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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapat | Structured version Visualization version GIF version |
Description: The projective map of an atom. (Contributed by NM, 25-Jan-2012.) |
Ref | Expression |
---|---|
pmapat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pmapat.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
pmapat | ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑀‘𝑃) = {𝑃}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | pmapat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | atbase 37718 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
4 | eqid 2736 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | pmapat.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
6 | 1, 4, 2, 5 | pmapval 38187 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑀‘𝑃) = {𝑞 ∈ 𝐴 ∣ 𝑞(le‘𝐾)𝑃}) |
7 | 3, 6 | sylan2 593 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑀‘𝑃) = {𝑞 ∈ 𝐴 ∣ 𝑞(le‘𝐾)𝑃}) |
8 | hlatl 37789 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
9 | 8 | ad2antrr 724 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝐾 ∈ AtLat) |
10 | simpr 485 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴) | |
11 | simplr 767 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝑃 ∈ 𝐴) | |
12 | 4, 2 | atcmp 37740 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑞 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑞(le‘𝐾)𝑃 ↔ 𝑞 = 𝑃)) |
13 | 9, 10, 11, 12 | syl3anc 1371 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → (𝑞(le‘𝐾)𝑃 ↔ 𝑞 = 𝑃)) |
14 | 13 | rabbidva 3412 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → {𝑞 ∈ 𝐴 ∣ 𝑞(le‘𝐾)𝑃} = {𝑞 ∈ 𝐴 ∣ 𝑞 = 𝑃}) |
15 | rabsn 4680 | . . 3 ⊢ (𝑃 ∈ 𝐴 → {𝑞 ∈ 𝐴 ∣ 𝑞 = 𝑃} = {𝑃}) | |
16 | 15 | adantl 482 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → {𝑞 ∈ 𝐴 ∣ 𝑞 = 𝑃} = {𝑃}) |
17 | 7, 14, 16 | 3eqtrd 2780 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑀‘𝑃) = {𝑃}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3405 {csn 4584 class class class wbr 5103 ‘cfv 6493 Basecbs 17075 lecple 17132 Atomscatm 37692 AtLatcal 37693 HLchlt 37779 pmapcpmap 37927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-proset 18176 df-poset 18194 df-plt 18211 df-glb 18228 df-p0 18306 df-lat 18313 df-covers 37695 df-ats 37696 df-atl 37727 df-cvlat 37751 df-hlat 37780 df-pmap 37934 |
This theorem is referenced by: elpmapat 38194 2polatN 38362 paddatclN 38379 |
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