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Mirrors > Home > MPE Home > Th. List > Mathboxes > atmod1i1 | Structured version Visualization version GIF version |
Description: Version of modular law pmod1i 39547 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 11-May-2012.) (Revised by Mario Carneiro, 10-May-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 |
---|---|
atmod1i1 | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑌) → (𝑃 ∨ (𝑋 ∧ 𝑌)) = ((𝑃 ∨ 𝑋) ∧ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐾 ∈ HL) | |
2 | simpr2 1192 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
3 | simpr1 1191 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑃 ∈ 𝐴) | |
4 | atmod.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | atmod.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
6 | atmod.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | eqid 2726 | . . . . . 6 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
8 | eqid 2726 | . . . . . 6 ⊢ (+𝑃‘𝐾) = (+𝑃‘𝐾) | |
9 | 4, 5, 6, 7, 8 | pmapjat2 39553 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((pmap‘𝐾)‘(𝑃 ∨ 𝑋)) = (((pmap‘𝐾)‘𝑃)(+𝑃‘𝐾)((pmap‘𝐾)‘𝑋))) |
10 | 1, 2, 3, 9 | syl3anc 1368 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((pmap‘𝐾)‘(𝑃 ∨ 𝑋)) = (((pmap‘𝐾)‘𝑃)(+𝑃‘𝐾)((pmap‘𝐾)‘𝑋))) |
11 | 4, 6 | atbase 38987 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
12 | atmod.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
13 | atmod.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
14 | 4, 12, 5, 13, 7, 8 | hlmod1i 39555 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑃 ≤ 𝑌 ∧ ((pmap‘𝐾)‘(𝑃 ∨ 𝑋)) = (((pmap‘𝐾)‘𝑃)(+𝑃‘𝐾)((pmap‘𝐾)‘𝑋))) → ((𝑃 ∨ 𝑋) ∧ 𝑌) = (𝑃 ∨ (𝑋 ∧ 𝑌)))) |
15 | 11, 14 | syl3anr1 1413 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑃 ≤ 𝑌 ∧ ((pmap‘𝐾)‘(𝑃 ∨ 𝑋)) = (((pmap‘𝐾)‘𝑃)(+𝑃‘𝐾)((pmap‘𝐾)‘𝑋))) → ((𝑃 ∨ 𝑋) ∧ 𝑌) = (𝑃 ∨ (𝑋 ∧ 𝑌)))) |
16 | 10, 15 | mpan2d 692 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑃 ≤ 𝑌 → ((𝑃 ∨ 𝑋) ∧ 𝑌) = (𝑃 ∨ (𝑋 ∧ 𝑌)))) |
17 | 16 | 3impia 1114 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑌) → ((𝑃 ∨ 𝑋) ∧ 𝑌) = (𝑃 ∨ (𝑋 ∧ 𝑌))) |
18 | 17 | eqcomd 2732 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑌) → (𝑃 ∨ (𝑋 ∧ 𝑌)) = ((𝑃 ∨ 𝑋) ∧ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 lecple 17273 joincjn 18336 meetcmee 18337 Atomscatm 38961 HLchlt 39048 pmapcpmap 39196 +𝑃cpadd 39494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-lat 18457 df-clat 18524 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-psubsp 39202 df-pmap 39203 df-padd 39495 |
This theorem is referenced by: atmod1i1m 39557 atmod2i1 39560 atmod3i1 39563 atmod4i1 39565 dalawlem6 39575 dalawlem11 39580 dalawlem12 39581 cdleme11g 39964 cdlemednpq 39998 cdleme20c 40010 cdleme22e 40043 cdleme22eALTN 40044 cdleme35c 40150 |
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