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Theorem islinei 34541
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 1063 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑄𝐴)
2 simpl3 1064 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑅𝐴)
3 simpr 477 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)}))
4 neeq1 2852 . . . . 5 (𝑞 = 𝑄 → (𝑞𝑟𝑄𝑟))
5 oveq1 6617 . . . . . . . 8 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
65breq2d 4630 . . . . . . 7 (𝑞 = 𝑄 → (𝑝 (𝑞 𝑟) ↔ 𝑝 (𝑄 𝑟)))
76rabbidv 3180 . . . . . 6 (𝑞 = 𝑄 → {𝑝𝐴𝑝 (𝑞 𝑟)} = {𝑝𝐴𝑝 (𝑄 𝑟)})
87eqeq2d 2631 . . . . 5 (𝑞 = 𝑄 → (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)}))
94, 8anbi12d 746 . . . 4 (𝑞 = 𝑄 → ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ (𝑄𝑟𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)})))
10 neeq2 2853 . . . . 5 (𝑟 = 𝑅 → (𝑄𝑟𝑄𝑅))
11 oveq2 6618 . . . . . . . 8 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1211breq2d 4630 . . . . . . 7 (𝑟 = 𝑅 → (𝑝 (𝑄 𝑟) ↔ 𝑝 (𝑄 𝑅)))
1312rabbidv 3180 . . . . . 6 (𝑟 = 𝑅 → {𝑝𝐴𝑝 (𝑄 𝑟)} = {𝑝𝐴𝑝 (𝑄 𝑅)})
1413eqeq2d 2631 . . . . 5 (𝑟 = 𝑅 → (𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)}))
1510, 14anbi12d 746 . . . 4 (𝑟 = 𝑅 → ((𝑄𝑟𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)}) ↔ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})))
169, 15rspc2ev 3312 . . 3 ((𝑄𝐴𝑅𝐴 ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
171, 2, 3, 16syl3anc 1323 . 2 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
18 simpl1 1062 . . 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 34540 . . 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 247 1 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911   class class class wbr 4618  cfv 5852  (class class class)co 6610  lecple 15880  joincjn 16876  Atomscatm 34065  Linesclines 34295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-lines 34302
This theorem is referenced by:  linepmap  34576
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