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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 2821 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | pmapat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | atbase 36419 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
4 | eqid 2821 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | pmapat.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
6 | 1, 4, 2, 5 | pmapval 36887 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑀‘𝑃) = {𝑞 ∈ 𝐴 ∣ 𝑞(le‘𝐾)𝑃}) |
7 | 3, 6 | sylan2 594 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑀‘𝑃) = {𝑞 ∈ 𝐴 ∣ 𝑞(le‘𝐾)𝑃}) |
8 | hlatl 36490 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
9 | 8 | ad2antrr 724 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝐾 ∈ AtLat) |
10 | simpr 487 | . . . 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 36441 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑞 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑞(le‘𝐾)𝑃 ↔ 𝑞 = 𝑃)) |
13 | 9, 10, 11, 12 | syl3anc 1367 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → (𝑞(le‘𝐾)𝑃 ↔ 𝑞 = 𝑃)) |
14 | 13 | rabbidva 3479 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → {𝑞 ∈ 𝐴 ∣ 𝑞(le‘𝐾)𝑃} = {𝑞 ∈ 𝐴 ∣ 𝑞 = 𝑃}) |
15 | rabsn 4651 | . . 3 ⊢ (𝑃 ∈ 𝐴 → {𝑞 ∈ 𝐴 ∣ 𝑞 = 𝑃} = {𝑃}) | |
16 | 15 | adantl 484 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → {𝑞 ∈ 𝐴 ∣ 𝑞 = 𝑃} = {𝑃}) |
17 | 7, 14, 16 | 3eqtrd 2860 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑀‘𝑃) = {𝑃}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 {csn 4561 class class class wbr 5059 ‘cfv 6350 Basecbs 16477 lecple 16566 Atomscatm 36393 AtLatcal 36394 HLchlt 36480 pmapcpmap 36627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-proset 17532 df-poset 17550 df-plt 17562 df-glb 17579 df-p0 17643 df-lat 17650 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-pmap 36634 |
This theorem is referenced by: elpmapat 36894 2polatN 37062 paddatclN 37079 |
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