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Mirrors > Home > MPE Home > Th. List > Mathboxes > metidv | Structured version Visualization version GIF version |
Description: 𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
Ref | Expression |
---|---|
metidv | ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2837 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋)) | |
2 | eleq1 2837 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) | |
3 | 1, 2 | bi2anan9 612 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))) |
4 | oveq12 6801 | . . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎𝐷𝑏) = (𝐴𝐷𝐵)) | |
5 | 4 | eqeq1d 2772 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎𝐷𝑏) = 0 ↔ (𝐴𝐷𝐵) = 0)) |
6 | 3, 5 | anbi12d 608 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) |
7 | eqid 2770 | . . . 4 ⊢ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)} | |
8 | 6, 7 | brabga 5122 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) |
9 | 8 | adantl 467 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) |
10 | metidval 30267 | . . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}) | |
11 | 10 | adantr 466 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (~Met‘𝐷) = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}) |
12 | 11 | breqd 4795 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ 𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵)) |
13 | ibar 512 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) | |
14 | 13 | adantl 467 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) |
15 | 9, 12, 14 | 3bitr4d 300 | 1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1630 ∈ wcel 2144 class class class wbr 4784 {copab 4844 ‘cfv 6031 (class class class)co 6792 0cc0 10137 PsMetcpsmet 19944 ~Metcmetid 30263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-fv 6039 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-map 8010 df-xr 10279 df-psmet 19952 df-metid 30265 |
This theorem is referenced by: metideq 30270 metider 30271 pstmfval 30273 pstmxmet 30274 |
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