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 37069 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 ∨ 𝑌)) = (𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌))))) |
7 | 6 | fveq2d 6677 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝐹‘(𝑋 ∨ 𝑌))) = (𝑁‘(𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))))) |
8 | simp1 1132 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ HL) | |
9 | hllat 36503 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
10 | 1, 2 | latjcl 17664 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
11 | 9, 10 | syl3an1 1159 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
12 | pmapocj.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
13 | 1, 12, 3, 5 | polpmapN 37052 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑌) ∈ 𝐵) → (𝑁‘(𝐹‘(𝑋 ∨ 𝑌))) = (𝐹‘( ⊥ ‘(𝑋 ∨ 𝑌)))) |
14 | 8, 11, 13 | syl2anc 586 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝐹‘(𝑋 ∨ 𝑌))) = (𝐹‘( ⊥ ‘(𝑋 ∨ 𝑌)))) |
15 | eqid 2824 | . . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
16 | 1, 15, 3 | pmapssat 36899 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾)) |
17 | 16 | 3adant3 1128 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾)) |
18 | 1, 15, 3 | pmapssat 36899 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) |
19 | 18 | 3adant2 1127 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) |
20 | 15, 4 | paddssat 36954 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) → ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) |
21 | 8, 17, 19, 20 | syl3anc 1367 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) |
22 | 15, 5 | 3polN 37056 | . . 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 586 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌))))) = (𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))) |
24 | 7, 14, 23 | 3eqtr3d 2867 | 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 1083 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 occoc 16576 joincjn 17557 meetcmee 17558 Latclat 17658 Atomscatm 36403 HLchlt 36490 pmapcpmap 36637 +𝑃cpadd 36935 ⊥𝑃cpolN 37042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-riotaBAD 36093 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-undef 7942 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-psubsp 36643 df-pmap 36644 df-padd 36936 df-polarityN 37043 |
This theorem is referenced by: (None) |
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