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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 30957 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 7420 | . . . . . 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 38414 | . . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) | |
4 | omllaw3.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾) | |
5 | eqid 2731 | . . . . . . . . 9 ⊢ (join‘𝐾) = (join‘𝐾) | |
6 | omllaw3.z | . . . . . . . . 9 ⊢ 0 = (0.‘𝐾) | |
7 | 4, 5, 6 | olj01 38399 | . . . . . . . 8 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋) |
8 | 3, 7 | sylan 579 | . . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋) |
9 | 8 | 3adant3 1131 | . . . . . 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 2772 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))) |
12 | 11 | adantrl 713 | . . 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 38417 | . . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))))) |
17 | 16 | imp 406 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))) |
18 | 17 | adantrr 714 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 )) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))) |
19 | 12, 18 | eqtr4d 2774 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 )) → 𝑋 = 𝑌) |
20 | 19 | ex 412 | 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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 Basecbs 17149 lecple 17209 occoc 17210 joincjn 18269 meetcmee 18270 0.cp0 18381 OLcol 38348 OMLcoml 38349 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-proset 18253 df-poset 18271 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-lat 18390 df-oposet 38350 df-ol 38352 df-oml 38353 |
This theorem is referenced by: omlfh1N 38432 atlatmstc 38493 |
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