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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 29364 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 36381 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) |
7 | eqcom 2830 | . . 3 ⊢ ((𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌 ↔ 𝑌 = (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌))) | |
8 | omllat 36380 | . . . . . . 7 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) | |
9 | 8 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) |
10 | omlop 36379 | . . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP) | |
11 | 1, 5 | opoccl 36332 | . . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
12 | 10, 11 | sylan 582 | . . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
13 | 12 | 3adant3 1128 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
14 | simp3 1134 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
15 | 1, 4 | latmcom 17687 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) = (𝑌 ∧ ( ⊥ ‘𝑋))) |
16 | 9, 13, 14, 15 | syl3anc 1367 | . . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) = (𝑌 ∧ ( ⊥ ‘𝑋))) |
17 | 16 | oveq2d 7174 | . . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))) |
18 | 17 | eqeq2d 2834 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 = (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) ↔ 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) |
19 | 7, 18 | syl5bb 285 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌 ↔ 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) |
20 | 6, 19 | sylibrd 261 | 1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 lecple 16574 occoc 16575 joincjn 17556 meetcmee 17557 Latclat 17657 OPcops 36310 OMLcoml 36313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-glb 17587 df-meet 17589 df-lat 17658 df-oposet 36314 df-ol 36316 df-oml 36317 |
This theorem is referenced by: omllaw5N 36385 cmtcomlemN 36386 cmtbr3N 36392 |
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