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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 2900 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋)) | |
2 | eleq1 2900 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) | |
3 | 1, 2 | bi2anan9 637 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))) |
4 | oveq12 7159 | . . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎𝐷𝑏) = (𝐴𝐷𝐵)) | |
5 | 4 | eqeq1d 2823 | . . . . 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 2821 | . . . 4 ⊢ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)} | |
8 | 6, 7 | brabga 5413 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) |
9 | 8 | adantl 484 | . 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 31125 | . . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}) | |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (~Met‘𝐷) = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}) |
12 | 11 | breqd 5069 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ 𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵)) |
13 | ibar 531 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) | |
14 | 13 | adantl 484 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) |
15 | 9, 12, 14 | 3bitr4d 313 | 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 {copab 5120 ‘cfv 6349 (class class class)co 7150 0cc0 10531 PsMetcpsmet 20523 ~Metcmetid 31121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-xr 10673 df-psmet 20531 df-metid 31123 |
This theorem is referenced by: metideq 31128 metider 31129 pstmfval 31131 pstmxmet 31132 |
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