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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 2737 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 2 | pmapat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 3 | 1, 2 | atbase 39290 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) | 
| 4 | eqid 2737 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 5 | pmapat.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
| 6 | 1, 4, 2, 5 | pmapval 39759 | . . 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 39361 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
| 9 | 8 | ad2antrr 726 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝐾 ∈ AtLat) | 
| 10 | simpr 484 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴) | |
| 11 | simplr 769 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝑃 ∈ 𝐴) | |
| 12 | 4, 2 | atcmp 39312 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑞 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑞(le‘𝐾)𝑃 ↔ 𝑞 = 𝑃)) | 
| 13 | 9, 10, 11, 12 | syl3anc 1373 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → (𝑞(le‘𝐾)𝑃 ↔ 𝑞 = 𝑃)) | 
| 14 | 13 | rabbidva 3443 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → {𝑞 ∈ 𝐴 ∣ 𝑞(le‘𝐾)𝑃} = {𝑞 ∈ 𝐴 ∣ 𝑞 = 𝑃}) | 
| 15 | rabsn 4721 | . . 3 ⊢ (𝑃 ∈ 𝐴 → {𝑞 ∈ 𝐴 ∣ 𝑞 = 𝑃} = {𝑃}) | |
| 16 | 15 | adantl 481 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → {𝑞 ∈ 𝐴 ∣ 𝑞 = 𝑃} = {𝑃}) | 
| 17 | 7, 14, 16 | 3eqtrd 2781 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑀‘𝑃) = {𝑃}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 {csn 4626 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Atomscatm 39264 AtLatcal 39265 HLchlt 39351 pmapcpmap 39499 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-proset 18340 df-poset 18359 df-plt 18375 df-glb 18392 df-p0 18470 df-lat 18477 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-pmap 39506 | 
| This theorem is referenced by: elpmapat 39766 2polatN 39934 paddatclN 39951 | 
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