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Mirrors > Home > MPE Home > Th. List > tmslem | Structured version Visualization version GIF version |
Description: Lemma for tmsbas 24412, tmsds 24413, and tmstopn 24414. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tmsval.m | ⊢ 𝑀 = {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} |
tmsval.k | ⊢ 𝐾 = (toMetSp‘𝐷) |
Ref | Expression |
---|---|
tmslem | ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6939 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) | |
2 | tmsval.m | . . . . 5 ⊢ 𝑀 = {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} | |
3 | basendxltdsndx 17376 | . . . . 5 ⊢ (Base‘ndx) < (dist‘ndx) | |
4 | dsndxnn 17375 | . . . . 5 ⊢ (dist‘ndx) ∈ ℕ | |
5 | 2, 3, 4 | 2strbas1 17214 | . . . 4 ⊢ (𝑋 ∈ dom ∞Met → 𝑋 = (Base‘𝑀)) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝑀)) |
7 | xmetf 24255 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
8 | ffn 6727 | . . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) | |
9 | fnresdm 6679 | . . . . 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 17374 | . . . . . 6 ⊢ dist = Slot (dist‘ndx) | |
12 | 2, 3, 4, 11 | 2strop1 17215 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝑀)) |
13 | 12 | reseq1d 5988 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
14 | 10, 13 | eqtr3d 2770 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
15 | tmsval.k | . . . 4 ⊢ 𝐾 = (toMetSp‘𝐷) | |
16 | 2, 15 | tmsval 24409 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
17 | 6, 14, 16 | setsmsbas 24401 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾)) |
18 | 6, 14, 16 | setsmsds 24403 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (dist‘𝑀) = (dist‘𝐾)) |
19 | 12, 18 | eqtrd 2768 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾)) |
20 | prex 5438 | . . . . 5 ⊢ {〈(Base‘ndx), 𝑋〉, 〈(dist‘ndx), 𝐷〉} ∈ V | |
21 | 2, 20 | eqeltri 2825 | . . . 4 ⊢ 𝑀 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑀 ∈ V) |
23 | 6, 14, 16, 22 | setsmstopn 24406 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
24 | 17, 19, 23 | 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 1533 ∈ wcel 2098 Vcvv 3473 {cpr 4634 〈cop 4638 × cxp 5680 dom cdm 5682 ↾ cres 5684 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 ℝ*cxr 11285 ndxcnx 17169 Basecbs 17187 distcds 17249 TopOpenctopn 17410 ∞Metcxmet 21271 MetOpencmopn 21276 toMetSpctms 24245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-tset 17259 df-ds 17262 df-rest 17411 df-topn 17412 df-topgen 17432 df-psmet 21278 df-xmet 21279 df-bl 21281 df-mopn 21282 df-top 22816 df-topon 22833 df-bases 22869 df-tms 24248 |
This theorem is referenced by: tmsbas 24412 tmsds 24413 tmstopn 24414 |
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