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Mathbox for Norm Megill |
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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 31410 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 1134 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML) | |
2 | simp2 1135 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
3 | omllat 38651 | . . . 4 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) | |
4 | omllaw5.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | omllaw5.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
6 | 4, 5 | latjcl 18422 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
7 | 3, 6 | syl3an1 1161 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
8 | 1, 2, 7 | 3jca 1126 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵)) |
9 | eqid 2727 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | 4, 9, 5 | latlej1 18431 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 𝑌)) |
11 | 3, 10 | syl3an1 1161 | . 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 38653 | . 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 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 Basecbs 17171 lecple 17231 occoc 17232 joincjn 18294 meetcmee 18295 Latclat 18414 OMLcoml 38584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-lub 18329 df-glb 18330 df-join 18331 df-meet 18332 df-lat 18415 df-oposet 38585 df-ol 38587 df-oml 38588 |
This theorem is referenced by: (None) |
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