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Theorem tgcgrtriv 27999
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 = (Baseβ€˜πΊ)
tkgeom.d βˆ’ = (distβ€˜πΊ)
tkgeom.i 𝐼 = (Itvβ€˜πΊ)
tkgeom.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tgcgrtriv.1 (πœ‘ β†’ 𝐴 ∈ 𝑃)
tgcgrtriv.2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
Assertion
Ref Expression
tgcgrtriv (πœ‘ β†’ (𝐴 βˆ’ 𝐴) = (𝐡 βˆ’ 𝐡))

Proof of Theorem tgcgrtriv
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 tkgeom.p . . . . 5 𝑃 = (Baseβ€˜πΊ)
2 tkgeom.d . . . . 5 βˆ’ = (distβ€˜πΊ)
3 tkgeom.i . . . . 5 𝐼 = (Itvβ€˜πΊ)
4 tkgeom.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
54ad2antrr 723 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐴 ∈ (𝐡𝐼π‘₯) ∧ (𝐴 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ 𝐺 ∈ TarskiG)
6 tgcgrtriv.1 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ 𝑃)
76ad2antrr 723 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐴 ∈ (𝐡𝐼π‘₯) ∧ (𝐴 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ 𝐴 ∈ 𝑃)
8 simplr 766 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐴 ∈ (𝐡𝐼π‘₯) ∧ (𝐴 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ π‘₯ ∈ 𝑃)
9 tgcgrtriv.2 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ 𝑃)
109ad2antrr 723 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐴 ∈ (𝐡𝐼π‘₯) ∧ (𝐴 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ 𝐡 ∈ 𝑃)
11 simprr 770 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐴 ∈ (𝐡𝐼π‘₯) ∧ (𝐴 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ (𝐴 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))
121, 2, 3, 5, 7, 8, 10, 11axtgcgrid 27978 . . . 4 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐴 ∈ (𝐡𝐼π‘₯) ∧ (𝐴 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ 𝐴 = π‘₯)
1312oveq2d 7428 . . 3 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐴 ∈ (𝐡𝐼π‘₯) ∧ (𝐴 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ (𝐴 βˆ’ 𝐴) = (𝐴 βˆ’ π‘₯))
1413, 11eqtrd 2771 . 2 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐴 ∈ (𝐡𝐼π‘₯) ∧ (𝐴 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ (𝐴 βˆ’ 𝐴) = (𝐡 βˆ’ 𝐡))
151, 2, 3, 4, 9, 6, 9, 9axtgsegcon 27979 . 2 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 (𝐴 ∈ (𝐡𝐼π‘₯) ∧ (𝐴 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)))
1614, 15r19.29a 3161 1 (πœ‘ β†’ (𝐴 βˆ’ 𝐴) = (𝐡 βˆ’ 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  β€˜cfv 6544  (class class class)co 7412  Basecbs 17149  distcds 17211  TarskiGcstrkg 27942  Itvcitv 27948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7415  df-trkgc 27963  df-trkgcb 27965  df-trkg 27968
This theorem is referenced by:  tgcgrextend  28000  tgcgrsub  28024  iscgrglt  28029  trgcgrg  28030  tgbtwnconn1lem3  28089  leg0  28107
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