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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 36377 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) | |
2 | eqid 2818 | . . . . 5 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | 2polat.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2818 | . . . . 5 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | 2polat.p | . . . . 5 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
6 | 2, 3, 4, 5 | polatN 36947 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = ((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) |
7 | 1, 6 | sylan 580 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = ((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) |
8 | 7 | fveq2d 6667 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘(𝑃‘{𝑄})) = (𝑃‘((pmap‘𝐾)‘((oc‘𝐾)‘𝑄)))) |
9 | hlop 36378 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | 10, 3 | atbase 36305 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
12 | 10, 2 | opoccl 36210 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑄) ∈ (Base‘𝐾)) |
13 | 9, 11, 12 | syl2an 595 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((oc‘𝐾)‘𝑄) ∈ (Base‘𝐾)) |
14 | 10, 2, 4, 5 | polpmapN 36928 | . . . 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 591 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) = ((pmap‘𝐾)‘((oc‘𝐾)‘((oc‘𝐾)‘𝑄)))) |
16 | 10, 2 | opococ 36211 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑄)) = 𝑄) |
17 | 9, 11, 16 | syl2an 595 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑄)) = 𝑄) |
18 | 17 | fveq2d 6667 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((oc‘𝐾)‘𝑄))) = ((pmap‘𝐾)‘𝑄)) |
19 | 3, 4 | pmapat 36779 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((pmap‘𝐾)‘𝑄) = {𝑄}) |
20 | 18, 19 | eqtrd 2853 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((oc‘𝐾)‘𝑄))) = {𝑄}) |
21 | 15, 20 | eqtrd 2853 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) = {𝑄}) |
22 | 8, 21 | eqtrd 2853 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘(𝑃‘{𝑄})) = {𝑄}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 ‘cfv 6348 Basecbs 16471 occoc 16561 OPcops 36188 OLcol 36190 Atomscatm 36279 HLchlt 36366 pmapcpmap 36513 ⊥𝑃cpolN 36918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-undef 7928 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-pmap 36520 df-polarityN 36919 |
This theorem is referenced by: atpsubclN 36961 |
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