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Mirrors > Home > MPE Home > Th. List > Mathboxes > atmod4i2 | Structured version Visualization version GIF version |
Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-Mar-2013.) |
Ref | Expression |
---|---|
atmod.b | ⊢ 𝐵 = (Base‘𝐾) |
atmod.l | ⊢ ≤ = (le‘𝐾) |
atmod.j | ⊢ ∨ = (join‘𝐾) |
atmod.m | ⊢ ∧ = (meet‘𝐾) |
atmod.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atmod4i2 | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((𝑃 ∧ 𝑌) ∨ 𝑋) = ((𝑃 ∨ 𝑋) ∧ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 36493 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | 1 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ Lat) |
3 | simp21 1202 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑃 ∈ 𝐴) | |
4 | atmod.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | atmod.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 4, 5 | atbase 36419 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
7 | 3, 6 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑃 ∈ 𝐵) |
8 | simp23 1204 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵) | |
9 | atmod.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
10 | 4, 9 | latmcl 17656 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃 ∧ 𝑌) ∈ 𝐵) |
11 | 2, 7, 8, 10 | syl3anc 1367 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑃 ∧ 𝑌) ∈ 𝐵) |
12 | simp22 1203 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵) | |
13 | atmod.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
14 | 4, 13 | latjcom 17663 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∧ 𝑌) ∨ 𝑋) = (𝑋 ∨ (𝑃 ∧ 𝑌))) |
15 | 2, 11, 12, 14 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((𝑃 ∧ 𝑌) ∨ 𝑋) = (𝑋 ∨ (𝑃 ∧ 𝑌))) |
16 | atmod.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
17 | 4, 16, 13, 9, 5 | atmod1i2 36989 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 ∨ (𝑃 ∧ 𝑌)) = ((𝑋 ∨ 𝑃) ∧ 𝑌)) |
18 | 4, 13 | latjcom 17663 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∨ 𝑃) = (𝑃 ∨ 𝑋)) |
19 | 2, 12, 7, 18 | syl3anc 1367 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 ∨ 𝑃) = (𝑃 ∨ 𝑋)) |
20 | 19 | oveq1d 7165 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((𝑋 ∨ 𝑃) ∧ 𝑌) = ((𝑃 ∨ 𝑋) ∧ 𝑌)) |
21 | 15, 17, 20 | 3eqtrd 2860 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((𝑃 ∧ 𝑌) ∨ 𝑋) = ((𝑃 ∨ 𝑋) ∧ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 lecple 16566 joincjn 17548 meetcmee 17549 Latclat 17649 Atomscatm 36393 HLchlt 36480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-psubsp 36633 df-pmap 36634 df-padd 36926 |
This theorem is referenced by: lhp2at0 37162 lhpelim 37167 cdleme2 37358 cdleme35d 37582 cdlemeg46frv 37655 cdlemg2fv2 37730 cdlemg2m 37734 cdlemg10bALTN 37766 cdlemh2 37946 cdlemk9 37969 cdlemk9bN 37970 |
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