Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isline2 Structured version   Visualization version   GIF version

Theorem isline2 38633
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 2732 . . 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 38598 . 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 483 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
7 eqid 2732 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
87, 3atbase 38147 . . . . . . . 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 38147 . . . . . . . 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 18388 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Baseβ€˜πΎ) ∧ π‘ž ∈ (Baseβ€˜πΎ)) β†’ (𝑝 ∨ π‘ž) ∈ (Baseβ€˜πΎ))
136, 9, 11, 12syl3anc 1371 . . . . . 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 38616 . . . . . 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 591 . . . . 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 2743 . . . 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 629 . . 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 3217 . 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070  {crab 3432   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  Latclat 18380  Atomscatm 38121  Linesclines 38353  pmapcpmap 38356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-lat 18381  df-ats 38125  df-lines 38360  df-pmap 38363
This theorem is referenced by:  isline3  38635  lncvrelatN  38640  linepsubclN  38810
  Copyright terms: Public domain W3C validator