![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 38605 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 ∨ 𝑌)) = (𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌))))) |
7 | 6 | fveq2d 6882 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝐹‘(𝑋 ∨ 𝑌))) = (𝑁‘(𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))))) |
8 | simp1 1136 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ HL) | |
9 | hllat 38038 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
10 | 1, 2 | latjcl 18374 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
11 | 9, 10 | syl3an1 1163 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
12 | pmapocj.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
13 | 1, 12, 3, 5 | polpmapN 38588 | . . 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 2731 | . . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
16 | 1, 15, 3 | pmapssat 38435 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾)) |
17 | 16 | 3adant3 1132 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾)) |
18 | 1, 15, 3 | pmapssat 38435 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) |
19 | 18 | 3adant2 1131 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) |
20 | 15, 4 | paddssat 38490 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) → ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) |
21 | 8, 17, 19, 20 | syl3anc 1371 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) |
22 | 15, 5 | 3polN 38592 | . . 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 2779 | 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 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3944 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 occoc 17187 joincjn 18246 meetcmee 18247 Latclat 18366 Atomscatm 37938 HLchlt 38025 pmapcpmap 38173 +𝑃cpadd 38471 ⊥𝑃cpolN 38578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-proset 18230 df-poset 18248 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18367 df-clat 18434 df-oposet 37851 df-ol 37853 df-oml 37854 df-covers 37941 df-ats 37942 df-atl 37973 df-cvlat 37997 df-hlat 38026 df-psubsp 38179 df-pmap 38180 df-padd 38472 df-polarityN 38579 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |