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Mirrors > Home > MPE Home > Th. List > Mathboxes > olm01 | Structured version Visualization version GIF version |
Description: Meet with lattice zero is zero. (chm0 27734 analog.) (Contributed by NM, 8-Nov-2011.) |
Ref | Expression |
---|---|
olm0.b | ⊢ 𝐵 = (Base‘𝐾) |
olm0.m | ⊢ ∧ = (meet‘𝐾) |
olm0.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
olm01 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olm0.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2610 | . 2 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | ollat 33518 | . . 3 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
4 | 3 | adantr 480 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) |
5 | simpr 476 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
6 | olop 33519 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) |
8 | olm0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
9 | 1, 8 | op0cl 33489 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
10 | 7, 9 | syl 17 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
11 | olm0.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
12 | 1, 11 | latmcl 16875 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∧ 0 ) ∈ 𝐵) |
13 | 4, 5, 10, 12 | syl3anc 1318 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 ) ∈ 𝐵) |
14 | 1, 2, 11 | latmle2 16900 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∧ 0 )(le‘𝐾) 0 ) |
15 | 4, 5, 10, 14 | syl3anc 1318 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 )(le‘𝐾) 0 ) |
16 | 1, 2, 8 | op0le 33491 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
17 | 6, 16 | sylan 487 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
18 | 1, 2 | latref 16876 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 0 ∈ 𝐵) → 0 (le‘𝐾) 0 ) |
19 | 4, 10, 18 | syl2anc 691 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾) 0 ) |
20 | 1, 2, 11 | latlem12 16901 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵)) → (( 0 (le‘𝐾)𝑋 ∧ 0 (le‘𝐾) 0 ) ↔ 0 (le‘𝐾)(𝑋 ∧ 0 ))) |
21 | 4, 10, 5, 10, 20 | syl13anc 1320 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (( 0 (le‘𝐾)𝑋 ∧ 0 (le‘𝐾) 0 ) ↔ 0 (le‘𝐾)(𝑋 ∧ 0 ))) |
22 | 17, 19, 21 | mpbi2and 958 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)(𝑋 ∧ 0 )) |
23 | 1, 2, 4, 13, 10, 15, 22 | latasymd 16880 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 meetcmee 16768 0.cp0 16860 Latclat 16868 OPcops 33477 OLcol 33479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-preset 16751 df-poset 16769 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-lat 16869 df-oposet 33481 df-ol 33483 |
This theorem is referenced by: olm02 33542 omlfh1N 33563 |
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