Theorem axtgcgrrflx 28696

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 28687	. . . . 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 4197	. . . . . 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 4197	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
4	2, 3	sstri 3954	. . . . 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 3991	. . . 4 ⊢ TarskiG ⊆ TarskiGC
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3943	. . 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 28688	. . . . 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 7418	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝑦) = (𝑋 − 𝑦))
18		oveq2 7419	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 − 𝑥) = (𝑦 − 𝑋))
19	17, 18	eqeq12d 2785	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝑦) = (𝑦 − 𝑥) ↔ (𝑋 − 𝑦) = (𝑦 − 𝑋)))
20		oveq2 7419	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 − 𝑦) = (𝑋 − 𝑌))
21		oveq1 7418	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 − 𝑋) = (𝑌 − 𝑋))
22	20, 21	eqeq12d 2785	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 − 𝑦) = (𝑦 − 𝑋) ↔ (𝑋 − 𝑌) = (𝑌 − 𝑋)))
23	19, 22	rspc2v 3601	. . 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))

Syntax hints:   → wi 4   ∧ wa 400   ∨ w3o 1100   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  {crab 3423  Vcvv 3463  [wsbc 3753   ∖ cdif 3910   ∩ cin 3912  {csn 4594  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17268  distcds 17318  TarskiGcstrkg 28661  TarskiG_Ccstrkgc 28662  TarskiG_Bcstrkgb 28663  TarskiG_CBcstrkgcb 28664  Itvcitv 28667  LineGclng 28668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414  df-trkgc 28682  df-trkg 28687

This theorem is referenced by:  tgcgrcomimp  28711  tgcgrcomr  28712  tgcgrcoml  28713  tgcgrcomlr  28714  tgbtwnconn1lem1  28806  tgbtwnconn1lem2  28807  tgbtwnconn1lem3  28808  miriso  28908  symquadlem  28927  midexlem  28930  footexALT  28956  footexlem1  28957  footexlem2  28958  colperpexlem1  28969  opphllem  28974  cgraswap  29087  isoas  29135  f1otrg  29160