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Theorem tglinecom 28657
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 28574 . . . . 5 (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵)
1211sselda 3994 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥𝐵)
1310necomd 2993 . . . . 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 28644 . . 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 28574 . . . . 5 (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵)
2120sselda 3994 . . . 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 28644 . . 3 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄))
2516, 24impbida 801 . 2 (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃)))
2625eqrdv 2732 1 (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  cfv 6562  (class class class)co 7430  Basecbs 17244  TarskiGcstrkg 28449  Itvcitv 28455  LineGclng 28456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-trkgc 28470  df-trkgb 28471  df-trkgcb 28472  df-trkg 28475
This theorem is referenced by:  tglinethru  28658  coltr3  28670  footeq  28746  colperpexlem3  28754  mideulem2  28756  opphllem  28757  midex  28759  opphllem3  28771  opphllem5  28773  lmicom  28810  lmiisolem  28818  lnperpex  28825  trgcopy  28826
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