Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > oldmj1 | Structured version Visualization version GIF version |
Description: De Morgan's law for join in an ortholattice. (chdmj1 29564 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 |
---|---|
oldmj1 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∨ 𝑌)) = (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oldmm1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | oldmm1.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | oldmm1.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
4 | oldmm1.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
5 | 1, 2, 3, 4 | oldmm4 36920 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) = (𝑋 ∨ 𝑌)) |
6 | 5 | fveq2d 6699 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)))) = ( ⊥ ‘(𝑋 ∨ 𝑌))) |
7 | olop 36914 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
8 | 7 | 3ad2ant1 1135 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP) |
9 | ollat 36913 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
10 | 9 | 3ad2ant1 1135 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) |
11 | 1, 4 | opoccl 36894 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
12 | 7, 11 | sylan 583 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
13 | 12 | 3adant3 1134 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
14 | 1, 4 | opoccl 36894 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
15 | 7, 14 | sylan 583 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
16 | 15 | 3adant2 1133 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
17 | 1, 3 | latmcl 17900 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)) ∈ 𝐵) |
18 | 10, 13, 16, 17 | syl3anc 1373 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)) ∈ 𝐵) |
19 | 1, 4 | opococ 36895 | . . 3 ⊢ ((𝐾 ∈ OP ∧ (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)) ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) |
20 | 8, 18, 19 | syl2anc 587 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) |
21 | 6, 20 | eqtr3d 2773 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∨ 𝑌)) = (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 occoc 16757 joincjn 17772 meetcmee 17773 Latclat 17891 OPcops 36872 OLcol 36874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-proset 17756 df-poset 17774 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-lat 17892 df-oposet 36876 df-ol 36878 |
This theorem is referenced by: oldmj2 36922 oldmj3 36923 cmtbr2N 36953 omlfh1N 36958 omlfh3N 36959 cvrexch 37120 poldmj1N 37628 lhpmod2i2 37738 lhpmod6i1 37739 djajN 38837 |
Copyright terms: Public domain | W3C validator |