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Theorem islinei 39217
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 483 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)}))
4 neeq1 2999 . . . . 5 (𝑞 = 𝑄 → (𝑞𝑟𝑄𝑟))
5 oveq1 7431 . . . . . . . 8 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
65breq2d 5162 . . . . . . 7 (𝑞 = 𝑄 → (𝑝 (𝑞 𝑟) ↔ 𝑝 (𝑄 𝑟)))
76rabbidv 3436 . . . . . 6 (𝑞 = 𝑄 → {𝑝𝐴𝑝 (𝑞 𝑟)} = {𝑝𝐴𝑝 (𝑄 𝑟)})
87eqeq2d 2738 . . . . 5 (𝑞 = 𝑄 → (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)}))
94, 8anbi12d 630 . . . 4 (𝑞 = 𝑄 → ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ (𝑄𝑟𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)})))
10 neeq2 3000 . . . . 5 (𝑟 = 𝑅 → (𝑄𝑟𝑄𝑅))
11 oveq2 7432 . . . . . . . 8 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1211breq2d 5162 . . . . . . 7 (𝑟 = 𝑅 → (𝑝 (𝑄 𝑟) ↔ 𝑝 (𝑄 𝑅)))
1312rabbidv 3436 . . . . . 6 (𝑟 = 𝑅 → {𝑝𝐴𝑝 (𝑄 𝑟)} = {𝑝𝐴𝑝 (𝑄 𝑅)})
1413eqeq2d 2738 . . . . 5 (𝑟 = 𝑅 → (𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)}))
1510, 14anbi12d 630 . . . 4 (𝑟 = 𝑅 → ((𝑄𝑟𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)}) ↔ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})))
169, 15rspc2ev 3622 . . 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 39216 . . 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 256 1 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2936  wrex 3066  {crab 3428   class class class wbr 5150  cfv 6551  (class class class)co 7424  lecple 17245  joincjn 18308  Atomscatm 38739  Linesclines 38971
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-lines 38978
This theorem is referenced by:  linepmap  39252
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