Theorem tgbtwntriv2 25182

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwntriv2.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwntriv2.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
tgbtwntriv2	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵))

Proof of Theorem tgbtwntriv2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 790	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐺 ∈ TarskiG)
7		tgbtwntriv2.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8	7	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 ∈ 𝑃)
9		simplr 788	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝑥 ∈ 𝑃)
10		simpr 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → (𝐵 − 𝑥) = (𝐵 − 𝐵))
11	2, 3, 4, 6, 8, 9, 8, 10	axtgcgrid 25162	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 = 𝑥)
12	11	adantrl 748	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 = 𝑥)
13	12	oveq2d 6565	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥))
14	1, 13	eleqtrrd 2691	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵))
15		tgbtwntriv2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
16	2, 3, 4, 5, 15, 7, 7, 7	axtgsegcon 25163	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)))
17	14, 16	r19.29a 3060	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-trkgc 25147  df-trkgcb 25149  df-trkg 25152

This theorem is referenced by:  tgbtwncom  25183  tgbtwntriv1  25186  tgcolg  25249  legid  25282  hlid  25304  lnhl  25310  tglinerflx2  25329  mirreu3  25349  mirconn  25373  symquadlem  25384  outpasch  25447  hlpasch  25448