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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapocjN | Structured version Visualization version GIF version |
Description: The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pmapocj.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapocj.j | ⊢ ∨ = (join‘𝐾) |
pmapocj.m | ⊢ ∧ = (meet‘𝐾) |
pmapocj.o | ⊢ ⊥ = (oc‘𝐾) |
pmapocj.f | ⊢ 𝐹 = (pmap‘𝐾) |
pmapocj.p | ⊢ + = (+𝑃‘𝐾) |
pmapocj.r | ⊢ 𝑁 = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
pmapocjN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘( ⊥ ‘(𝑋 ∨ 𝑌))) = (𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapocj.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | pmapocj.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | pmapocj.f | . . . 4 ⊢ 𝐹 = (pmap‘𝐾) | |
4 | pmapocj.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
5 | pmapocj.r | . . . 4 ⊢ 𝑁 = (⊥𝑃‘𝐾) | |
6 | 1, 2, 3, 4, 5 | pmapj2N 39912 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 ∨ 𝑌)) = (𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌))))) |
7 | 6 | fveq2d 6911 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝐹‘(𝑋 ∨ 𝑌))) = (𝑁‘(𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))))) |
8 | simp1 1135 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ HL) | |
9 | hllat 39345 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
10 | 1, 2 | latjcl 18497 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
11 | 9, 10 | syl3an1 1162 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
12 | pmapocj.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
13 | 1, 12, 3, 5 | polpmapN 39895 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑌) ∈ 𝐵) → (𝑁‘(𝐹‘(𝑋 ∨ 𝑌))) = (𝐹‘( ⊥ ‘(𝑋 ∨ 𝑌)))) |
14 | 8, 11, 13 | syl2anc 584 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝐹‘(𝑋 ∨ 𝑌))) = (𝐹‘( ⊥ ‘(𝑋 ∨ 𝑌)))) |
15 | eqid 2735 | . . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
16 | 1, 15, 3 | pmapssat 39742 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾)) |
17 | 16 | 3adant3 1131 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾)) |
18 | 1, 15, 3 | pmapssat 39742 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) |
19 | 18 | 3adant2 1130 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) |
20 | 15, 4 | paddssat 39797 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) → ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) |
21 | 8, 17, 19, 20 | syl3anc 1370 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) |
22 | 15, 5 | 3polN 39899 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) → (𝑁‘(𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌))))) = (𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))) |
23 | 8, 21, 22 | syl2anc 584 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌))))) = (𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))) |
24 | 7, 14, 23 | 3eqtr3d 2783 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘( ⊥ ‘(𝑋 ∨ 𝑌))) = (𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 occoc 17306 joincjn 18369 meetcmee 18370 Latclat 18489 Atomscatm 39245 HLchlt 39332 pmapcpmap 39480 +𝑃cpadd 39778 ⊥𝑃cpolN 39885 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-polarityN 39886 |
This theorem is referenced by: (None) |
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