decadd

Description: Add two numerals M and N (no 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 \|-
	decadd.e	$⊢ A + C = E$
	decadd.f	$⊢ B + D = F$
Assertion	decadd	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		decadd.e	$⊢ A + C = E$
8		decadd.f	$⊢ B + D = F$
9		10nn0	$⊢ 10 \in ℕ_{0}$
10		dfdec10	Could not format ; A B = ( ( ; 1 0 x. A ) + B ) : No typesetting found for \|- ; A B = ( ( ; 1 0 x. A ) + B ) with typecode \|-
11	5 10	eqtri	$⊢ M = 10 A + B$
12		dfdec10	Could not format ; C D = ( ( ; 1 0 x. C ) + D ) : No typesetting found for \|- ; C D = ( ( ; 1 0 x. C ) + D ) with typecode \|-
13	6 12	eqtri	$⊢ N = 10 C + D$
14	9 1 2 3 4 11 13 7 8	numadd	$⊢ M + N = 10 E + F$
15		dfdec10	Could not format ; E F = ( ( ; 1 0 x. E ) + F ) : No typesetting found for \|- ; E F = ( ( ; 1 0 x. E ) + F ) with typecode \|-
16	14 15	eqtr4i	Could not format ( M + N ) = ; E F : No typesetting found for \|- ( M + N ) = ; E F with typecode \|-

Metamath Proof Explorer