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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 29620 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 36943 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) |
7 | eqcom 2743 | . . 3 ⊢ ((𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌 ↔ 𝑌 = (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌))) | |
8 | omllat 36942 | . . . . . . 7 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) | |
9 | 8 | 3ad2ant1 1135 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) |
10 | omlop 36941 | . . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP) | |
11 | 1, 5 | opoccl 36894 | . . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
12 | 10, 11 | sylan 583 | . . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
13 | 12 | 3adant3 1134 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
14 | simp3 1140 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
15 | 1, 4 | latmcom 17923 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) = (𝑌 ∧ ( ⊥ ‘𝑋))) |
16 | 9, 13, 14, 15 | syl3anc 1373 | . . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) = (𝑌 ∧ ( ⊥ ‘𝑋))) |
17 | 16 | oveq2d 7207 | . . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))) |
18 | 17 | eqeq2d 2747 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 = (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) ↔ 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) |
19 | 7, 18 | syl5bb 286 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌 ↔ 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) |
20 | 6, 19 | sylibrd 262 | 1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 lecple 16756 occoc 16757 joincjn 17772 meetcmee 17773 Latclat 17891 OPcops 36872 OMLcoml 36875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-glb 17807 df-meet 17809 df-lat 17892 df-oposet 36876 df-ol 36878 df-oml 36879 |
This theorem is referenced by: omllaw5N 36947 cmtcomlemN 36948 cmtbr3N 36954 |
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