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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2polatN | Structured version Visualization version GIF version |
Description: Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2polat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
2polat.p | ⊢ 𝑃 = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
2polatN | ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘(𝑃‘{𝑄})) = {𝑄}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlol 33666 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) | |
2 | eqid 2610 | . . . . 5 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | 2polat.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2610 | . . . . 5 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | 2polat.p | . . . . 5 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
6 | 2, 3, 4, 5 | polatN 34235 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = ((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) |
7 | 1, 6 | sylan 487 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = ((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) |
8 | 7 | fveq2d 6107 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘(𝑃‘{𝑄})) = (𝑃‘((pmap‘𝐾)‘((oc‘𝐾)‘𝑄)))) |
9 | hlop 33667 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | 10, 3 | atbase 33594 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
12 | 10, 2 | opoccl 33499 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑄) ∈ (Base‘𝐾)) |
13 | 9, 11, 12 | syl2an 493 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((oc‘𝐾)‘𝑄) ∈ (Base‘𝐾)) |
14 | 10, 2, 4, 5 | polpmapN 34216 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ ((oc‘𝐾)‘𝑄) ∈ (Base‘𝐾)) → (𝑃‘((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) = ((pmap‘𝐾)‘((oc‘𝐾)‘((oc‘𝐾)‘𝑄)))) |
15 | 13, 14 | syldan 486 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) = ((pmap‘𝐾)‘((oc‘𝐾)‘((oc‘𝐾)‘𝑄)))) |
16 | 10, 2 | opococ 33500 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑄)) = 𝑄) |
17 | 9, 11, 16 | syl2an 493 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑄)) = 𝑄) |
18 | 17 | fveq2d 6107 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((oc‘𝐾)‘𝑄))) = ((pmap‘𝐾)‘𝑄)) |
19 | 3, 4 | pmapat 34067 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((pmap‘𝐾)‘𝑄) = {𝑄}) |
20 | 18, 19 | eqtrd 2644 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((oc‘𝐾)‘𝑄))) = {𝑄}) |
21 | 15, 20 | eqtrd 2644 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) = {𝑄}) |
22 | 8, 21 | eqtrd 2644 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘(𝑃‘{𝑄})) = {𝑄}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 ‘cfv 5804 Basecbs 15695 occoc 15776 OPcops 33477 OLcol 33479 Atomscatm 33568 HLchlt 33655 pmapcpmap 33801 ⊥𝑃cpolN 34206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-undef 7286 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-pmap 33808 df-polarityN 34207 |
This theorem is referenced by: atpsubclN 34249 |
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