| 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 39179 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) |
| 3 | olop 39180 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
| 4 | olj0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
| 5 | olj0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
| 6 | 4, 5 | op0cl 39150 | . . . . 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 18382 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = (𝑋 ∨ 0 )) |
| 12 | 2, 8, 9, 11 | syl3anc 1373 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = (𝑋 ∨ 0 )) |
| 13 | 4, 10, 5 | olj01 39191 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
| 14 | 12, 13 | eqtrd 2764 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ‘cfv 6499 (class class class)co 7369 Basecbs 17155 joincjn 18248 0.cp0 18358 Latclat 18366 OPcops 39138 OLcol 39140 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-proset 18231 df-poset 18250 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-lat 18367 df-oposet 39142 df-ol 39144 |
| This theorem is referenced by: atle 39403 athgt 39423 pmapjat1 39820 atmod1i1m 39825 llnexchb2lem 39835 lhp2at0 39999 lhpelim 40004 4atex2-0aOLDN 40045 cdleme2 40195 cdleme15b 40242 cdleme22cN 40309 cdleme22d 40310 cdleme35d 40419 cdlemeg46frv 40492 cdlemg2fv2 40567 cdlemg2m 40571 cdlemg10bALTN 40603 cdlemh2 40783 cdlemh 40784 cdlemk9 40806 cdlemk9bN 40807 dia2dimlem1 41031 |
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