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Mathbox for Norm Megill |
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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 38596 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) |
3 | olop 38597 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
4 | olj0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | olj0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
6 | 4, 5 | op0cl 38567 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
7 | 3, 6 | syl 17 | . . . 4 ⊢ (𝐾 ∈ OL → 0 ∈ 𝐵) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
9 | simpr 484 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | olj0.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
11 | 4, 10 | latjcom 18412 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = (𝑋 ∨ 0 )) |
12 | 2, 8, 9, 11 | syl3anc 1368 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = (𝑋 ∨ 0 )) |
13 | 4, 10, 5 | olj01 38608 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
14 | 12, 13 | eqtrd 2766 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 joincjn 18276 0.cp0 18388 Latclat 18396 OPcops 38555 OLcol 38557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18260 df-poset 18278 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-lat 18397 df-oposet 38559 df-ol 38561 |
This theorem is referenced by: atle 38820 athgt 38840 pmapjat1 39237 atmod1i1m 39242 llnexchb2lem 39252 lhp2at0 39416 lhpelim 39421 4atex2-0aOLDN 39462 cdleme2 39612 cdleme15b 39659 cdleme22cN 39726 cdleme22d 39727 cdleme35d 39836 cdlemeg46frv 39909 cdlemg2fv2 39984 cdlemg2m 39988 cdlemg10bALTN 40020 cdlemh2 40200 cdlemh 40201 cdlemk9 40223 cdlemk9bN 40224 dia2dimlem1 40448 |
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