Theorem axtgcgrrflx 27980

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 27971	. . . . 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 4227	. . . . . 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 4227	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
4	2, 3	sstri 3990	. . . . 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 4015	. . . 4 ⊢ TarskiG ⊆ TarskiGC
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3979	. . 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 27972	. . . . 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 495	. . . 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 493	. . 3 ⊢ (𝐺 ∈ TarskiGC → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥))
14	7, 13	syl 17	. 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 2746	. . . 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 2746	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 − 𝑦) = (𝑦 − 𝑋) ↔ (𝑋 − 𝑌) = (𝑌 − 𝑋)))
23	19, 22	rspc2v 3621	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) → (𝑋 − 𝑌) = (𝑌 − 𝑋)))
24	15, 16, 23	syl2anc 582	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) → (𝑋 − 𝑌) = (𝑌 − 𝑋)))
25	14, 24	mpd 15	1 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑌 − 𝑋))

Syntax hints:   → wi 4   ∧ wa 394   ∨ w3o 1084   = wceq 1539   ∈ wcel 2104  {cab 2707  ∀wral 3059  {crab 3430  Vcvv 3472  [wsbc 3776   ∖ cdif 3944   ∩ cin 3946  {csn 4627  ‘cfv 6542  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17148  distcds 17210  TarskiGcstrkg 27945  TarskiG_Ccstrkgc 27946  TarskiG_Bcstrkgb 27947  TarskiG_CBcstrkgcb 27948  Itvcitv 27951  LineGclng 27952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-trkgc 27966  df-trkg 27971

This theorem is referenced by:  tgcgrcomimp  27995  tgcgrcomr  27996  tgcgrcoml  27997  tgcgrcomlr  27998  tgbtwnconn1lem1  28090  tgbtwnconn1lem2  28091  tgbtwnconn1lem3  28092  miriso  28188  symquadlem  28207  midexlem  28210  footexALT  28236  footexlem1  28237  footexlem2  28238  colperpexlem1  28248  opphllem  28253  cgraswap  28338  isoas  28382  f1otrg  28389