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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 36348 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) |
3 | olop 36349 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
4 | olj0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | olj0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
6 | 4, 5 | op0cl 36319 | . . . . 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 17668 | . . 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 36360 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
14 | 12, 13 | eqtrd 2856 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 joincjn 17553 0.cp0 17646 Latclat 17654 OPcops 36307 OLcol 36309 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-proset 17537 df-poset 17555 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-lat 17655 df-oposet 36311 df-ol 36313 |
This theorem is referenced by: atle 36571 athgt 36591 pmapjat1 36988 atmod1i1m 36993 llnexchb2lem 37003 lhp2at0 37167 lhpelim 37172 4atex2-0aOLDN 37213 cdleme2 37363 cdleme15b 37410 cdleme22cN 37477 cdleme22d 37478 cdleme35d 37587 cdlemeg46frv 37660 cdlemg2fv2 37735 cdlemg2m 37739 cdlemg10bALTN 37771 cdlemh2 37951 cdlemh 37952 cdlemk9 37974 cdlemk9bN 37975 dia2dimlem1 38199 |
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