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Theorem tglinecom 25330
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 25247 . . . . 5 (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵)
1211sselda 3568 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥𝐵)
1310necomd 2837 . . . . 5 (𝜑𝑄𝑃)
1413adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄𝑃)
15 simpr 476 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄))
161, 2, 3, 5, 7, 9, 12, 14, 15lncom 25317 . . 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 25247 . . . . 5 (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵)
2120sselda 3568 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥𝐵)
2210adantr 480 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃𝑄)
23 simpr 476 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑄𝐿𝑃))
241, 2, 3, 17, 18, 19, 21, 22, 23lncom 25317 . . 3 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄))
2516, 24impbida 873 . 2 (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃)))
2625eqrdv 2608 1 (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wne 2780  cfv 5804  (class class class)co 6549  Basecbs 15695  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152
This theorem is referenced by:  tglinethru  25331  coltr3  25343  footeq  25416  colperpexlem3  25424  mideulem2  25426  opphllem  25427  midex  25429  opphllem3  25441  opphllem5  25443  lmicom  25480  lmiisolem  25488  lnperpex  25495  trgcopy  25496
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