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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > oldmm4 | Structured version Visualization version GIF version |
Description: De Morgan's law for meet in an ortholattice. (chdmm4 28984 analog.) (Contributed by NM, 6-Nov-2011.) |
Ref | Expression |
---|---|
oldmm1.b | ⊢ 𝐵 = (Base‘𝐾) |
oldmm1.j | ⊢ ∨ = (join‘𝐾) |
oldmm1.m | ⊢ ∧ = (meet‘𝐾) |
oldmm1.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
oldmm4 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) = (𝑋 ∨ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olop 35831 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
2 | oldmm1.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
3 | oldmm1.o | . . . . . 6 ⊢ ⊥ = (oc‘𝐾) | |
4 | 2, 3 | opoccl 35811 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
5 | 1, 4 | sylan 580 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
6 | 5 | 3adant2 1122 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
7 | oldmm1.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
8 | oldmm1.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
9 | 2, 7, 8, 3 | oldmm2 35835 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) = (𝑋 ∨ ( ⊥ ‘( ⊥ ‘𝑌)))) |
10 | 6, 9 | syld3an3 1400 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) = (𝑋 ∨ ( ⊥ ‘( ⊥ ‘𝑌)))) |
11 | 2, 3 | opococ 35812 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
12 | 1, 11 | sylan 580 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
13 | 12 | 3adant2 1122 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
14 | 13 | oveq2d 7023 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘( ⊥ ‘𝑌))) = (𝑋 ∨ 𝑌)) |
15 | 10, 14 | eqtrd 2829 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) = (𝑋 ∨ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1078 = wceq 1520 ∈ wcel 2079 ‘cfv 6217 (class class class)co 7007 Basecbs 16300 occoc 16390 joincjn 17371 meetcmee 17372 OPcops 35789 OLcol 35791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-proset 17355 df-poset 17373 df-lub 17401 df-glb 17402 df-join 17403 df-meet 17404 df-lat 17473 df-oposet 35793 df-ol 35795 |
This theorem is referenced by: oldmj1 35838 omlfh3N 35876 pmapj2N 36546 djhlj 38018 |
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