Theorem axtgcgrrflx 26553

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 26544	. . . . 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 4143	. . . . . 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 4143	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
4	2, 3	sstri 3910	. . . . 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 3935	. . . 4 ⊢ TarskiG ⊆ TarskiGC
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sseldi 3899	. . 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 26545	. . . . 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 500	. . . 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 498	. . 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 7220	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝑦) = (𝑋 − 𝑦))
18		oveq2 7221	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 − 𝑥) = (𝑦 − 𝑋))
19	17, 18	eqeq12d 2753	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝑦) = (𝑦 − 𝑥) ↔ (𝑋 − 𝑦) = (𝑦 − 𝑋)))
20		oveq2 7221	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 − 𝑦) = (𝑋 − 𝑌))
21		oveq1 7220	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 − 𝑋) = (𝑌 − 𝑋))
22	20, 21	eqeq12d 2753	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 − 𝑦) = (𝑦 − 𝑋) ↔ (𝑋 − 𝑌) = (𝑌 − 𝑋)))
23	19, 22	rspc2v 3547	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) → (𝑋 − 𝑌) = (𝑌 − 𝑋)))
24	15, 16, 23	syl2anc 587	. 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 399   ∨ w3o 1088   = wceq 1543   ∈ wcel 2110  {cab 2714  ∀wral 3061  {crab 3065  Vcvv 3408  [wsbc 3694   ∖ cdif 3863   ∩ cin 3865  {csn 4541  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  Basecbs 16760  distcds 16811  TarskiGcstrkg 26521  TarskiG_Ccstrkgc 26522  TarskiG_Bcstrkgb 26523  TarskiG_CBcstrkgcb 26524  Itvcitv 26527  LineGclng 26528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-trkgc 26539  df-trkg 26544

This theorem is referenced by:  tgcgrcomimp  26568  tgcgrcomr  26569  tgcgrcoml  26570  tgcgrcomlr  26571  tgbtwnconn1lem1  26663  tgbtwnconn1lem2  26664  tgbtwnconn1lem3  26665  miriso  26761  symquadlem  26780  midexlem  26783  footexALT  26809  footexlem1  26810  footexlem2  26811  colperpexlem1  26821  opphllem  26826  cgraswap  26911  isoas  26955  f1otrg  26962