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Theorem isline 34844
Description: The predicate "is a line". (Contributed by NM, 19-Sep-2011.)
Hypotheses
Ref Expression
isline.l = (le‘𝐾)
isline.j = (join‘𝐾)
isline.a 𝐴 = (Atoms‘𝐾)
isline.n 𝑁 = (Lines‘𝐾)
Assertion
Ref Expression
isline (𝐾𝐷 → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
Distinct variable groups:   𝑞,𝑝,𝑟,𝐴   𝐾,𝑝,𝑞,𝑟   𝑋,𝑞,𝑟
Allowed substitution hints:   𝐷(𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   𝑁(𝑟,𝑞,𝑝)   𝑋(𝑝)

Proof of Theorem isline
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isline.l . . . 4 = (le‘𝐾)
2 isline.j . . . 4 = (join‘𝐾)
3 isline.a . . . 4 𝐴 = (Atoms‘𝐾)
4 isline.n . . . 4 𝑁 = (Lines‘𝐾)
51, 2, 3, 4lineset 34843 . . 3 (𝐾𝐷𝑁 = {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
65eleq2d 2685 . 2 (𝐾𝐷 → (𝑋𝑁𝑋 ∈ {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})}))
7 fvex 6188 . . . . . . . . 9 (Atoms‘𝐾) ∈ V
83, 7eqeltri 2695 . . . . . . . 8 𝐴 ∈ V
98rabex 4804 . . . . . . 7 {𝑝𝐴𝑝 (𝑞 𝑟)} ∈ V
10 eleq1 2687 . . . . . . 7 (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} → (𝑋 ∈ V ↔ {𝑝𝐴𝑝 (𝑞 𝑟)} ∈ V))
119, 10mpbiri 248 . . . . . 6 (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} → 𝑋 ∈ V)
1211adantl 482 . . . . 5 ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V)
1312a1i 11 . . . 4 ((𝑞𝐴𝑟𝐴) → ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V))
1413rexlimivv 3032 . . 3 (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V)
15 eqeq1 2624 . . . . 5 (𝑥 = 𝑋 → (𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
1615anbi2d 739 . . . 4 (𝑥 = 𝑋 → ((𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
17162rexbidv 3053 . . 3 (𝑥 = 𝑋 → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
1814, 17elab3 3352 . 2 (𝑋 ∈ {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
196, 18syl6bb 276 1 (𝐾𝐷 → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  {cab 2606  wne 2791  wrex 2910  {crab 2913  Vcvv 3195   class class class wbr 4644  cfv 5876  (class class class)co 6635  lecple 15929  joincjn 16925  Atomscatm 34369  Linesclines 34599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-lines 34606
This theorem is referenced by:  islinei  34845  linepsubN  34857  isline2  34879
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