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Theorem linepmap 36947
Description: A line described with a projective map. (Contributed by NM, 3-Feb-2012.)
Hypotheses
Ref Expression
isline2.j = (join‘𝐾)
isline2.a 𝐴 = (Atoms‘𝐾)
isline2.n 𝑁 = (Lines‘𝐾)
isline2.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
linepmap (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑀‘(𝑃 𝑄)) ∈ 𝑁)

Proof of Theorem linepmap
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 simpl1 1187 . . 3 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝐾 ∈ Lat)
2 simpl2 1188 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝑃𝐴)
3 eqid 2820 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
4 isline2.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4atbase 36461 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
62, 5syl 17 . . . 4 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝑃 ∈ (Base‘𝐾))
7 simpl3 1189 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝑄𝐴)
83, 4atbase 36461 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
97, 8syl 17 . . . 4 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝑄 ∈ (Base‘𝐾))
10 isline2.j . . . . 5 = (join‘𝐾)
113, 10latjcl 17640 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
121, 6, 9, 11syl3anc 1367 . . 3 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 𝑄) ∈ (Base‘𝐾))
13 eqid 2820 . . . 4 (le‘𝐾) = (le‘𝐾)
14 isline2.m . . . 4 𝑀 = (pmap‘𝐾)
153, 13, 4, 14pmapval 36929 . . 3 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑀‘(𝑃 𝑄)) = {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)})
161, 12, 15syl2anc 586 . 2 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑀‘(𝑃 𝑄)) = {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)})
17 eqid 2820 . . 3 {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)} = {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)}
18 isline2.n . . . 4 𝑁 = (Lines‘𝐾)
1913, 10, 4, 18islinei 36912 . . 3 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄 ∧ {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)} = {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)})) → {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)} ∈ 𝑁)
2017, 19mpanr2 702 . 2 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)} ∈ 𝑁)
2116, 20eqeltrd 2911 1 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑀‘(𝑃 𝑄)) ∈ 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1537  wcel 2114  wne 3006  {crab 3129   class class class wbr 5042  cfv 6331  (class class class)co 7133  Basecbs 16462  lecple 16551  joincjn 17533  Latclat 17634  Atomscatm 36435  Linesclines 36666  pmapcpmap 36669
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 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
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 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-lub 17563  df-glb 17564  df-join 17565  df-meet 17566  df-lat 17635  df-ats 36439  df-lines 36673  df-pmap 36676
This theorem is referenced by:  cdleme3h  37407  cdleme7ga  37420
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