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Theorem linepmap 39159
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 2726 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 isline2.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
53, 4atbase 38672 . . . . 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 38672 . . . . 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 18404 . . . 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 2726 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
14 isline2.m . . . 4 𝑀 = (pmapβ€˜πΎ)
153, 13, 4, 14pmapval 39141 . . 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 583 . 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 2726 . . 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 39124 . . 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 701 . 2 (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑃 ∨ 𝑄)} ∈ 𝑁)
2116, 20eqeltrd 2827 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  {crab 3426   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  Latclat 18396  Atomscatm 38646  Linesclines 38878  pmapcpmap 38881
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-lat 18397  df-ats 38650  df-lines 38885  df-pmap 38888
This theorem is referenced by:  cdleme3h  39619  cdleme7ga  39632
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