![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 12543 | . . . 4 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ) | |
4 | 1re 7975 | . . . . . 6 ⊢ 1 ∈ ℝ | |
5 | 1lt9 9142 | . . . . . 6 ⊢ 1 < 9 | |
6 | 4, 5 | ltneii 8073 | . . . . 5 ⊢ 1 ≠ 9 |
7 | basendx 12541 | . . . . . 6 ⊢ (Base‘ndx) = 1 | |
8 | tsetndx 12669 | . . . . . 6 ⊢ (TopSet‘ndx) = 9 | |
9 | 7, 8 | neeq12i 2377 | . . . . 5 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9) |
10 | 6, 9 | mpbir 146 | . . . 4 ⊢ (Base‘ndx) ≠ (TopSet‘ndx) |
11 | 9nn 9106 | . . . . 5 ⊢ 9 ∈ ℕ | |
12 | 8, 11 | eqeltri 2262 | . . . 4 ⊢ (TopSet‘ndx) ∈ ℕ |
13 | 3, 10, 12 | setsslnid 12538 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (MetOpen‘𝐷) ∈ 𝑊) → (Base‘𝑀) = (Base‘(𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))) |
14 | 1, 2, 13 | syl2anc 411 | . 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 5534 | . 2 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))) |
18 | 14, 15, 17 | 3eqtr4d 2232 | 1 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ⟨cop 3610 × cxp 4639 ↾ cres 4643 ‘cfv 5231 (class class class)co 5891 1c1 7831 ℕcn 8938 9c9 8996 ndxcnx 12483 sSet csts 12484 Basecbs 12486 TopSetcts 12567 distcds 12570 MetOpencmopn 13821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-i2m1 7935 ax-0lt1 7936 ax-0id 7938 ax-rnegex 7939 ax-pre-ltirr 7942 ax-pre-lttrn 7944 ax-pre-ltadd 7946 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-iota 5193 df-fun 5233 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8013 df-mnf 8014 df-ltxr 8016 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-5 9000 df-6 9001 df-7 9002 df-8 9003 df-9 9004 df-ndx 12489 df-slot 12490 df-base 12492 df-sets 12493 df-tset 12580 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |