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Theorem tglinecom 25437
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 25354 . . . . 5 (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵)
1211sselda 3584 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥𝐵)
1310necomd 2845 . . . . 5 (𝜑𝑄𝑃)
1413adantr 481 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄𝑃)
15 simpr 477 . . . 4 ((𝜑𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄))
161, 2, 3, 5, 7, 9, 12, 14, 15lncom 25424 . . 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 25354 . . . . 5 (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵)
2120sselda 3584 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥𝐵)
2210adantr 481 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃𝑄)
23 simpr 477 . . . 4 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑄𝐿𝑃))
241, 2, 3, 17, 18, 19, 21, 22, 23lncom 25424 . . 3 ((𝜑𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄))
2516, 24impbida 876 . 2 (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃)))
2625eqrdv 2619 1 (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  cfv 5849  (class class class)co 6607  Basecbs 15784  TarskiGcstrkg 25236  Itvcitv 25242  LineGclng 25243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-iota 5812  df-fun 5851  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-trkgc 25254  df-trkgb 25255  df-trkgcb 25256  df-trkg 25259
This theorem is referenced by:  tglinethru  25438  coltr3  25450  footeq  25523  colperpexlem3  25531  mideulem2  25533  opphllem  25534  midex  25536  opphllem3  25548  opphllem5  25550  lmicom  25587  lmiisolem  25595  lnperpex  25602  trgcopy  25603
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