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Mirrors > Home > MPE Home > Th. List > Mathboxes > latmrot | Structured version Visualization version GIF version |
Description: Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.) |
Ref | Expression |
---|---|
olmass.b | ⊢ 𝐵 = (Base‘𝐾) |
olmass.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latmrot | ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = ((𝑍 ∧ 𝑋) ∧ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ollat 33518 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat) |
3 | simpr1 1060 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
4 | simpr2 1061 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
5 | olmass.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
6 | olmass.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
7 | 5, 6 | latmcl 16875 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵) |
8 | 2, 3, 4, 7 | syl3anc 1318 | . . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ 𝑌) ∈ 𝐵) |
9 | simpr3 1062 | . . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵) | |
10 | 5, 6 | latmcom 16898 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = (𝑍 ∧ (𝑋 ∧ 𝑌))) |
11 | 2, 8, 9, 10 | syl3anc 1318 | . 2 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = (𝑍 ∧ (𝑋 ∧ 𝑌))) |
12 | simpl 472 | . . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ OL) | |
13 | 5, 6 | latmassOLD 33534 | . . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑍 ∧ 𝑋) ∧ 𝑌) = (𝑍 ∧ (𝑋 ∧ 𝑌))) |
14 | 12, 9, 3, 4, 13 | syl13anc 1320 | . 2 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 ∧ 𝑋) ∧ 𝑌) = (𝑍 ∧ (𝑋 ∧ 𝑌))) |
15 | 11, 14 | eqtr4d 2647 | 1 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = ((𝑍 ∧ 𝑋) ∧ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 meetcmee 16768 Latclat 16868 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-lat 16869 df-oposet 33481 df-ol 33483 |
This theorem is referenced by: cdleme15b 34580 |
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