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Theorem tgbtwntriv2 25602
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 811 . . 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 764 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐺 ∈ TarskiG)
7 tgbtwntriv2.2 . . . . . . 7 (𝜑𝐵𝑃)
87ad2antrr 764 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐵𝑃)
9 simplr 809 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝑥𝑃)
10 simpr 479 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → (𝐵 𝑥) = (𝐵 𝐵))
112, 3, 4, 6, 8, 9, 8, 10axtgcgrid 25582 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐵 = 𝑥)
1211adantrl 754 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 = 𝑥)
1312oveq2d 6830 . . 3 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥))
141, 13eleqtrrd 2842 . 2 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵))
15 tgbtwntriv2.1 . . 3 (𝜑𝐴𝑃)
162, 3, 4, 5, 15, 7, 7, 7axtgsegcon 25583 . 2 (𝜑 → ∃𝑥𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵)))
1714, 16r19.29a 3216 1 (𝜑𝐵 ∈ (𝐴𝐼𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6814  Basecbs 16079  distcds 16172  TarskiGcstrkg 25549  Itvcitv 25555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817  df-trkgc 25567  df-trkgcb 25569  df-trkg 25572
This theorem is referenced by:  tgbtwncom  25603  tgbtwntriv1  25606  tgcolg  25669  legid  25702  hlid  25724  lnhl  25730  tglinerflx2  25749  mirreu3  25769  mirconn  25793  symquadlem  25804  outpasch  25867  hlpasch  25868
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