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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapjat2 | Structured version Visualization version GIF version |
Description: The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.) |
Ref | Expression |
---|---|
pmapjat.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapjat.j | ⊢ ∨ = (join‘𝐾) |
pmapjat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pmapjat.m | ⊢ 𝑀 = (pmap‘𝐾) |
pmapjat.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
pmapjat2 | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = ((𝑀‘𝑄) + (𝑀‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapjat.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | pmapjat.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | pmapjat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | pmapjat.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
5 | pmapjat.p | . . 3 ⊢ + = (+𝑃‘𝐾) | |
6 | 1, 2, 3, 4, 5 | pmapjat1 37414 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑋 ∨ 𝑄)) = ((𝑀‘𝑋) + (𝑀‘𝑄))) |
7 | hllat 36924 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
8 | 7 | 3ad2ant1 1131 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Lat) |
9 | 1, 3 | atbase 36850 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
10 | 9 | 3ad2ant3 1133 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ 𝐵) |
11 | simp2 1135 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
12 | 1, 2 | latjcom 17720 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄)) |
13 | 8, 10, 11, 12 | syl3anc 1369 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄)) |
14 | 13 | fveq2d 6655 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = (𝑀‘(𝑋 ∨ 𝑄))) |
15 | simp1 1134 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ HL) | |
16 | 1, 3, 4 | pmapssat 37320 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐵) → (𝑀‘𝑄) ⊆ 𝐴) |
17 | 15, 10, 16 | syl2anc 588 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘𝑄) ⊆ 𝐴) |
18 | 1, 3, 4 | pmapssat 37320 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐴) |
19 | 18 | 3adant3 1130 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘𝑋) ⊆ 𝐴) |
20 | 3, 5 | paddcom 37374 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑀‘𝑄) ⊆ 𝐴 ∧ (𝑀‘𝑋) ⊆ 𝐴) → ((𝑀‘𝑄) + (𝑀‘𝑋)) = ((𝑀‘𝑋) + (𝑀‘𝑄))) |
21 | 8, 17, 19, 20 | syl3anc 1369 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → ((𝑀‘𝑄) + (𝑀‘𝑋)) = ((𝑀‘𝑋) + (𝑀‘𝑄))) |
22 | 6, 14, 21 | 3eqtr4d 2804 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = ((𝑀‘𝑄) + (𝑀‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ⊆ wss 3854 ‘cfv 6328 (class class class)co 7143 Basecbs 16526 joincjn 17605 Latclat 17706 Atomscatm 36824 HLchlt 36911 pmapcpmap 37058 +𝑃cpadd 37356 |
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-fal 1552 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-mpo 7148 df-1st 7686 df-2nd 7687 df-proset 17589 df-poset 17607 df-plt 17619 df-lub 17635 df-glb 17636 df-join 17637 df-meet 17638 df-p0 17700 df-lat 17707 df-clat 17769 df-oposet 36737 df-ol 36739 df-oml 36740 df-covers 36827 df-ats 36828 df-atl 36859 df-cvlat 36883 df-hlat 36912 df-pmap 37065 df-padd 37357 |
This theorem is referenced by: atmod1i1 37418 |
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