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Mirrors > Home > ILE Home > Th. List > setsmsbasg | 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)〉)) |
setsmsbasg.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
setsmsbasg.d | ⊢ (𝜑 → (MetOpen‘𝐷) ∈ 𝑊) |
Ref | Expression |
---|---|
setsmsbasg | ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsmsbasg.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
2 | setsmsbasg.d | . . 3 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ 𝑊) | |
3 | baseslid 12393 | . . . 4 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ) | |
4 | 1re 7889 | . . . . . 6 ⊢ 1 ∈ ℝ | |
5 | 1lt9 9052 | . . . . . 6 ⊢ 1 < 9 | |
6 | 4, 5 | ltneii 7986 | . . . . 5 ⊢ 1 ≠ 9 |
7 | basendx 12391 | . . . . . 6 ⊢ (Base‘ndx) = 1 | |
8 | tsetndx 12485 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
9 | 7, 8 | neeq12i 2351 | . . . . 5 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9) |
10 | 6, 9 | mpbir 145 | . . . 4 ⊢ (Base‘ndx) ≠ (TopSet‘ndx) |
11 | 9nn 9016 | . . . . 5 ⊢ 9 ∈ ℕ | |
12 | 8, 11 | eqeltri 2237 | . . . 4 ⊢ (TopSet‘ndx) ∈ ℕ |
13 | 3, 10, 12 | setsslnid 12388 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (MetOpen‘𝐷) ∈ 𝑊) → (Base‘𝑀) = (Base‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
14 | 1, 2, 13 | syl2anc 409 | . 2 ⊢ (𝜑 → (Base‘𝑀) = (Base‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
15 | setsms.x | . 2 ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) | |
16 | setsms.k | . . 3 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
17 | 16 | fveq2d 5484 | . 2 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉))) |
18 | 14, 15, 17 | 3eqtr4d 2207 | 1 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 〈cop 3573 × cxp 4596 ↾ cres 4600 ‘cfv 5182 (class class class)co 5836 1c1 7745 ℕcn 8848 9c9 8906 ndxcnx 12334 sSet csts 12335 Basecbs 12337 TopSetcts 12405 distcds 12408 MetOpencmopn 12532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-ltxr 7929 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-ndx 12340 df-slot 12341 df-base 12343 df-sets 12344 df-tset 12418 |
This theorem is referenced by: (None) |
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