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Mirrors > Home > MPE Home > Th. List > Mathboxes > omllaw5N | Structured version Visualization version GIF version |
Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 29390 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
omllaw5.b | ⊢ 𝐵 = (Base‘𝐾) |
omllaw5.j | ⊢ ∨ = (join‘𝐾) |
omllaw5.m | ⊢ ∧ = (meet‘𝐾) |
omllaw5.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
omllaw5N | ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ (𝑋 ∨ 𝑌))) = (𝑋 ∨ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML) | |
2 | simp2 1133 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
3 | omllat 36393 | . . . 4 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) | |
4 | omllaw5.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | omllaw5.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
6 | 4, 5 | latjcl 17661 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
7 | 3, 6 | syl3an1 1159 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
8 | 1, 2, 7 | 3jca 1124 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵)) |
9 | eqid 2821 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | 4, 9, 5 | latlej1 17670 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 𝑌)) |
11 | 3, 10 | syl3an1 1159 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 𝑌)) |
12 | omllaw5.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
13 | omllaw5.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
14 | 4, 9, 5, 12, 13 | omllaw2N 36395 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵) → (𝑋(le‘𝐾)(𝑋 ∨ 𝑌) → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ (𝑋 ∨ 𝑌))) = (𝑋 ∨ 𝑌))) |
15 | 8, 11, 14 | sylc 65 | 1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ (𝑋 ∨ 𝑌))) = (𝑋 ∨ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 lecple 16572 occoc 16573 joincjn 17554 meetcmee 17555 Latclat 17655 OMLcoml 36326 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-lat 17656 df-oposet 36327 df-ol 36329 df-oml 36330 |
This theorem is referenced by: (None) |
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