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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 2726 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | 2polpmap.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
4 | 2polpmap.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
5 | 1, 2, 3, 4 | polpmapN 39296 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘(𝑀‘𝑋)) = (𝑀‘((oc‘𝐾)‘𝑋))) |
6 | 5 | fveq2d 6889 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(𝑀‘𝑋))) = ( ⊥ ‘(𝑀‘((oc‘𝐾)‘𝑋)))) |
7 | hlop 38745 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
8 | 1, 2 | opoccl 38577 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵) |
9 | 7, 8 | sylan 579 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵) |
10 | 1, 2, 3, 4 | polpmapN 39296 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵) → ( ⊥ ‘(𝑀‘((oc‘𝐾)‘𝑋))) = (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘𝑋)))) |
11 | 9, 10 | syldan 590 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘(𝑀‘((oc‘𝐾)‘𝑋))) = (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘𝑋)))) |
12 | 1, 2 | opococ 38578 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑋)) = 𝑋) |
13 | 7, 12 | sylan 579 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑋)) = 𝑋) |
14 | 13 | fveq2d 6889 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘((oc‘𝐾)‘((oc‘𝐾)‘𝑋))) = (𝑀‘𝑋)) |
15 | 6, 11, 14 | 3eqtrd 2770 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(𝑀‘𝑋))) = (𝑀‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 Basecbs 17153 occoc 17214 OPcops 38555 HLchlt 38733 pmapcpmap 38881 ⊥𝑃cpolN 39286 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-pmap 38888 df-polarityN 39287 |
This theorem is referenced by: pmapsubclN 39330 ispsubcl2N 39331 |
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