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 35215 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 ∨ 𝑌)) = (𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌))))) |
7 | 6 | fveq2d 6195 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝐹‘(𝑋 ∨ 𝑌))) = (𝑁‘(𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))))) |
8 | simp1 1061 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ HL) | |
9 | hllat 34650 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
10 | 1, 2 | latjcl 17051 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
11 | 9, 10 | syl3an1 1359 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
12 | pmapocj.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
13 | 1, 12, 3, 5 | polpmapN 35198 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑌) ∈ 𝐵) → (𝑁‘(𝐹‘(𝑋 ∨ 𝑌))) = (𝐹‘( ⊥ ‘(𝑋 ∨ 𝑌)))) |
14 | 8, 11, 13 | syl2anc 693 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝐹‘(𝑋 ∨ 𝑌))) = (𝐹‘( ⊥ ‘(𝑋 ∨ 𝑌)))) |
15 | eqid 2622 | . . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
16 | 1, 15, 3 | pmapssat 35045 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾)) |
17 | 16 | 3adant3 1081 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾)) |
18 | 1, 15, 3 | pmapssat 35045 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) |
19 | 18 | 3adant2 1080 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) |
20 | 15, 4 | paddssat 35100 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) → ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) |
21 | 8, 17, 19, 20 | syl3anc 1326 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) |
22 | 15, 5 | 3polN 35202 | . . 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 693 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘(𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌))))) = (𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))) |
24 | 7, 14, 23 | 3eqtr3d 2664 | 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 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 occoc 15949 joincjn 16944 meetcmee 16945 Latclat 17045 Atomscatm 34550 HLchlt 34637 pmapcpmap 34783 +𝑃cpadd 35081 ⊥𝑃cpolN 35188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-undef 7399 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-polarityN 35189 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |