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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatj4 | Structured version Visualization version GIF version |
Description: Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 17762 for atoms. (Contributed by NM, 9-Aug-2012.) |
Ref | Expression |
---|---|
hlatjcom.j | ⊢ ∨ = (join‘𝐾) |
hlatjcom.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
hlatj4 | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑃 ∨ 𝑅) ∨ (𝑄 ∨ 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 36924 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | 1 | 3ad2ant1 1131 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ Lat) |
3 | simp2l 1197 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑃 ∈ 𝐴) | |
4 | eqid 2759 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | hlatjcom.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 4, 5 | atbase 36850 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
7 | 3, 6 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑃 ∈ (Base‘𝐾)) |
8 | simp2r 1198 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ∈ 𝐴) | |
9 | 4, 5 | atbase 36850 | . . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
10 | 8, 9 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ∈ (Base‘𝐾)) |
11 | simp3l 1199 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑅 ∈ 𝐴) | |
12 | 4, 5 | atbase 36850 | . . 3 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
13 | 11, 12 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑅 ∈ (Base‘𝐾)) |
14 | simp3r 1200 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑆 ∈ 𝐴) | |
15 | 4, 5 | atbase 36850 | . . 3 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾)) |
16 | 14, 15 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑆 ∈ (Base‘𝐾)) |
17 | hlatjcom.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
18 | 4, 17 | latj4 17762 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑃 ∨ 𝑅) ∨ (𝑄 ∨ 𝑆))) |
19 | 2, 7, 10, 13, 16, 18 | syl122anc 1377 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑃 ∨ 𝑅) ∨ (𝑄 ∨ 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ‘cfv 6328 (class class class)co 7143 Basecbs 16526 joincjn 17605 Latclat 17706 Atomscatm 36824 HLchlt 36911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-proset 17589 df-poset 17607 df-lub 17635 df-glb 17636 df-join 17637 df-meet 17638 df-lat 17707 df-ats 36828 df-atl 36859 df-cvlat 36883 df-hlat 36912 |
This theorem is referenced by: 4atlem4b 37161 4atlem11 37170 dalem2 37222 dalem23 37257 cdleme16c 37841 |
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