Theorem numsuc

Description: The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014)

	Ref	Expression
Hypotheses	numnncl.1	⊢ 𝑇 ∈ ℕ0
	numnncl.2	⊢ 𝐴 ∈ ℕ0
	numcl.2	⊢ 𝐵 ∈ ℕ0
	numsuc.4	⊢ ( 𝐵 + 1 ) = 𝐶
	numsuc.5	⊢ 𝑁 = ( ( 𝑇 · 𝐴 ) + 𝐵 )
Assertion	numsuc	⊢ ( 𝑁 + 1 ) = ( ( 𝑇 · 𝐴 ) + 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		numnncl.1	⊢ 𝑇 ∈ ℕ0
2		numnncl.2	⊢ 𝐴 ∈ ℕ0
3		numcl.2	⊢ 𝐵 ∈ ℕ0
4		numsuc.4	⊢ ( 𝐵 + 1 ) = 𝐶
5		numsuc.5	⊢ 𝑁 = ( ( 𝑇 · 𝐴 ) + 𝐵 )
6	5	oveq1i	⊢ ( 𝑁 + 1 ) = ( ( ( 𝑇 · 𝐴 ) + 𝐵 ) + 1 )
7	1 2	nn0mulcli	⊢ ( 𝑇 · 𝐴 ) ∈ ℕ0
8	7	nn0cni	⊢ ( 𝑇 · 𝐴 ) ∈ ℂ
9	3	nn0cni	⊢ 𝐵 ∈ ℂ
10		ax-1cn	⊢ 1 ∈ ℂ
11	8 9 10	addassi	⊢ ( ( ( 𝑇 · 𝐴 ) + 𝐵 ) + 1 ) = ( ( 𝑇 · 𝐴 ) + ( 𝐵 + 1 ) )
12	4	oveq2i	⊢ ( ( 𝑇 · 𝐴 ) + ( 𝐵 + 1 ) ) = ( ( 𝑇 · 𝐴 ) + 𝐶 )
13	6 11 12	3eqtri	⊢ ( 𝑁 + 1 ) = ( ( 𝑇 · 𝐴 ) + 𝐶 )