![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > latasymd | Structured version Visualization version GIF version |
Description: Deduce equality from lattice ordering. (eqssd 3932 analog.) (Contributed by NM, 18-Nov-2011.) |
Ref | Expression |
---|---|
latasymd.b | ⊢ 𝐵 = (Base‘𝐾) |
latasymd.l | ⊢ ≤ = (le‘𝐾) |
latasymd.3 | ⊢ (𝜑 → 𝐾 ∈ Lat) |
latasymd.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
latasymd.5 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
latasymd.6 | ⊢ (𝜑 → 𝑋 ≤ 𝑌) |
latasymd.7 | ⊢ (𝜑 → 𝑌 ≤ 𝑋) |
Ref | Expression |
---|---|
latasymd | ⊢ (𝜑 → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latasymd.6 | . 2 ⊢ (𝜑 → 𝑋 ≤ 𝑌) | |
2 | latasymd.7 | . 2 ⊢ (𝜑 → 𝑌 ≤ 𝑋) | |
3 | latasymd.3 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Lat) | |
4 | latasymd.4 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | latasymd.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | latasymd.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
7 | latasymd.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
8 | 6, 7 | latasymb 17656 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
9 | 3, 4, 5, 8 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) |
10 | 1, 2, 9 | mpbi2and 711 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 lecple 16564 Latclat 17647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-dm 5529 df-iota 6283 df-fv 6332 df-proset 17530 df-poset 17548 df-lat 17648 |
This theorem is referenced by: latjidm 17676 latmidm 17688 latjass 17697 oldmm1 36513 olj01 36521 olm01 36532 cvlcvr1 36635 llnmlplnN 36835 2llnjaN 36862 2lplnja 36915 cdlema1N 37087 hlmod1i 37152 lautj 37389 lautm 37390 cdleme19a 37599 cdleme28b 37667 trljco 38036 dochvalr 38653 |
Copyright terms: Public domain | W3C validator |