MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tglinecom Structured version   Visualization version   GIF version

Theorem tglinecom 27405
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 481 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝐺 ∈ TarskiG)
6 tglineelsb2.2 . . . . 5 (𝜑𝑄𝐵)
76adantr 481 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄𝐵)
8 tglineelsb2.1 . . . . 5 (𝜑𝑃𝐵)
98adantr 481 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑃𝐵)
10 tglineelsb2.4 . . . . . 6 (𝜑𝑃𝑄)
111, 3, 2, 4, 8, 6, 10tglnssp 27322 . . . . 5 (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵)
1211sselda 3942 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥𝐵)
1310necomd 2997 . . . . 5 (𝜑𝑄𝑃)
1413adantr 481 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄𝑃)
15 simpr 485 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄))
161, 2, 3, 5, 7, 9, 12, 14, 15lncom 27392 . . 3 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑄𝐿𝑃))
174adantr 481 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝐺 ∈ TarskiG)
188adantr 481 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃𝐵)
196adantr 481 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑄𝐵)
201, 3, 2, 4, 6, 8, 13tglnssp 27322 . . . . 5 (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵)
2120sselda 3942 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥𝐵)
2210adantr 481 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃𝑄)
23 simpr 485 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑄𝐿𝑃))
241, 2, 3, 17, 18, 19, 21, 22, 23lncom 27392 . . 3 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄))
2516, 24impbida 799 . 2 (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃)))
2625eqrdv 2735 1 (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2941  cfv 6493  (class class class)co 7351  Basecbs 17042  TarskiGcstrkg 27197  Itvcitv 27203  LineGclng 27204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-trkgc 27218  df-trkgb 27219  df-trkgcb 27220  df-trkg 27223
This theorem is referenced by:  tglinethru  27406  coltr3  27418  footeq  27494  colperpexlem3  27502  mideulem2  27504  opphllem  27505  midex  27507  opphllem3  27519  opphllem5  27521  lmicom  27558  lmiisolem  27566  lnperpex  27573  trgcopy  27574
  Copyright terms: Public domain W3C validator