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Theorem tglinecom 28643
Description: Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
Hypotheses
Ref Expression
tglineelsb2.p 𝐵 = (Base‘𝐺)
tglineelsb2.i 𝐼 = (Itv‘𝐺)
tglineelsb2.l 𝐿 = (LineG‘𝐺)
tglineelsb2.g (𝜑𝐺 ∈ TarskiG)
tglineelsb2.1 (𝜑𝑃𝐵)
tglineelsb2.2 (𝜑𝑄𝐵)
tglineelsb2.4 (𝜑𝑃𝑄)
Assertion
Ref Expression
tglinecom (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))

Proof of Theorem tglinecom
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tglineelsb2.p . . . 4 𝐵 = (Base‘𝐺)
2 tglineelsb2.i . . . 4 𝐼 = (Itv‘𝐺)
3 tglineelsb2.l . . . 4 𝐿 = (LineG‘𝐺)
4 tglineelsb2.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝐺 ∈ TarskiG)
6 tglineelsb2.2 . . . . 5 (𝜑𝑄𝐵)
76adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄𝐵)
8 tglineelsb2.1 . . . . 5 (𝜑𝑃𝐵)
98adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑃𝐵)
10 tglineelsb2.4 . . . . . 6 (𝜑𝑃𝑄)
111, 3, 2, 4, 8, 6, 10tglnssp 28560 . . . . 5 (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵)
1211sselda 3983 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥𝐵)
1310necomd 2996 . . . . 5 (𝜑𝑄𝑃)
1413adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄𝑃)
15 simpr 484 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄))
161, 2, 3, 5, 7, 9, 12, 14, 15lncom 28630 . . 3 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑄𝐿𝑃))
174adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝐺 ∈ TarskiG)
188adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃𝐵)
196adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑄𝐵)
201, 3, 2, 4, 6, 8, 13tglnssp 28560 . . . . 5 (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵)
2120sselda 3983 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥𝐵)
2210adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃𝑄)
23 simpr 484 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑄𝐿𝑃))
241, 2, 3, 17, 18, 19, 21, 22, 23lncom 28630 . . 3 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄))
2516, 24impbida 801 . 2 (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃)))
2625eqrdv 2735 1 (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  cfv 6561  (class class class)co 7431  Basecbs 17247  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461
This theorem is referenced by:  tglinethru  28644  coltr3  28656  footeq  28732  colperpexlem3  28740  mideulem2  28742  opphllem  28743  midex  28745  opphllem3  28757  opphllem5  28759  lmicom  28796  lmiisolem  28804  lnperpex  28811  trgcopy  28812
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