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Theorem islinei 36984
Description: Condition implying "is a line". (Contributed by NM, 3-Feb-2012.)
Hypotheses
Ref Expression
isline.l = (le‘𝐾)
isline.j = (join‘𝐾)
isline.a 𝐴 = (Atoms‘𝐾)
isline.n 𝑁 = (Lines‘𝐾)
Assertion
Ref Expression
islinei (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑋𝑁)
Distinct variable groups:   𝐴,𝑝   𝐾,𝑝   𝑄,𝑝   𝑅,𝑝
Allowed substitution hints:   𝐷(𝑝)   (𝑝)   (𝑝)   𝑁(𝑝)   𝑋(𝑝)

Proof of Theorem islinei
Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1189 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑄𝐴)
2 simpl3 1190 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑅𝐴)
3 simpr 488 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)}))
4 neeq1 3076 . . . . 5 (𝑞 = 𝑄 → (𝑞𝑟𝑄𝑟))
5 oveq1 7156 . . . . . . . 8 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
65breq2d 5064 . . . . . . 7 (𝑞 = 𝑄 → (𝑝 (𝑞 𝑟) ↔ 𝑝 (𝑄 𝑟)))
76rabbidv 3465 . . . . . 6 (𝑞 = 𝑄 → {𝑝𝐴𝑝 (𝑞 𝑟)} = {𝑝𝐴𝑝 (𝑄 𝑟)})
87eqeq2d 2835 . . . . 5 (𝑞 = 𝑄 → (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)}))
94, 8anbi12d 633 . . . 4 (𝑞 = 𝑄 → ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ (𝑄𝑟𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)})))
10 neeq2 3077 . . . . 5 (𝑟 = 𝑅 → (𝑄𝑟𝑄𝑅))
11 oveq2 7157 . . . . . . . 8 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1211breq2d 5064 . . . . . . 7 (𝑟 = 𝑅 → (𝑝 (𝑄 𝑟) ↔ 𝑝 (𝑄 𝑅)))
1312rabbidv 3465 . . . . . 6 (𝑟 = 𝑅 → {𝑝𝐴𝑝 (𝑄 𝑟)} = {𝑝𝐴𝑝 (𝑄 𝑅)})
1413eqeq2d 2835 . . . . 5 (𝑟 = 𝑅 → (𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)}))
1510, 14anbi12d 633 . . . 4 (𝑟 = 𝑅 → ((𝑄𝑟𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)}) ↔ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})))
169, 15rspc2ev 3621 . . 3 ((𝑄𝐴𝑅𝐴 ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
171, 2, 3, 16syl3anc 1368 . 2 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
18 simpl1 1188 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝐾𝐷)
19 isline.l . . . 4 = (le‘𝐾)
20 isline.j . . . 4 = (join‘𝐾)
21 isline.a . . . 4 𝐴 = (Atoms‘𝐾)
22 isline.n . . . 4 𝑁 = (Lines‘𝐾)
2319, 20, 21, 22isline 36983 . . 3 (𝐾𝐷 → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
2418, 23syl 17 . 2 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
2517, 24mpbird 260 1 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wrex 3134  {crab 3137   class class class wbr 5052  cfv 6343  (class class class)co 7149  lecple 16572  joincjn 17554  Atomscatm 36507  Linesclines 36738
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-lines 36745
This theorem is referenced by:  linepmap  37019
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