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Mirrors > Home > MPE Home > Th. List > Mathboxes > olj01 | Structured version Visualization version GIF version |
Description: An ortholattice element joined with zero equals itself. (chj0 29276 analog.) (Contributed by NM, 19-Oct-2011.) |
Ref | Expression |
---|---|
olj0.b | ⊢ 𝐵 = (Base‘𝐾) |
olj0.j | ⊢ ∨ = (join‘𝐾) |
olj0.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
olj01 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olop 36352 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
2 | olj0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | olj0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
4 | 2, 3 | op0cl 36322 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝐾 ∈ OL → 0 ∈ 𝐵) |
6 | 5 | adantr 483 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
7 | eqid 2823 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | ollat 36351 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
9 | 8 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝐾 ∈ Lat) |
10 | olj0.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
11 | 2, 10 | latjcl 17663 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) ∈ 𝐵) |
12 | 8, 11 | syl3an1 1159 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) ∈ 𝐵) |
13 | simp2 1133 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
14 | 2, 7 | latref 17665 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
15 | 8, 14 | sylan 582 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
16 | 15 | 3adant3 1128 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
17 | 2, 7, 3 | op0le 36324 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
18 | 1, 17 | sylan 582 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
19 | 18 | 3adant3 1128 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
20 | simp3 1134 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 0 ∈ 𝐵) | |
21 | 2, 7, 10 | latjle12 17674 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋(le‘𝐾)𝑋 ∧ 0 (le‘𝐾)𝑋) ↔ (𝑋 ∨ 0 )(le‘𝐾)𝑋)) |
22 | 9, 13, 20, 13, 21 | syl13anc 1368 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → ((𝑋(le‘𝐾)𝑋 ∧ 0 (le‘𝐾)𝑋) ↔ (𝑋 ∨ 0 )(le‘𝐾)𝑋)) |
23 | 16, 19, 22 | mpbi2and 710 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 )(le‘𝐾)𝑋) |
24 | 2, 7, 10 | latlej1 17672 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 0 )) |
25 | 8, 24 | syl3an1 1159 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 0 )) |
26 | 2, 7, 9, 12, 13, 23, 25 | latasymd 17669 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
27 | 6, 26 | mpd3an3 1458 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 lecple 16574 joincjn 17556 0.cp0 17649 Latclat 17657 OPcops 36310 OLcol 36312 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-proset 17540 df-poset 17558 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-lat 17658 df-oposet 36314 df-ol 36316 |
This theorem is referenced by: olj02 36364 olm11 36365 omllaw3 36383 omlspjN 36399 2at0mat0 36663 lhp2at0nle 37173 lhple 37180 cdlemc6 37334 cdleme3c 37368 cdleme7e 37385 cdlemednpq 37437 cdlemefrs29pre00 37533 cdlemefrs29bpre0 37534 cdlemefrs29cpre1 37536 cdleme32fva 37575 cdleme42ke 37623 cdlemg12e 37785 cdlemg31d 37838 trljco 37878 cdlemkid2 38062 dihvalcqat 38377 dihmeetlem7N 38448 dihjatc1 38449 djh01 38550 |
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