Theorem axtgcgrrflx 28924

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 28915	. . . . 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 4182	. . . . . 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 4182	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
4	2, 3	sstri 3940	. . . . 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 3977	. . . 4 ⊢ TarskiG ⊆ TarskiGC
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3929	. . 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 28916	. . . . 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 503	. . . 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 500	. . 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 7427	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝑦) = (𝑋 − 𝑦))
18		oveq2 7428	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 − 𝑥) = (𝑦 − 𝑋))
19	17, 18	eqeq12d 2777	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝑦) = (𝑦 − 𝑥) ↔ (𝑋 − 𝑦) = (𝑦 − 𝑋)))
20		oveq2 7428	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 − 𝑦) = (𝑋 − 𝑌))
21		oveq1 7427	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 − 𝑋) = (𝑌 − 𝑋))
22	20, 21	eqeq12d 2777	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 − 𝑦) = (𝑦 − 𝑋) ↔ (𝑋 − 𝑌) = (𝑌 − 𝑋)))
23	19, 22	rspc2v 3587	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) → (𝑋 − 𝑌) = (𝑌 − 𝑋)))
24	15, 16, 23	syl2anc 596	. 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 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  {crab 3413  Vcvv 3451  [wsbc 3739   ∖ cdif 3896   ∩ cin 3898  {csn 4584  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  Basecbs 17387  distcds 17437  TarskiGcstrkg 28889  TarskiG_Ccstrkgc 28890  TarskiG_Bcstrkgb 28891  TarskiG_CBcstrkgcb 28892  Itvcitv 28895  LineGclng 28896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-trkgc 28910  df-trkg 28915

This theorem is used by:  tgcgrcomimp  28939  tgcgrcomr  28940  tgcgrcoml  28941  tgcgrcomlr  28942  tgbtwnconn1lem1  29035  tgbtwnconn1lem2  29036  tgbtwnconn1lem3  29037  miriso  29142  symquadlem  29161  midexlem  29164  footexALT  29193  footexlem1  29194  footexlem2  29195  colperpexlem1  29206  opphllem  29211  cgraswap  29327  isoas  29409  tgaltai  29445  f1otrg  29448