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Mirrors > Home > MPE Home > Th. List > Mathboxes > llnmod1i2 | Structured version Visualization version GIF version |
Description: Version of modular law pmod1i 37518 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join 𝑃 ∨ 𝑄). (Contributed by NM, 16-Sep-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 |
---|---|
llnmod1i2 | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≤ 𝑌) → (𝑋 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑌)) = ((𝑋 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1192 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ HL) | |
2 | simpl2 1193 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵) | |
3 | simprl 771 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴) | |
4 | simprr 773 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴) | |
5 | atmod.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
6 | atmod.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
7 | atmod.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | eqid 2739 | . . . . . 6 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
9 | eqid 2739 | . . . . . 6 ⊢ (+𝑃‘𝐾) = (+𝑃‘𝐾) | |
10 | 5, 6, 7, 8, 9 | pmapjlln1 37525 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((pmap‘𝐾)‘(𝑋 ∨ (𝑃 ∨ 𝑄))) = (((pmap‘𝐾)‘𝑋)(+𝑃‘𝐾)((pmap‘𝐾)‘(𝑃 ∨ 𝑄)))) |
11 | 1, 2, 3, 4, 10 | syl13anc 1373 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((pmap‘𝐾)‘(𝑋 ∨ (𝑃 ∨ 𝑄))) = (((pmap‘𝐾)‘𝑋)(+𝑃‘𝐾)((pmap‘𝐾)‘(𝑃 ∨ 𝑄)))) |
12 | 1 | hllatd 37034 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat) |
13 | 5, 7 | atbase 36959 | . . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
14 | 3, 13 | syl 17 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐵) |
15 | 5, 7 | atbase 36959 | . . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
16 | 4, 15 | syl 17 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐵) |
17 | 5, 6 | latjcl 17790 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵) |
18 | 12, 14, 16, 17 | syl3anc 1372 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵) |
19 | simpl3 1194 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑌 ∈ 𝐵) | |
20 | atmod.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
21 | atmod.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
22 | 5, 20, 6, 21, 8, 9 | hlmod1i 37526 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ ((pmap‘𝐾)‘(𝑋 ∨ (𝑃 ∨ 𝑄))) = (((pmap‘𝐾)‘𝑋)(+𝑃‘𝐾)((pmap‘𝐾)‘(𝑃 ∨ 𝑄)))) → ((𝑋 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑌) = (𝑋 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑌)))) |
23 | 1, 2, 18, 19, 22 | syl13anc 1373 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≤ 𝑌 ∧ ((pmap‘𝐾)‘(𝑋 ∨ (𝑃 ∨ 𝑄))) = (((pmap‘𝐾)‘𝑋)(+𝑃‘𝐾)((pmap‘𝐾)‘(𝑃 ∨ 𝑄)))) → ((𝑋 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑌) = (𝑋 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑌)))) |
24 | 11, 23 | mpan2d 694 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ≤ 𝑌 → ((𝑋 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑌) = (𝑋 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑌)))) |
25 | 24 | 3impia 1118 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≤ 𝑌) → ((𝑋 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑌) = (𝑋 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑌))) |
26 | 25 | eqcomd 2745 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≤ 𝑌) → (𝑋 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑌)) = ((𝑋 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 class class class wbr 5040 ‘cfv 6350 (class class class)co 7183 Basecbs 16599 lecple 16688 joincjn 17683 meetcmee 17684 Latclat 17784 Atomscatm 36933 HLchlt 37020 pmapcpmap 37167 +𝑃cpadd 37465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-1st 7727 df-2nd 7728 df-proset 17667 df-poset 17685 df-plt 17697 df-lub 17713 df-glb 17714 df-join 17715 df-meet 17716 df-p0 17778 df-lat 17785 df-clat 17847 df-oposet 36846 df-ol 36848 df-oml 36849 df-covers 36936 df-ats 36937 df-atl 36968 df-cvlat 36992 df-hlat 37021 df-psubsp 37173 df-pmap 37174 df-padd 37466 |
This theorem is referenced by: llnmod2i2 37533 dalawlem12 37552 |
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