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Theorem isline2 39279
Description: Definition of line in terms of projective map. (Contributed by NM, 25-Jan-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
isline2 (𝐾 ∈ Lat β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (𝑝 β‰  π‘ž ∧ 𝑋 = (π‘€β€˜(𝑝 ∨ π‘ž)))))
Distinct variable groups:   π‘ž,𝑝,𝐴   𝐾,𝑝,π‘ž   𝑋,𝑝,π‘ž
Allowed substitution hints:   ∨ (π‘ž,𝑝)   𝑀(π‘ž,𝑝)   𝑁(π‘ž,𝑝)

Proof of Theorem isline2
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
2 isline2.j . . 3 ∨ = (joinβ€˜πΎ)
3 isline2.a . . 3 𝐴 = (Atomsβ€˜πΎ)
4 isline2.n . . 3 𝑁 = (Linesβ€˜πΎ)
51, 2, 3, 4isline 39244 . 2 (𝐾 ∈ Lat β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (𝑝 β‰  π‘ž ∧ 𝑋 = {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑝 ∨ π‘ž)})))
6 simpl 481 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
7 eqid 2728 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
87, 3atbase 38793 . . . . . . . 8 (𝑝 ∈ 𝐴 β†’ 𝑝 ∈ (Baseβ€˜πΎ))
98ad2antrl 726 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
107, 3atbase 38793 . . . . . . . 8 (π‘ž ∈ 𝐴 β†’ π‘ž ∈ (Baseβ€˜πΎ))
1110ad2antll 727 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ π‘ž ∈ (Baseβ€˜πΎ))
127, 2latjcl 18438 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Baseβ€˜πΎ) ∧ π‘ž ∈ (Baseβ€˜πΎ)) β†’ (𝑝 ∨ π‘ž) ∈ (Baseβ€˜πΎ))
136, 9, 11, 12syl3anc 1368 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ (𝑝 ∨ π‘ž) ∈ (Baseβ€˜πΎ))
14 isline2.m . . . . . . 7 𝑀 = (pmapβ€˜πΎ)
157, 1, 3, 14pmapval 39262 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑝 ∨ π‘ž) ∈ (Baseβ€˜πΎ)) β†’ (π‘€β€˜(𝑝 ∨ π‘ž)) = {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑝 ∨ π‘ž)})
1613, 15syldan 589 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ (π‘€β€˜(𝑝 ∨ π‘ž)) = {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑝 ∨ π‘ž)})
1716eqeq2d 2739 . . . 4 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ (𝑋 = (π‘€β€˜(𝑝 ∨ π‘ž)) ↔ 𝑋 = {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑝 ∨ π‘ž)}))
1817anbi2d 628 . . 3 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ ((𝑝 β‰  π‘ž ∧ 𝑋 = (π‘€β€˜(𝑝 ∨ π‘ž))) ↔ (𝑝 β‰  π‘ž ∧ 𝑋 = {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑝 ∨ π‘ž)})))
19182rexbidva 3215 . 2 (𝐾 ∈ Lat β†’ (βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (𝑝 β‰  π‘ž ∧ 𝑋 = (π‘€β€˜(𝑝 ∨ π‘ž))) ↔ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (𝑝 β‰  π‘ž ∧ 𝑋 = {π‘Ÿ ∈ 𝐴 ∣ π‘Ÿ(leβ€˜πΎ)(𝑝 ∨ π‘ž)})))
205, 19bitr4d 281 1 (𝐾 ∈ Lat β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (𝑝 β‰  π‘ž ∧ 𝑋 = (π‘€β€˜(𝑝 ∨ π‘ž)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆƒwrex 3067  {crab 3430   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  joincjn 18310  Latclat 18430  Atomscatm 38767  Linesclines 38999  pmapcpmap 39002
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 7746
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 18345  df-glb 18346  df-join 18347  df-meet 18348  df-lat 18431  df-ats 38771  df-lines 39006  df-pmap 39009
This theorem is referenced by:  isline3  39281  lncvrelatN  39286  linepsubclN  39456
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