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Mirrors > Home > MPE Home > Th. List > Mathboxes > olj02 | Structured version Visualization version GIF version |
Description: An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.) |
Ref | Expression |
---|---|
olj0.b | ⊢ 𝐵 = (Base‘𝐾) |
olj0.j | ⊢ ∨ = (join‘𝐾) |
olj0.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
olj02 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ollat 36376 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) |
3 | olop 36377 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
4 | olj0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | olj0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
6 | 4, 5 | op0cl 36347 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
7 | 3, 6 | syl 17 | . . . 4 ⊢ (𝐾 ∈ OL → 0 ∈ 𝐵) |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
9 | simpr 487 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | olj0.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
11 | 4, 10 | latjcom 17647 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = (𝑋 ∨ 0 )) |
12 | 2, 8, 9, 11 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = (𝑋 ∨ 0 )) |
13 | 4, 10, 5 | olj01 36388 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
14 | 12, 13 | eqtrd 2855 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6336 (class class class)co 7137 Basecbs 16461 joincjn 17532 0.cp0 17625 Latclat 17633 OPcops 36335 OLcol 36337 |
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 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 |
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 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-id 5441 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 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-proset 17516 df-poset 17534 df-lub 17562 df-glb 17563 df-join 17564 df-meet 17565 df-p0 17627 df-lat 17634 df-oposet 36339 df-ol 36341 |
This theorem is referenced by: atle 36599 athgt 36619 pmapjat1 37016 atmod1i1m 37021 llnexchb2lem 37031 lhp2at0 37195 lhpelim 37200 4atex2-0aOLDN 37241 cdleme2 37391 cdleme15b 37438 cdleme22cN 37505 cdleme22d 37506 cdleme35d 37615 cdlemeg46frv 37688 cdlemg2fv2 37763 cdlemg2m 37767 cdlemg10bALTN 37799 cdlemh2 37979 cdlemh 37980 cdlemk9 38002 cdlemk9bN 38003 dia2dimlem1 38227 |
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