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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 2829 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋)) | |
| 2 | eleq1 2829 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) | |
| 3 | 1, 2 | bi2anan9 638 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))) |
| 4 | oveq12 7440 | . . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎𝐷𝑏) = (𝐴𝐷𝐵)) | |
| 5 | 4 | eqeq1d 2739 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎𝐷𝑏) = 0 ↔ (𝐴𝐷𝐵) = 0)) |
| 6 | 3, 5 | anbi12d 632 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) |
| 7 | eqid 2737 | . . . 4 ⊢ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)} | |
| 8 | 6, 7 | brabga 5539 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) |
| 9 | 8 | adantl 481 | . 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 33889 | . . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}) | |
| 11 | 10 | adantr 480 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (~Met‘𝐷) = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}) |
| 12 | 11 | breqd 5154 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ 𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵)) |
| 13 | ibar 528 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) | |
| 14 | 13 | adantl 481 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) |
| 15 | 9, 12, 14 | 3bitr4d 311 | 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 {copab 5205 ‘cfv 6561 (class class class)co 7431 0cc0 11155 PsMetcpsmet 21348 ~Metcmetid 33885 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-xr 11299 df-psmet 21356 df-metid 33887 |
| This theorem is referenced by: metideq 33892 metider 33893 pstmfval 33895 pstmxmet 33896 |
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