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Mathbox for Norm Megill |
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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 31459 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 38931 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) = (𝑋 ∨ 𝑌)) |
6 | 5 | fveq2d 6897 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)))) = ( ⊥ ‘(𝑋 ∨ 𝑌))) |
7 | olop 38925 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
8 | 7 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP) |
9 | ollat 38924 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
10 | 9 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) |
11 | 1, 4 | opoccl 38905 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
12 | 7, 11 | sylan 578 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
13 | 12 | 3adant3 1129 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
14 | 1, 4 | opoccl 38905 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
15 | 7, 14 | sylan 578 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
16 | 15 | 3adant2 1128 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
17 | 1, 3 | latmcl 18460 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)) ∈ 𝐵) |
18 | 10, 13, 16, 17 | syl3anc 1368 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)) ∈ 𝐵) |
19 | 1, 4 | opococ 38906 | . . 3 ⊢ ((𝐾 ∈ OP ∧ (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)) ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) |
20 | 8, 18, 19 | syl2anc 582 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) |
21 | 6, 20 | eqtr3d 2768 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∨ 𝑌)) = (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 occoc 17269 joincjn 18331 meetcmee 18332 Latclat 18451 OPcops 38883 OLcol 38885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-proset 18315 df-poset 18333 df-lub 18366 df-glb 18367 df-join 18368 df-meet 18369 df-lat 18452 df-oposet 38887 df-ol 38889 |
This theorem is referenced by: oldmj2 38933 oldmj3 38934 cmtbr2N 38964 omlfh1N 38969 omlfh3N 38970 cvrexch 39132 poldmj1N 39640 lhpmod2i2 39750 lhpmod6i1 39751 djajN 40849 |
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