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Theorem tgbtwntriv2 25376
Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwntriv2.1 (𝜑𝐴𝑃)
tgbtwntriv2.2 (𝜑𝐵𝑃)
Assertion
Ref Expression
tgbtwntriv2 (𝜑𝐵 ∈ (𝐴𝐼𝐵))

Proof of Theorem tgbtwntriv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simprl 794 . . 3 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 ∈ (𝐴𝐼𝑥))
2 tkgeom.p . . . . . 6 𝑃 = (Base‘𝐺)
3 tkgeom.d . . . . . 6 = (dist‘𝐺)
4 tkgeom.i . . . . . 6 𝐼 = (Itv‘𝐺)
5 tkgeom.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
65ad2antrr 762 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐺 ∈ TarskiG)
7 tgbtwntriv2.2 . . . . . . 7 (𝜑𝐵𝑃)
87ad2antrr 762 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐵𝑃)
9 simplr 792 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝑥𝑃)
10 simpr 477 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → (𝐵 𝑥) = (𝐵 𝐵))
112, 3, 4, 6, 8, 9, 8, 10axtgcgrid 25356 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐵 = 𝑥)
1211adantrl 752 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 = 𝑥)
1312oveq2d 6663 . . 3 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥))
141, 13eleqtrrd 2703 . 2 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵))
15 tgbtwntriv2.1 . . 3 (𝜑𝐴𝑃)
162, 3, 4, 5, 15, 7, 7, 7axtgsegcon 25357 . 2 (𝜑 → ∃𝑥𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵)))
1714, 16r19.29a 3076 1 (𝜑𝐵 ∈ (𝐴𝐼𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  cfv 5886  (class class class)co 6647  Basecbs 15851  distcds 15944  TarskiGcstrkg 25323  Itvcitv 25329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-trkgc 25341  df-trkgcb 25343  df-trkg 25346
This theorem is referenced by:  tgbtwncom  25377  tgbtwntriv1  25380  tgcolg  25443  legid  25476  hlid  25498  lnhl  25504  tglinerflx2  25523  mirreu3  25543  mirconn  25567  symquadlem  25578  outpasch  25641  hlpasch  25642
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