Mathbox for Norm Megill |
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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 36499 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) | |
2 | eqid 2823 | . . . . 5 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | 2polat.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2823 | . . . . 5 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | 2polat.p | . . . . 5 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
6 | 2, 3, 4, 5 | polatN 37069 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = ((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) |
7 | 1, 6 | sylan 582 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = ((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) |
8 | 7 | fveq2d 6676 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘(𝑃‘{𝑄})) = (𝑃‘((pmap‘𝐾)‘((oc‘𝐾)‘𝑄)))) |
9 | hlop 36500 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | eqid 2823 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | 10, 3 | atbase 36427 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
12 | 10, 2 | opoccl 36332 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑄) ∈ (Base‘𝐾)) |
13 | 9, 11, 12 | syl2an 597 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((oc‘𝐾)‘𝑄) ∈ (Base‘𝐾)) |
14 | 10, 2, 4, 5 | polpmapN 37050 | . . . 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 593 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) = ((pmap‘𝐾)‘((oc‘𝐾)‘((oc‘𝐾)‘𝑄)))) |
16 | 10, 2 | opococ 36333 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑄)) = 𝑄) |
17 | 9, 11, 16 | syl2an 597 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((oc‘𝐾)‘((oc‘𝐾)‘𝑄)) = 𝑄) |
18 | 17 | fveq2d 6676 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((oc‘𝐾)‘𝑄))) = ((pmap‘𝐾)‘𝑄)) |
19 | 3, 4 | pmapat 36901 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((pmap‘𝐾)‘𝑄) = {𝑄}) |
20 | 18, 19 | eqtrd 2858 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ((pmap‘𝐾)‘((oc‘𝐾)‘((oc‘𝐾)‘𝑄))) = {𝑄}) |
21 | 15, 20 | eqtrd 2858 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘((pmap‘𝐾)‘((oc‘𝐾)‘𝑄))) = {𝑄}) |
22 | 8, 21 | eqtrd 2858 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘(𝑃‘{𝑄})) = {𝑄}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4569 ‘cfv 6357 Basecbs 16485 occoc 16575 OPcops 36310 OLcol 36312 Atomscatm 36401 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: atpsubclN 37083 |
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