Theorem tgcgrcomr 28820

Description: Congruence commutes on the RHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.)

Hypotheses

Ref	Expression
tkgeom.p	⊢ 𝑃 = (Base‘𝐺)
tkgeom.d	⊢ − = (dist‘𝐺)
tkgeom.i	⊢ 𝐼 = (Itv‘𝐺)
tkgeom.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgcgrcomr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgcgrcomr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgcgrcomr.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgcgrcomr.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgcgrcomr.6	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))

Assertion

Ref	Expression
tgcgrcomr	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐶))

Proof of Theorem tgcgrcomr

Step	Hyp	Ref	Expression
1		tgcgrcomr.6	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
2		tkgeom.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
3		tkgeom.d	. . 3 ⊢ − = (dist‘𝐺)
4		tkgeom.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
5		tkgeom.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		tgcgrcomr.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7		tgcgrcomr.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
8	2, 3, 4, 5, 6, 7	axtgcgrrflx 28804	. 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
9	1, 8	eqtrd 2795	1 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  distcds 17352  TarskiGcstrkg 28769  Itvcitv 28775

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 2732  ax-nul 5263

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  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 6489  df-fv 6541  df-ov 7417  df-trkgc 28790  df-trkg 28795

This theorem is used by:  tgbtwnconn1lem1  28915  dfcgra2  29218