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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 28846 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 6918 | . . . . . 6 ⊢ ((𝑌 ∧ ( ⊥ ‘𝑋)) = 0 → (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))) = (𝑋(join‘𝐾) 0 )) | |
2 | 1 | adantl 475 | . . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))) = (𝑋(join‘𝐾) 0 )) |
3 | omlol 35314 | . . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) | |
4 | omllaw3.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾) | |
5 | eqid 2825 | . . . . . . . . 9 ⊢ (join‘𝐾) = (join‘𝐾) | |
6 | omllaw3.z | . . . . . . . . 9 ⊢ 0 = (0.‘𝐾) | |
7 | 4, 5, 6 | olj01 35299 | . . . . . . . 8 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋) |
8 | 3, 7 | sylan 575 | . . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋) |
9 | 8 | 3adant3 1166 | . . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋) |
10 | 9 | adantr 474 | . . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → (𝑋(join‘𝐾) 0 ) = 𝑋) |
11 | 2, 10 | eqtr2d 2862 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))) |
12 | 11 | adantrl 707 | . . 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 35317 | . . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))))) |
17 | 16 | imp 397 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))) |
18 | 17 | adantrr 708 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 )) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))) |
19 | 12, 18 | eqtr4d 2864 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 )) → 𝑋 = 𝑌) |
20 | 19 | ex 403 | 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 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 lecple 16319 occoc 16320 joincjn 17304 meetcmee 17305 0.cp0 17397 OLcol 35248 OMLcoml 35249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-proset 17288 df-poset 17306 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-lat 17406 df-oposet 35250 df-ol 35252 df-oml 35253 |
This theorem is referenced by: omlfh1N 35332 atlatmstc 35393 |
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