Theorem declt

Description: Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015) (Revised by AV, 6-Sep-2021)

	Ref	Expression
Hypotheses	declt.a	|- A e. NN0
	declt.b	|- B e. NN0
	declt.c	|- C e. NN
	declt.l	|- B < C
Assertion	declt	|- ; A B < ; A C

Proof

Step	Hyp	Ref	Expression
1		declt.a	|- A e. NN0
2		declt.b	|- B e. NN0
3		declt.c	|- C e. NN
4		declt.l	|- B < C
5		10nn	|- ; 1 0 e. NN
6	5 1 2 3 4	numlt	|- ( ( ; 1 0 x. A ) + B ) < ( ( ; 1 0 x. A ) + C )
7		dfdec10	|- ; A B = ( ( ; 1 0 x. A ) + B )
8		dfdec10	|- ; A C = ( ( ; 1 0 x. A ) + C )
9	6 7 8	3brtr4i	|- ; A B < ; A C