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Mirrors > Home > MPE Home > Th. List > Mathboxes > oldmj2 | Structured version Visualization version GIF version |
Description: De Morgan's law for join in an ortholattice. (chdmj2 29301 analog.) (Contributed by NM, 7-Nov-2011.) |
Ref | Expression |
---|---|
oldmm1.b | ⊢ 𝐵 = (Base‘𝐾) |
oldmm1.j | ⊢ ∨ = (join‘𝐾) |
oldmm1.m | ⊢ ∧ = (meet‘𝐾) |
oldmm1.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
oldmj2 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ ( ⊥ ‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olop 36344 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
2 | oldmm1.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
3 | oldmm1.o | . . . . . 6 ⊢ ⊥ = (oc‘𝐾) | |
4 | 2, 3 | opoccl 36324 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
5 | 1, 4 | sylan 582 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
6 | 5 | 3adant3 1128 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
7 | oldmm1.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
8 | oldmm1.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
9 | 2, 7, 8, 3 | oldmj1 36351 | . . 3 ⊢ ((𝐾 ∈ OL ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∨ 𝑌)) = (( ⊥ ‘( ⊥ ‘𝑋)) ∧ ( ⊥ ‘𝑌))) |
10 | 6, 9 | syld3an2 1407 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∨ 𝑌)) = (( ⊥ ‘( ⊥ ‘𝑋)) ∧ ( ⊥ ‘𝑌))) |
11 | 2, 3 | opococ 36325 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
12 | 1, 11 | sylan 582 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
13 | 12 | 3adant3 1128 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
14 | 13 | oveq1d 7165 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑋)) ∧ ( ⊥ ‘𝑌)) = (𝑋 ∧ ( ⊥ ‘𝑌))) |
15 | 10, 14 | eqtrd 2856 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ ( ⊥ ‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 occoc 16567 joincjn 17548 meetcmee 17549 OPcops 36302 OLcol 36304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-proset 17532 df-poset 17550 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-lat 17650 df-oposet 36306 df-ol 36308 |
This theorem is referenced by: oldmj4 36354 latmassOLD 36359 cmtcomlemN 36378 |
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