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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2polpmapN | Structured version Visualization version GIF version |
Description: Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2polpmap.b | ⊢ 𝐵 = (Base‘𝐾) |
2polpmap.m | ⊢ 𝑀 = (pmap‘𝐾) |
2polpmap.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
2polpmapN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(𝑀‘𝑋))) = (𝑀‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2polpmap.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2725 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | 2polpmap.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
4 | 2polpmap.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
5 | 1, 2, 3, 4 | polpmapN 39441 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘(𝑀‘𝑋)) = (𝑀‘((oc‘𝐾)‘𝑋))) |
6 | 5 | fveq2d 6896 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(𝑀‘𝑋))) = ( ⊥ ‘(𝑀‘((oc‘𝐾)‘𝑋)))) |
7 | hlop 38890 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
8 | 1, 2 | opoccl 38722 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵) |
9 | 7, 8 | sylan 578 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵) |
10 | 1, 2, 3, 4 | polpmapN 39441 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵) → ( ⊥ ‘(𝑀‘((oc‘𝐾)‘𝑋))) = (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘𝑋)))) |
11 | 9, 10 | syldan 589 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘(𝑀‘((oc‘𝐾)‘𝑋))) = (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘𝑋)))) |
12 | 1, 2 | opococ 38723 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑋)) = 𝑋) |
13 | 7, 12 | sylan 578 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑋)) = 𝑋) |
14 | 13 | fveq2d 6896 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘𝑋))) = (𝑀‘𝑋)) |
15 | 6, 11, 14 | 3eqtrd 2769 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(𝑀‘𝑋))) = (𝑀‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6543 Basecbs 17179 occoc 17240 OPcops 38700 HLchlt 38878 pmapcpmap 39026 ⊥𝑃cpolN 39431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-oposet 38704 df-ol 38706 df-oml 38707 df-covers 38794 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 df-pmap 39033 df-polarityN 39432 |
This theorem is referenced by: pmapsubclN 39475 ispsubcl2N 39476 |
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