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Theorem islinei 39723
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 1191 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑄𝐴)
2 simpl3 1192 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑅𝐴)
3 simpr 484 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)}))
4 neeq1 3001 . . . . 5 (𝑞 = 𝑄 → (𝑞𝑟𝑄𝑟))
5 oveq1 7438 . . . . . . . 8 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
65breq2d 5160 . . . . . . 7 (𝑞 = 𝑄 → (𝑝 (𝑞 𝑟) ↔ 𝑝 (𝑄 𝑟)))
76rabbidv 3441 . . . . . 6 (𝑞 = 𝑄 → {𝑝𝐴𝑝 (𝑞 𝑟)} = {𝑝𝐴𝑝 (𝑄 𝑟)})
87eqeq2d 2746 . . . . 5 (𝑞 = 𝑄 → (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)}))
94, 8anbi12d 632 . . . 4 (𝑞 = 𝑄 → ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ (𝑄𝑟𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)})))
10 neeq2 3002 . . . . 5 (𝑟 = 𝑅 → (𝑄𝑟𝑄𝑅))
11 oveq2 7439 . . . . . . . 8 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1211breq2d 5160 . . . . . . 7 (𝑟 = 𝑅 → (𝑝 (𝑄 𝑟) ↔ 𝑝 (𝑄 𝑅)))
1312rabbidv 3441 . . . . . 6 (𝑟 = 𝑅 → {𝑝𝐴𝑝 (𝑄 𝑟)} = {𝑝𝐴𝑝 (𝑄 𝑅)})
1413eqeq2d 2746 . . . . 5 (𝑟 = 𝑅 → (𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)}))
1510, 14anbi12d 632 . . . 4 (𝑟 = 𝑅 → ((𝑄𝑟𝑋 = {𝑝𝐴𝑝 (𝑄 𝑟)}) ↔ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})))
169, 15rspc2ev 3635 . . 3 ((𝑄𝐴𝑅𝐴 ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
171, 2, 3, 16syl3anc 1370 . 2 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
18 simpl1 1190 . . 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 39722 . . 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 257 1 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433   class class class wbr 5148  cfv 6563  (class class class)co 7431  lecple 17305  joincjn 18369  Atomscatm 39245  Linesclines 39477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-lines 39484
This theorem is referenced by:  linepmap  39758
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