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Mirrors > Home > MPE Home > Th. List > Mathboxes > oldmm3N | Structured version Visualization version GIF version |
Description: De Morgan's law for meet in an ortholattice. (chdmm3 29306 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
oldmm1.b | ⊢ 𝐵 = (Base‘𝐾) |
oldmm1.j | ⊢ ∨ = (join‘𝐾) |
oldmm1.m | ⊢ ∧ = (meet‘𝐾) |
oldmm1.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
oldmm3N | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olop 36352 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
2 | 1 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP) |
3 | simp3 1134 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
4 | oldmm1.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | oldmm1.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
6 | 4, 5 | opoccl 36332 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
7 | 2, 3, 6 | syl2anc 586 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
8 | oldmm1.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
9 | oldmm1.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
10 | 4, 8, 9, 5 | oldmm1 36355 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘( ⊥ ‘𝑌)))) |
11 | 7, 10 | syld3an3 1405 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘( ⊥ ‘𝑌)))) |
12 | 4, 5 | opococ 36333 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
13 | 2, 3, 12 | syl2anc 586 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
14 | 13 | oveq2d 7174 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ ( ⊥ ‘( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌)) |
15 | 11, 14 | eqtrd 2858 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 occoc 16575 joincjn 17556 meetcmee 17557 OPcops 36310 OLcol 36312 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-proset 17540 df-poset 17558 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-lat 17658 df-oposet 36314 df-ol 36316 |
This theorem is referenced by: cmtbr3N 36392 lhprelat3N 37178 |
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