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Mirrors > Home > MPE Home > Th. List > setsmsbasOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of setsmsbas 24394 as of 12-Nov-2024. The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
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 |
---|---|
setsmsbasOLD | ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseid 17177 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
2 | 1re 11239 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 1lt9 12443 | . . . . 5 ⊢ 1 < 9 | |
4 | 2, 3 | ltneii 11352 | . . . 4 ⊢ 1 ≠ 9 |
5 | basendx 17183 | . . . . 5 ⊢ (Base‘ndx) = 1 | |
6 | tsetndx 17327 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
7 | 5, 6 | neeq12i 2997 | . . . 4 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9) |
8 | 4, 7 | mpbir 230 | . . 3 ⊢ (Base‘ndx) ≠ (TopSet‘ndx) |
9 | 1, 8 | setsnid 17172 | . 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 6894 | . 2 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))) |
13 | 9, 10, 12 | 3eqtr4a 2791 | 1 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ≠ wne 2930 ⟨cop 4631 × cxp 5671 ↾ cres 5675 ‘cfv 6543 (class class class)co 7413 1c1 11134 9c9 12299 sSet csts 17126 ndxcnx 17156 Basecbs 17174 TopSetcts 17233 distcds 17236 MetOpencmopn 21268 |
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 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-tset 17246 |
This theorem is referenced by: (None) |
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