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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omllaw2N | Structured version Visualization version GIF version |
Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 31613 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
omllaw.b | ⊢ 𝐵 = (Base‘𝐾) |
omllaw.l | ⊢ ≤ = (le‘𝐾) |
omllaw.j | ⊢ ∨ = (join‘𝐾) |
omllaw.m | ⊢ ∧ = (meet‘𝐾) |
omllaw.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
omllaw2N | ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omllaw.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | omllaw.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | omllaw.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | omllaw.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
5 | omllaw.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
6 | 1, 2, 3, 4, 5 | omllaw 39224 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) |
7 | eqcom 2741 | . . 3 ⊢ ((𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌 ↔ 𝑌 = (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌))) | |
8 | omllat 39223 | . . . . . . 7 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) | |
9 | 8 | 3ad2ant1 1132 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) |
10 | omlop 39222 | . . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP) | |
11 | 1, 5 | opoccl 39175 | . . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
12 | 10, 11 | sylan 580 | . . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
13 | 12 | 3adant3 1131 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
14 | simp3 1137 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
15 | 1, 4 | latmcom 18520 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) = (𝑌 ∧ ( ⊥ ‘𝑋))) |
16 | 9, 13, 14, 15 | syl3anc 1370 | . . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) = (𝑌 ∧ ( ⊥ ‘𝑋))) |
17 | 16 | oveq2d 7446 | . . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))) |
18 | 17 | eqeq2d 2745 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 = (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) ↔ 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) |
19 | 7, 18 | bitrid 283 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌 ↔ 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) |
20 | 6, 19 | sylibrd 259 | 1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 lecple 17304 occoc 17305 joincjn 18368 meetcmee 18369 Latclat 18488 OPcops 39153 OMLcoml 39156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-glb 18404 df-meet 18406 df-lat 18489 df-oposet 39157 df-ol 39159 df-oml 39160 |
This theorem is referenced by: omllaw5N 39228 cmtcomlemN 39229 cmtbr3N 39235 |
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