Theorem axtgcgrrflx 28742

Description: Axiom of reflexivity of congruence, Axiom A1 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.)

Hypotheses

Ref	Expression
axtrkg.p	⊢ 𝑃 = (Base‘𝐺)
axtrkg.d	⊢ − = (dist‘𝐺)
axtrkg.i	⊢ 𝐼 = (Itv‘𝐺)
axtrkg.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
axtgcgrrflx.1	⊢ (𝜑 → 𝑋 ∈ 𝑃)
axtgcgrrflx.2	⊢ (𝜑 → 𝑌 ∈ 𝑃)

Assertion

Ref	Expression
axtgcgrrflx	⊢ (𝜑 → (𝑋 − 𝑌) = (𝑌 − 𝑋))

Proof of Theorem axtgcgrrflx

Dummy variables 𝑓 𝑖 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-trkg 28733	. . . . 5 ⊢ TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
2		inss1 4188	. . . . . 6 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGC ∩ TarskiGB)
3		inss1 4188	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
4	2, 3	sstri 3945	. . . . 5 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGC
5	1, 4	eqsstri 3982	. . . 4 ⊢ TarskiG ⊆ TarskiGC
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3934	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiGC)
8		axtrkg.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
9		axtrkg.d	. . . . . 6 ⊢ − = (dist‘𝐺)
10		axtrkg.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
11	8, 9, 10	istrkgc 28734	. . . . 5 ⊢ (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦))))
12	11	simprbi 502	. . . 4 ⊢ (𝐺 ∈ TarskiGC → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦)))
13	12	simpld 499	. . 3 ⊢ (𝐺 ∈ TarskiGC → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥))
14	7, 13	syl 18	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥))
15		axtgcgrrflx.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
16		axtgcgrrflx.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
17		oveq1 7419	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝑦) = (𝑋 − 𝑦))
18		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 − 𝑥) = (𝑦 − 𝑋))
19	17, 18	eqeq12d 2778	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝑦) = (𝑦 − 𝑥) ↔ (𝑋 − 𝑦) = (𝑦 − 𝑋)))
20		oveq2 7420	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 − 𝑦) = (𝑋 − 𝑌))
21		oveq1 7419	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 − 𝑋) = (𝑌 − 𝑋))
22	20, 21	eqeq12d 2778	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 − 𝑦) = (𝑦 − 𝑋) ↔ (𝑋 − 𝑌) = (𝑌 − 𝑋)))
23	19, 22	rspc2v 3591	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) → (𝑋 − 𝑌) = (𝑌 − 𝑋)))
24	15, 16, 23	syl2anc 595	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) → (𝑋 − 𝑌) = (𝑌 − 𝑋)))
25	14, 24	mpd 16	1 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑌 − 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∨ w3o 1101   = wceq 1569   ∈ wcel 2142  {cab 2740  ∀wral 3078  {crab 3415  Vcvv 3454  [wsbc 3743   ∖ cdif 3901   ∩ cin 3903  {csn 4588  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414  Basecbs 17275  distcds 17325  TarskiGcstrkg 28707  TarskiG_Ccstrkgc 28708  TarskiG_Bcstrkgb 28709  TarskiG_CBcstrkgcb 28710  Itvcitv 28713  LineGclng 28714

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5268

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415  df-trkgc 28728  df-trkg 28733

This theorem is used by:  tgcgrcomimp  28757  tgcgrcomr  28758  tgcgrcoml  28759  tgcgrcomlr  28760  tgbtwnconn1lem1  28852  tgbtwnconn1lem2  28853  tgbtwnconn1lem3  28854  miriso  28958  symquadlem  28977  midexlem  28980  footexALT  29009  footexlem1  29010  footexlem2  29011  colperpexlem1  29022  opphllem  29027  cgraswap  29142  isoas  29192  tgaltai  29228  f1otrg  29231