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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 29161 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 1116 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML) | |
2 | simp2 1117 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
3 | omllat 35771 | . . . 4 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) | |
4 | omllaw5.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | omllaw5.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
6 | 4, 5 | latjcl 17509 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
7 | 3, 6 | syl3an1 1143 | . . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
8 | 1, 2, 7 | 3jca 1108 | . 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵)) |
9 | eqid 2772 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | 4, 9, 5 | latlej1 17518 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 𝑌)) |
11 | 3, 10 | syl3an1 1143 | . 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 35773 | . 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 1068 = wceq 1507 ∈ wcel 2048 class class class wbr 4923 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 lecple 16418 occoc 16419 joincjn 17402 meetcmee 17403 Latclat 17503 OMLcoml 35704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-lub 17432 df-glb 17433 df-join 17434 df-meet 17435 df-lat 17504 df-oposet 35705 df-ol 35707 df-oml 35708 |
This theorem is referenced by: (None) |
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