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Theorem tmslem 22506

Description: Lemma for tmsbas 22507, tmsds 22508, and tmstopn 22509. (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 6361	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2		tmsval.m	. . . . 5 ⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}
3		df-ds 16171	. . . . 5 ⊢ dist = Slot ;12
4		1nn 11232	. . . . . 6 ⊢ 1 ∈ ℕ
5		2nn0 11510	. . . . . 6 ⊢ 2 ∈ ℕ₀
6		1nn0 11509	. . . . . 6 ⊢ 1 ∈ ℕ₀
7		1lt10 11881	. . . . . 6 ⊢ 1 < ;10
8	4, 5, 6, 7	declti 11747	. . . . 5 ⊢ 1 < ;12
9		2nn 11386	. . . . . 6 ⊢ 2 ∈ ℕ
10	6, 9	decnncl 11719	. . . . 5 ⊢ ;12 ∈ ℕ
11	2, 3, 8, 10	2strbas 16191	. . . 4 ⊢ (𝑋 ∈ dom ∞Met → 𝑋 = (Base‘𝑀))
12	1, 11	syl 17	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝑀))
13		xmetf 22353	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
14		ffn 6185	. . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → 𝐷 Fn (𝑋 × 𝑋))
15		fnresdm 6140	. . . . 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 16192	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝑀))
18	17	reseq1d 5533	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
19	16, 18	eqtr3d 2806	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
20		tmsval.k	. . . 4 ⊢ 𝐾 = (toMetSp‘𝐷)
21	2, 20	tmsval 22505	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
22	12, 19, 21	setsmsbas 22499	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾))
23	12, 19, 21	setsmsds 22500	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝑀) = (dist‘𝐾))
24	17, 23	eqtrd 2804	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾))
25		prex 5037	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩} ∈ V
26	2, 25	eqeltri 2845	. . . 4 ⊢ 𝑀 ∈ V
27	26	a1i 11	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑀 ∈ V)
28	12, 19, 21, 27	setsmstopn 22502	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾))
29	22, 24, 28	3jca 1121	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 1070 = wceq 1630 ∈ wcel 2144 Vcvv 3349 {cpr 4316 ⟨cop 4320 × cxp 5247 dom cdm 5249 ↾ cres 5251 Fn wfn 6026 ⟶wf 6027 ‘cfv 6031 1c1 10138 ℝ^*cxr 10274 2c2 11271 cdc 11694 ndxcnx 16060 Basecbs 16063 distcds 16157 TopOpenctopn 16289 ∞Metcxmt 19945 MetOpencmopn 19950 toMetSpctmt 22343

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-sup 8503 df-inf 8504 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-uz 11888 df-q 11991 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-fz 12533 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-tset 16167 df-ds 16171 df-rest 16290 df-topn 16291 df-topgen 16311 df-psmet 19952 df-xmet 19953 df-bl 19955 df-mopn 19956 df-top 20918 df-topon 20935 df-bases 20970 df-tms 22346

This theorem is referenced by: tmsbas 22507 tmsds 22508 tmstopn 22509

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