Theorem tgcgreqb

Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019)

	Ref	Expression
Hypotheses	tkgeom.p	|- P = ( Base ` G )
	tkgeom.d	|- .- = ( dist ` G )
	tkgeom.i	|- I = ( Itv ` G )
	tkgeom.g	|- ( ph -> G e. TarskiG )
	tgcgrcomlr.a	|- ( ph -> A e. P )
	tgcgrcomlr.b	|- ( ph -> B e. P )
	tgcgrcomlr.c	|- ( ph -> C e. P )
	tgcgrcomlr.d	|- ( ph -> D e. P )
	tgcgrcomlr.6	|- ( ph -> ( A .- B ) = ( C .- D ) )
Assertion	tgcgreqb	|- ( ph -> ( A = B <-> 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	|- ( ph -> G e. TarskiG )
5		tgcgrcomlr.a	|- ( ph -> A e. P )
6		tgcgrcomlr.b	|- ( ph -> B e. P )
7		tgcgrcomlr.c	|- ( ph -> C e. P )
8		tgcgrcomlr.d	|- ( ph -> D e. P )
9		tgcgrcomlr.6	|- ( ph -> ( A .- B ) = ( C .- D ) )
10	4	adantr	|- ( ( ph /\ A = B ) -> G e. TarskiG )
11	7	adantr	|- ( ( ph /\ A = B ) -> C e. P )
12	8	adantr	|- ( ( ph /\ A = B ) -> D e. P )
13	6	adantr	|- ( ( ph /\ A = B ) -> B e. P )
14	9	adantr	|- ( ( ph /\ A = B ) -> ( A .- B ) = ( C .- D ) )
15		simpr	|- ( ( ph /\ A = B ) -> A = B )
16	15	oveq1d	|- ( ( ph /\ A = B ) -> ( A .- B ) = ( B .- B ) )
17	14 16	eqtr3d	|- ( ( ph /\ A = B ) -> ( C .- D ) = ( B .- B ) )
18	1 2 3 10 11 12 13 17	axtgcgrid	|- ( ( ph /\ A = B ) -> C = D )
19	4	adantr	|- ( ( ph /\ C = D ) -> G e. TarskiG )
20	5	adantr	|- ( ( ph /\ C = D ) -> A e. P )
21	6	adantr	|- ( ( ph /\ C = D ) -> B e. P )
22	8	adantr	|- ( ( ph /\ C = D ) -> D e. P )
23	9	adantr	|- ( ( ph /\ C = D ) -> ( A .- B ) = ( C .- D ) )
24		simpr	|- ( ( ph /\ C = D ) -> C = D )
25	24	oveq1d	|- ( ( ph /\ C = D ) -> ( C .- D ) = ( D .- D ) )
26	23 25	eqtrd	|- ( ( ph /\ C = D ) -> ( A .- B ) = ( D .- D ) )
27	1 2 3 19 20 21 22 26	axtgcgrid	|- ( ( ph /\ C = D ) -> A = B )
28	18 27	impbida	|- ( ph -> ( A = B <-> C = D ) )