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Theorem linepmap 39288
Description: A line described with a projective map. (Contributed by NM, 3-Feb-2012.)
Hypotheses
Ref Expression
isline2.j ∨ = (joinβ€˜πΎ)
isline2.a 𝐴 = (Atomsβ€˜πΎ)
isline2.n 𝑁 = (Linesβ€˜πΎ)
isline2.m 𝑀 = (pmapβ€˜πΎ)
Assertion
Ref Expression
linepmap (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (π‘€β€˜(𝑃 ∨ 𝑄)) ∈ 𝑁)

Proof of Theorem linepmap
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 simpl1 1188 . . 3 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝐾 ∈ Lat)
2 simpl2 1189 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝑃 ∈ 𝐴)
3 eqid 2728 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 isline2.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
53, 4atbase 38801 . . . . 5 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
62, 5syl 17 . . . 4 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
7 simpl3 1190 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝑄 ∈ 𝐴)
83, 4atbase 38801 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
97, 8syl 17 . . . 4 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
10 isline2.j . . . . 5 ∨ = (joinβ€˜πΎ)
113, 10latjcl 18440 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
121, 6, 9, 11syl3anc 1368 . . 3 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
13 eqid 2728 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
14 isline2.m . . . 4 𝑀 = (pmapβ€˜πΎ)
153, 13, 4, 14pmapval 39270 . . 3 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ (π‘€β€˜(𝑃 ∨ 𝑄)) = {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑃 ∨ 𝑄)})
161, 12, 15syl2anc 582 . 2 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (π‘€β€˜(𝑃 ∨ 𝑄)) = {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑃 ∨ 𝑄)})
17 eqid 2728 . . 3 {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑃 ∨ 𝑄)} = {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑃 ∨ 𝑄)}
18 isline2.n . . . 4 𝑁 = (Linesβ€˜πΎ)
1913, 10, 4, 18islinei 39253 . . 3 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑃 ∨ 𝑄)} = {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑃 ∨ 𝑄)})) β†’ {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑃 ∨ 𝑄)} ∈ 𝑁)
2017, 19mpanr2 702 . 2 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑃 ∨ 𝑄)} ∈ 𝑁)
2116, 20eqeltrd 2829 1 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (π‘€β€˜(𝑃 ∨ 𝑄)) ∈ 𝑁)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  {crab 3430   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  lecple 17249  joincjn 18312  Latclat 18432  Atomscatm 38775  Linesclines 39007  pmapcpmap 39010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-lub 18347  df-glb 18348  df-join 18349  df-meet 18350  df-lat 18433  df-ats 38779  df-lines 39014  df-pmap 39017
This theorem is referenced by:  cdleme3h  39748  cdleme7ga  39761
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