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Mirrors > Home > MPE Home > Th. List > setsmsdsOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of setsmsds 24357 as of 11-Nov-2024. The distance function 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 |
---|---|
setsmsdsOLD | ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dsid 17352 | . . 3 ⊢ dist = Slot (dist‘ndx) | |
2 | 9re 12327 | . . . . 5 ⊢ 9 ∈ ℝ | |
3 | 1nn 12239 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | 2nn0 12505 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
5 | 9nn0 12512 | . . . . . 6 ⊢ 9 ∈ ℕ0 | |
6 | 9lt10 12824 | . . . . . 6 ⊢ 9 < ;10 | |
7 | 3, 4, 5, 6 | declti 12731 | . . . . 5 ⊢ 9 < ;12 |
8 | 2, 7 | gtneii 11342 | . . . 4 ⊢ ;12 ≠ 9 |
9 | dsndx 17351 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
10 | tsetndx 17318 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
11 | 9, 10 | neeq12i 3002 | . . . 4 ⊢ ((dist‘ndx) ≠ (TopSet‘ndx) ↔ ;12 ≠ 9) |
12 | 8, 11 | mpbir 230 | . . 3 ⊢ (dist‘ndx) ≠ (TopSet‘ndx) |
13 | 1, 12 | setsnid 17163 | . 2 ⊢ (dist‘𝑀) = (dist‘(𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) |
14 | setsms.k | . . 3 ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) | |
15 | 14 | fveq2d 6895 | . 2 ⊢ (𝜑 → (dist‘𝐾) = (dist‘(𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))) |
16 | 13, 15 | eqtr4id 2786 | 1 ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ≠ wne 2935 ⟨cop 4630 × cxp 5670 ↾ cres 5674 ‘cfv 6542 (class class class)co 7414 1c1 11125 2c2 12283 9c9 12290 ;cdc 12693 sSet csts 17117 ndxcnx 17147 Basecbs 17165 TopSetcts 17224 distcds 17227 MetOpencmopn 21249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-sets 17118 df-slot 17136 df-ndx 17148 df-tset 17237 df-ds 17240 |
This theorem is referenced by: (None) |
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