decaddc

Description: Add two numerals M and N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014) (Revised by AV, 6-Sep-2021)

	Ref	Expression
Hypotheses	decma.a	$⊢ A \in ℕ_{0}$
	decma.b	$⊢ B \in ℕ_{0}$
	decma.c	$⊢ C \in ℕ_{0}$
	decma.d	$⊢ D \in ℕ_{0}$
	decma.m	No typesetting found for \|- M = ; A B with typecode \|-
	decma.n	No typesetting found for \|- N = ; C D with typecode \|-
	decaddc.e	$⊢ A + C + 1 = E$
	decaddc.f	$⊢ F \in ℕ_{0}$
	decaddc.2	No typesetting found for \|- ( B + D ) = ; 1 F with typecode \|-
Assertion	decaddc	Could not format assertion : No typesetting found for \|- ( M + N ) = ; E F with typecode \|-

Step	Hyp	Ref	Expression
1		decma.a	$⊢ A \in ℕ_{0}$
2		decma.b	$⊢ B \in ℕ_{0}$
3		decma.c	$⊢ C \in ℕ_{0}$
4		decma.d	$⊢ D \in ℕ_{0}$
5		decma.m	Could not format M = ; A B : No typesetting found for \|- M = ; A B with typecode \|-
6		decma.n	Could not format N = ; C D : No typesetting found for \|- N = ; C D with typecode \|-
7		decaddc.e	$⊢ A + C + 1 = E$
8		decaddc.f	$⊢ F \in ℕ_{0}$
9		decaddc.2	Could not format ( B + D ) = ; 1 F : No typesetting found for \|- ( B + D ) = ; 1 F with typecode \|-
10		10nn0	$⊢ 10 \in ℕ_{0}$
11		dfdec10	Could not format ; A B = ( ( ; 1 0 x. A ) + B ) : No typesetting found for \|- ; A B = ( ( ; 1 0 x. A ) + B ) with typecode \|-
12	5 11	eqtri	$⊢ M = 10 A + B$
13		dfdec10	Could not format ; C D = ( ( ; 1 0 x. C ) + D ) : No typesetting found for \|- ; C D = ( ( ; 1 0 x. C ) + D ) with typecode \|-
14	6 13	eqtri	$⊢ N = 10 C + D$
15		dfdec10	Could not format ; 1 F = ( ( ; 1 0 x. 1 ) + F ) : No typesetting found for \|- ; 1 F = ( ( ; 1 0 x. 1 ) + F ) with typecode \|-
16	9 15	eqtri	$⊢ B + D = 10 \cdot 1 + F$
17	10 1 2 3 4 12 14 8 7 16	numaddc	$⊢ M + N = 10 E + F$
18		dfdec10	Could not format ; E F = ( ( ; 1 0 x. E ) + F ) : No typesetting found for \|- ; E F = ( ( ; 1 0 x. E ) + F ) with typecode \|-
19	17 18	eqtr4i	Could not format ( M + N ) = ; E F : No typesetting found for \|- ( M + N ) = ; E F with typecode \|-

Metamath Proof Explorer