| Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
| 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 31604 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 39244 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) |
| 7 | eqcom 2744 | . . 3 ⊢ ((𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌 ↔ 𝑌 = (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌))) | |
| 8 | omllat 39243 | . . . . . . 7 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) | |
| 9 | 8 | 3ad2ant1 1134 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) |
| 10 | omlop 39242 | . . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP) | |
| 11 | 1, 5 | opoccl 39195 | . . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
| 12 | 10, 11 | sylan 580 | . . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
| 13 | 12 | 3adant3 1133 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
| 14 | simp3 1139 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
| 15 | 1, 4 | latmcom 18508 | . . . . . 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 7447 | . . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))) |
| 18 | 17 | eqeq2d 2748 | . . 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 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 occoc 17305 joincjn 18357 meetcmee 18358 Latclat 18476 OPcops 39173 OMLcoml 39176 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-glb 18392 df-meet 18394 df-lat 18477 df-oposet 39177 df-ol 39179 df-oml 39180 |
| This theorem is referenced by: omllaw5N 39248 cmtcomlemN 39249 cmtbr3N 39255 |
| Copyright terms: Public domain | W3C validator |