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Theorem tgcgrtriv 26197
Description: Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgcgrtriv.1 (𝜑𝐴𝑃)
tgcgrtriv.2 (𝜑𝐵𝑃)
Assertion
Ref Expression
tgcgrtriv (𝜑 → (𝐴 𝐴) = (𝐵 𝐵))

Proof of Theorem tgcgrtriv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tkgeom.p . . . . 5 𝑃 = (Base‘𝐺)
2 tkgeom.d . . . . 5 = (dist‘𝐺)
3 tkgeom.i . . . . 5 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 722 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → 𝐺 ∈ TarskiG)
6 tgcgrtriv.1 . . . . . 6 (𝜑𝐴𝑃)
76ad2antrr 722 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → 𝐴𝑃)
8 simplr 765 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → 𝑥𝑃)
9 tgcgrtriv.2 . . . . . 6 (𝜑𝐵𝑃)
109ad2antrr 722 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → 𝐵𝑃)
11 simprr 769 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → (𝐴 𝑥) = (𝐵 𝐵))
121, 2, 3, 5, 7, 8, 10, 11axtgcgrid 26176 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → 𝐴 = 𝑥)
1312oveq2d 7161 . . 3 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → (𝐴 𝐴) = (𝐴 𝑥))
1413, 11eqtrd 2853 . 2 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → (𝐴 𝐴) = (𝐵 𝐵))
151, 2, 3, 4, 9, 6, 9, 9axtgsegcon 26177 . 2 (𝜑 → ∃𝑥𝑃 (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵)))
1614, 15r19.29a 3286 1 (𝜑 → (𝐴 𝐴) = (𝐵 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  Itvcitv 26149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-trkgc 26161  df-trkgcb 26163  df-trkg 26166
This theorem is referenced by:  tgcgrextend  26198  tgcgrsub  26222  iscgrglt  26227  trgcgrg  26228  tgbtwnconn1lem3  26287  leg0  26305
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