Theorem tgcgrcoml

Description: Congruence commutes on the LHS. 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)

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

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tkgeom.d	⊢ − = ( dist ‘ 𝐺 )
3		tkgeom.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tkgeom.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgcgrcomr.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgcgrcomr.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		tgcgrcomr.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		tgcgrcomr.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
9		tgcgrcomr.6	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) )
10	1 2 3 4 5 6	axtgcgrrflx	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐴 ) )
11	10 9	eqtr3d	⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) = ( 𝐶 − 𝐷 ) )