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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > linepsubclN | Structured version Visualization version GIF version |
Description: A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
linepsubcl.n | ⊢ 𝑁 = (Lines‘𝐾) |
linepsubcl.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
linepsubclN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 35384 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2799 | . . . . 5 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | eqid 2799 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
4 | linepsubcl.n | . . . . 5 ⊢ 𝑁 = (Lines‘𝐾) | |
5 | eqid 2799 | . . . . 5 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
6 | 2, 3, 4, 5 | isline2 35795 | . . . 4 ⊢ (𝐾 ∈ Lat → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))))) |
7 | 1, 6 | syl 17 | . . 3 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))))) |
8 | 1 | adantr 473 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝐾 ∈ Lat) |
9 | eqid 2799 | . . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | 9, 3 | atbase 35310 | . . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾)) |
11 | 10 | ad2antrl 720 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑝 ∈ (Base‘𝐾)) |
12 | 9, 3 | atbase 35310 | . . . . . . . . 9 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾)) |
13 | 12 | ad2antll 721 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑞 ∈ (Base‘𝐾)) |
14 | 9, 2 | latjcl 17366 | . . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)) |
15 | 8, 11, 13, 14 | syl3anc 1491 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)) |
16 | linepsubcl.c | . . . . . . . 8 ⊢ 𝐶 = (PSubCl‘𝐾) | |
17 | 9, 5, 16 | pmapsubclN 35967 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶) |
18 | 15, 17 | syldan 586 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶) |
19 | eleq1a 2873 | . . . . . 6 ⊢ (((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶 → (𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) → 𝑋 ∈ 𝐶)) | |
20 | 18, 19 | syl 17 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → (𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) → 𝑋 ∈ 𝐶)) |
21 | 20 | adantld 485 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((𝑝 ≠ 𝑞 ∧ 𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋 ∈ 𝐶)) |
22 | 21 | rexlimdvva 3219 | . . 3 ⊢ (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋 ∈ 𝐶)) |
23 | 7, 22 | sylbid 232 | . 2 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐶)) |
24 | 23 | imp 396 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∃wrex 3090 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 joincjn 17259 Latclat 17360 Atomscatm 35284 HLchlt 35371 Linesclines 35515 pmapcpmap 35518 PSubClcpscN 35955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-riotaBAD 34974 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-undef 7637 df-proset 17243 df-poset 17261 df-plt 17273 df-lub 17289 df-glb 17290 df-join 17291 df-meet 17292 df-p0 17354 df-p1 17355 df-lat 17361 df-clat 17423 df-oposet 35197 df-ol 35199 df-oml 35200 df-covers 35287 df-ats 35288 df-atl 35319 df-cvlat 35343 df-hlat 35372 df-lines 35522 df-pmap 35525 df-polarityN 35924 df-psubclN 35956 |
This theorem is referenced by: (None) |
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