1, 5	syl 17	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝑀))
7		xmetf 24296	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
8		ffn 6670	. . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → 𝐷 Fn (𝑋 × 𝑋))
9		fnresdm 6619	. . . . 5 ⊢ (𝐷 Fn (𝑋 × 𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
10	7, 8, 9	3syl 18	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
11		dsid 17351	. . . . . 6 ⊢ dist = Slot (dist‘ndx)
12	2, 3, 4, 11	2strop 17201	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝑀))
13	12	reseq1d 5945	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
14	10, 13	eqtr3d 2774	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
15		tmsval.k	. . . 4 ⊢ 𝐾 = (toMetSp‘𝐷)
16	2, 15	tmsval 24448	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
17	6, 14, 16	setsmsbas 24442	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾))
18	6, 14, 16	setsmsds 24443	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝑀) = (dist‘𝐾))
19	12, 18	eqtrd 2772	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾))
20		prex 5381	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩} ∈ V
21	2, 20	eqeltri 2833	. . . 4 ⊢ 𝑀 ∈ V
22	21	a1i 11	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑀 ∈ V)
23	6, 14, 16, 22	setsmstopn 24445	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾))
24	17, 19, 23	3jca 1129	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 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3430 {cpr 4570 ⟨cop 4574 × cxp 5630 dom cdm 5632 ↾ cres 5634 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 ℝ^*cxr 11180 ndxcnx 17165 Basecbs 17181 distcds 17231 TopOpenctopn 17386 ∞Metcxmet 21339 MetOpencmopn 21344 toMetSpctms 24286

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7691 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7820 df-1st 7944 df-2nd 7945 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-pnf 11183 df-mnf 11184 df-xr 11185 df-ltxr 11186 df-le 11187 df-sub 11381 df-neg 11382 df-div 11810 df-nn 12177 df-2 12246 df-3 12247 df-4 12248 df-5 12249 df-6 12250 df-7 12251 df-8 12252 df-9 12253 df-n0 12440 df-z 12527 df-dec 12647 df-uz 12791 df-q 12901 df-rp 12945 df-xneg 13065 df-xadd 13066 df-xmul 13067 df-fz 13464 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17182 df-tset 17241 df-ds 17244 df-rest 17387 df-topn 17388 df-topgen 17408 df-psmet 21346 df-xmet 21347 df-bl 21349 df-mopn 21350 df-top 22861 df-topon 22878 df-bases 22913 df-tms 24289

This theorem is referenced by: tmsbas 24450 tmsds 24451 tmstopn 24452

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