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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 35367 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
2 | 1 | adantr 474 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) |
3 | olop 35368 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
4 | olj0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | olj0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
6 | 4, 5 | op0cl 35338 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
7 | 3, 6 | syl 17 | . . . 4 ⊢ (𝐾 ∈ OL → 0 ∈ 𝐵) |
8 | 7 | adantr 474 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
9 | simpr 479 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | olj0.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
11 | 4, 10 | latjcom 17445 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = (𝑋 ∨ 0 )) |
12 | 2, 8, 9, 11 | syl3anc 1439 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = (𝑋 ∨ 0 )) |
13 | 4, 10, 5 | olj01 35379 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
14 | 12, 13 | eqtrd 2814 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 joincjn 17330 0.cp0 17423 Latclat 17431 OPcops 35326 OLcol 35328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-proset 17314 df-poset 17332 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-lat 17432 df-oposet 35330 df-ol 35332 |
This theorem is referenced by: atle 35590 athgt 35610 pmapjat1 36007 atmod1i1m 36012 llnexchb2lem 36022 lhp2at0 36186 lhpelim 36191 4atex2-0aOLDN 36232 cdleme2 36382 cdleme15b 36429 cdleme22cN 36496 cdleme22d 36497 cdleme35d 36606 cdlemeg46frv 36679 cdlemg2fv2 36754 cdlemg2m 36758 cdlemg10bALTN 36790 cdlemh2 36970 cdlemh 36971 cdlemk9 36993 cdlemk9bN 36994 dia2dimlem1 37218 |
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