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Mirrors > Home > MPE Home > Th. List > setsmsbas | Structured version Visualization version GIF version |
Description: The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
setsms.x | ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) |
setsms.d | ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
setsms.k | ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
Ref | Expression |
---|---|
setsmsbas | ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseid 16315 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
2 | 1re 10376 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 1lt9 11588 | . . . . 5 ⊢ 1 < 9 | |
4 | 2, 3 | ltneii 10489 | . . . 4 ⊢ 1 ≠ 9 |
5 | basendx 16319 | . . . . 5 ⊢ (Base‘ndx) = 1 | |
6 | tsetndx 16432 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
7 | 5, 6 | neeq12i 3034 | . . . 4 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9) |
8 | 4, 7 | mpbir 223 | . . 3 ⊢ (Base‘ndx) ≠ (TopSet‘ndx) |
9 | 1, 8 | setsnid 16311 | . 2 ⊢ (Base‘𝑀) = (Base‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
10 | setsms.x | . 2 ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) | |
11 | setsms.k | . . 3 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
12 | 11 | fveq2d 6450 | . 2 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
13 | 9, 10, 12 | 3eqtr4a 2839 | 1 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ≠ wne 2968 〈cop 4403 × cxp 5353 ↾ cres 5357 ‘cfv 6135 (class class class)co 6922 1c1 10273 9c9 11437 ndxcnx 16252 sSet csts 16253 Basecbs 16255 TopSetcts 16344 distcds 16347 MetOpencmopn 20132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-tset 16357 |
This theorem is referenced by: setsmstopn 22691 setsxms 22692 setsms 22693 tmslem 22695 |
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