Theorem tgcgrcomr 26743

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 26727	. 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
9	1, 8	eqtrd 2778	1 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐶))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  distcds 16897  TarskiGcstrkg 26693  Itvcitv 26699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-trkgc 26713  df-trkg 26718

This theorem is referenced by:  tgbtwnconn1lem1  26837  dfcgra2  27095