Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > polpmapN | Structured version Visualization version GIF version |
Description: The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polpmap.b | ⊢ 𝐵 = (Base‘𝐾) |
polpmap.o | ⊢ ⊥ = (oc‘𝐾) |
polpmap.m | ⊢ 𝑀 = (pmap‘𝐾) |
polpmap.p | ⊢ 𝑃 = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
polpmapN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑃‘(𝑀‘𝑋)) = (𝑀‘( ⊥ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | polpmap.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2823 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | polpmap.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
4 | 1, 2, 3 | pmapssat 36897 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ (Atoms‘𝐾)) |
5 | eqid 2823 | . . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
6 | polpmap.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
7 | polpmap.p | . . . 4 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
8 | 5, 6, 2, 3, 7 | polval2N 37044 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑀‘𝑋) ⊆ (Atoms‘𝐾)) → (𝑃‘(𝑀‘𝑋)) = (𝑀‘( ⊥ ‘((lub‘𝐾)‘(𝑀‘𝑋))))) |
9 | 4, 8 | syldan 593 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑃‘(𝑀‘𝑋)) = (𝑀‘( ⊥ ‘((lub‘𝐾)‘(𝑀‘𝑋))))) |
10 | eqid 2823 | . . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | 1, 10, 2, 3 | pmapval 36895 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑋}) |
12 | 11 | fveq2d 6676 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((lub‘𝐾)‘(𝑀‘𝑋)) = ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑋})) |
13 | hlomcmat 36503 | . . . . . 6 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) | |
14 | 1, 10, 5, 2 | atlatmstc 36457 | . . . . . 6 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑋}) = 𝑋) |
15 | 13, 14 | sylan 582 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((lub‘𝐾)‘{𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑋}) = 𝑋) |
16 | 12, 15 | eqtrd 2858 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((lub‘𝐾)‘(𝑀‘𝑋)) = 𝑋) |
17 | 16 | fveq2d 6676 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘((lub‘𝐾)‘(𝑀‘𝑋))) = ( ⊥ ‘𝑋)) |
18 | 17 | fveq2d 6676 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘( ⊥ ‘((lub‘𝐾)‘(𝑀‘𝑋)))) = (𝑀‘( ⊥ ‘𝑋))) |
19 | 9, 18 | eqtrd 2858 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑃‘(𝑀‘𝑋)) = (𝑀‘( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {crab 3144 ⊆ wss 3938 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 lecple 16574 occoc 16575 lubclub 17554 CLatccla 17719 OMLcoml 36313 Atomscatm 36401 AtLatcal 36402 HLchlt 36488 pmapcpmap 36635 ⊥𝑃cpolN 37040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-undef 7941 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-pmap 36642 df-polarityN 37041 |
This theorem is referenced by: 2polpmapN 37051 2polvalN 37052 3polN 37054 pmapj2N 37067 pmapocjN 37068 2polatN 37070 poml4N 37091 pmapojoinN 37106 |
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