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Theorem isline2 39157
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 2726 . . 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 39122 . 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 482 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
7 eqid 2726 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
87, 3atbase 38671 . . . . . . . 8 (𝑝 ∈ 𝐴 β†’ 𝑝 ∈ (Baseβ€˜πΎ))
98ad2antrl 725 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
107, 3atbase 38671 . . . . . . . 8 (π‘ž ∈ 𝐴 β†’ π‘ž ∈ (Baseβ€˜πΎ))
1110ad2antll 726 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ π‘ž ∈ (Baseβ€˜πΎ))
127, 2latjcl 18401 . . . . . . 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 39140 . . . . . 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 590 . . . . 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 2737 . . . 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 3211 . 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 282 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064  {crab 3426   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  lecple 17210  joincjn 18273  Latclat 18393  Atomscatm 38645  Linesclines 38877  pmapcpmap 38880
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 7721
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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-lat 18394  df-ats 38649  df-lines 38884  df-pmap 38887
This theorem is referenced by:  isline3  39159  lncvrelatN  39164  linepsubclN  39334
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