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Theorem islinei 39743
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 1192 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑄𝐴)
2 simpl3 1193 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑅𝐴)
3 simpr 484 . . 3 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)}))
4 neeq1 3002 . . . . 5 (𝑞 = 𝑄 → (𝑞𝑟𝑄𝑟))
5 oveq1 7439 . . . . . . . 8 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
65breq2d 5154 . . . . . . 7 (𝑞 = 𝑄 → (𝑝 (𝑞 𝑟) ↔ 𝑝 (𝑄 𝑟)))
76rabbidv 3443 . . . . . 6 (𝑞 = 𝑄 → {𝑝𝐴𝑝 (𝑞 𝑟)} = {𝑝𝐴𝑝 (𝑄 𝑟)})
87eqeq2d 2747 . . . . 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 3003 . . . . 5 (𝑟 = 𝑅 → (𝑄𝑟𝑄𝑅))
11 oveq2 7440 . . . . . . . 8 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
1211breq2d 5154 . . . . . . 7 (𝑟 = 𝑅 → (𝑝 (𝑄 𝑟) ↔ 𝑝 (𝑄 𝑅)))
1312rabbidv 3443 . . . . . 6 (𝑟 = 𝑅 → {𝑝𝐴𝑝 (𝑄 𝑟)} = {𝑝𝐴𝑝 (𝑄 𝑅)})
1413eqeq2d 2747 . . . . 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 3634 . . 3 ((𝑄𝐴𝑅𝐴 ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
171, 2, 3, 16syl3anc 1372 . 2 (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
18 simpl1 1191 . . 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 39742 . . 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 1539  wcel 2107  wne 2939  wrex 3069  {crab 3435   class class class wbr 5142  cfv 6560  (class class class)co 7432  lecple 17305  joincjn 18358  Atomscatm 39265  Linesclines 39497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-lines 39504
This theorem is referenced by:  linepmap  39778
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