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Mathbox for Thierry Arnoux |
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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 2832 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋)) | |
2 | eleq1 2832 | . . . . . 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 7457 | . . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎𝐷𝑏) = (𝐴𝐷𝐵)) | |
5 | 4 | eqeq1d 2742 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎𝐷𝑏) = 0 ↔ (𝐴𝐷𝐵) = 0)) |
6 | 3, 5 | anbi12d 631 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0))) |
7 | eqid 2740 | . . . 4 ⊢ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)} | |
8 | 6, 7 | brabga 5553 | . . 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 33836 | . . . 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 5177 | . 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 1537 ∈ wcel 2108 class class class wbr 5166 {copab 5228 ‘cfv 6573 (class class class)co 7448 0cc0 11184 PsMetcpsmet 21371 ~Metcmetid 33832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-map 8886 df-xr 11328 df-psmet 21379 df-metid 33834 |
This theorem is referenced by: metideq 33839 metider 33840 pstmfval 33842 pstmxmet 33843 |
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