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Mirrors > Home > MPE Home > Th. List > Mathboxes > omllaw3 | Structured version Visualization version GIF version |
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 28423 analog.) (Contributed by NM, 19-Oct-2011.) |
Ref | Expression |
---|---|
omllaw3.b | ⊢ 𝐵 = (Base‘𝐾) |
omllaw3.l | ⊢ ≤ = (le‘𝐾) |
omllaw3.m | ⊢ ∧ = (meet‘𝐾) |
omllaw3.o | ⊢ ⊥ = (oc‘𝐾) |
omllaw3.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
omllaw3 | ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6698 | . . . . . 6 ⊢ ((𝑌 ∧ ( ⊥ ‘𝑋)) = 0 → (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))) = (𝑋(join‘𝐾) 0 )) | |
2 | 1 | adantl 481 | . . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))) = (𝑋(join‘𝐾) 0 )) |
3 | omlol 34845 | . . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) | |
4 | omllaw3.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾) | |
5 | eqid 2651 | . . . . . . . . 9 ⊢ (join‘𝐾) = (join‘𝐾) | |
6 | omllaw3.z | . . . . . . . . 9 ⊢ 0 = (0.‘𝐾) | |
7 | 4, 5, 6 | olj01 34830 | . . . . . . . 8 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋) |
8 | 3, 7 | sylan 487 | . . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋) |
9 | 8 | 3adant3 1101 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋) |
10 | 9 | adantr 480 | . . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → (𝑋(join‘𝐾) 0 ) = 𝑋) |
11 | 2, 10 | eqtr2d 2686 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))) |
12 | 11 | adantrl 752 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 )) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))) |
13 | omllaw3.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
14 | omllaw3.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
15 | omllaw3.o | . . . . . 6 ⊢ ⊥ = (oc‘𝐾) | |
16 | 4, 13, 5, 14, 15 | omllaw 34848 | . . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))))) |
17 | 16 | imp 444 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))) |
18 | 17 | adantrr 753 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 )) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))) |
19 | 12, 18 | eqtr4d 2688 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 )) → 𝑋 = 𝑌) |
20 | 19 | ex 449 | 1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 lecple 15995 occoc 15996 joincjn 16991 meetcmee 16992 0.cp0 17084 OLcol 34779 OMLcoml 34780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-preset 16975 df-poset 16993 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-lat 17093 df-oposet 34781 df-ol 34783 df-oml 34784 |
This theorem is referenced by: omlfh1N 34863 atlatmstc 34924 |
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