decmul2c

Description: The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014) (Revised by AV, 6-Sep-2021)

	Ref	Expression
Hypotheses	decmul1.p	$⊢ P \in ℕ_{0}$
	decmul1.a	$⊢ A \in ℕ_{0}$
	decmul1.b	$⊢ B \in ℕ_{0}$
	decmul1.n	No typesetting found for \|- N = ; A B with typecode \|-
	decmul1.0	$⊢ D \in ℕ_{0}$
	decmul1c.e	$⊢ E \in ℕ_{0}$
	decmul2c.c	$⊢ P A + E = C$
	decmul2c.2	No typesetting found for \|- ( P x. B ) = ; E D with typecode \|-
Assertion	decmul2c	Could not format assertion : No typesetting found for \|- ( P x. N ) = ; C D with typecode \|-

Step	Hyp	Ref	Expression
1		decmul1.p	$⊢ P \in ℕ_{0}$
2		decmul1.a	$⊢ A \in ℕ_{0}$
3		decmul1.b	$⊢ B \in ℕ_{0}$
4		decmul1.n	Could not format N = ; A B : No typesetting found for \|- N = ; A B with typecode \|-
5		decmul1.0	$⊢ D \in ℕ_{0}$
6		decmul1c.e	$⊢ E \in ℕ_{0}$
7		decmul2c.c	$⊢ P A + E = C$
8		decmul2c.2	Could not format ( P x. B ) = ; E D : No typesetting found for \|- ( P x. B ) = ; E D with typecode \|-
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	4 10	eqtri	$⊢ N = 10 A + B$
12		dfdec10	Could not format ; E D = ( ( ; 1 0 x. E ) + D ) : No typesetting found for \|- ; E D = ( ( ; 1 0 x. E ) + D ) with typecode \|-
13	8 12	eqtri	$⊢ P B = 10 E + D$
14	9 1 2 3 11 5 6 7 13	nummul2c	$⊢ P \cdot N = 10 C + D$
15		dfdec10	Could not format ; C D = ( ( ; 1 0 x. C ) + D ) : No typesetting found for \|- ; C D = ( ( ; 1 0 x. C ) + D ) with typecode \|-
16	14 15	eqtr4i	Could not format ( P x. N ) = ; C D : No typesetting found for \|- ( P x. N ) = ; C D with typecode \|-

Metamath Proof Explorer