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Theorem islinei 38232
Description: Condition implying "is a line". (Contributed by NM, 3-Feb-2012.)
Hypotheses
Ref Expression
isline.l ≀ = (leβ€˜πΎ)
isline.j ∨ = (joinβ€˜πΎ)
isline.a 𝐴 = (Atomsβ€˜πΎ)
isline.n 𝑁 = (Linesβ€˜πΎ)
Assertion
Ref Expression
islinei (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})) β†’ 𝑋 ∈ 𝑁)
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 π‘ž π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1193 . . 3 (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})) β†’ 𝑄 ∈ 𝐴)
2 simpl3 1194 . . 3 (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})) β†’ 𝑅 ∈ 𝐴)
3 simpr 486 . . 3 (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})) β†’ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)}))
4 neeq1 3007 . . . . 5 (π‘ž = 𝑄 β†’ (π‘ž β‰  π‘Ÿ ↔ 𝑄 β‰  π‘Ÿ))
5 oveq1 7369 . . . . . . . 8 (π‘ž = 𝑄 β†’ (π‘ž ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))
65breq2d 5122 . . . . . . 7 (π‘ž = 𝑄 β†’ (𝑝 ≀ (π‘ž ∨ π‘Ÿ) ↔ 𝑝 ≀ (𝑄 ∨ π‘Ÿ)))
76rabbidv 3418 . . . . . 6 (π‘ž = 𝑄 β†’ {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ π‘Ÿ)})
87eqeq2d 2748 . . . . 5 (π‘ž = 𝑄 β†’ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ π‘Ÿ)}))
94, 8anbi12d 632 . . . 4 (π‘ž = 𝑄 β†’ ((π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}) ↔ (𝑄 β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ π‘Ÿ)})))
10 neeq2 3008 . . . . 5 (π‘Ÿ = 𝑅 β†’ (𝑄 β‰  π‘Ÿ ↔ 𝑄 β‰  𝑅))
11 oveq2 7370 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (𝑄 ∨ π‘Ÿ) = (𝑄 ∨ 𝑅))
1211breq2d 5122 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (𝑝 ≀ (𝑄 ∨ π‘Ÿ) ↔ 𝑝 ≀ (𝑄 ∨ 𝑅)))
1312rabbidv 3418 . . . . . 6 (π‘Ÿ = 𝑅 β†’ {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ π‘Ÿ)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})
1413eqeq2d 2748 . . . . 5 (π‘Ÿ = 𝑅 β†’ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ π‘Ÿ)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)}))
1510, 14anbi12d 632 . . . 4 (π‘Ÿ = 𝑅 β†’ ((𝑄 β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ π‘Ÿ)}) ↔ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})))
169, 15rspc2ev 3595 . . 3 ((𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})) β†’ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}))
171, 2, 3, 16syl3anc 1372 . 2 (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})) β†’ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}))
18 simpl1 1192 . . 3 (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})) β†’ 𝐾 ∈ 𝐷)
19 isline.l . . . 4 ≀ = (leβ€˜πΎ)
20 isline.j . . . 4 ∨ = (joinβ€˜πΎ)
21 isline.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
22 isline.n . . . 4 𝑁 = (Linesβ€˜πΎ)
2319, 20, 21, 22isline 38231 . . 3 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
2418, 23syl 17 . 2 (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})) β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
2517, 24mpbird 257 1 (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (𝑄 ∨ 𝑅)})) β†’ 𝑋 ∈ 𝑁)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074  {crab 3410   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  lecple 17147  joincjn 18207  Atomscatm 37754  Linesclines 37986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-lines 37993
This theorem is referenced by:  linepmap  38267
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