Theorem mulgnn

Description: Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014)

	Ref	Expression
Hypotheses	mulgnn.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mulgnn.p	⊢ + = ( +g ‘ 𝐺 )
	mulgnn.t	⊢ · = ( .g ‘ 𝐺 )
	mulgnn.s	⊢ 𝑆 = seq 1 ( + , ( ℕ × { 𝑋 } ) )
Assertion	mulgnn	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) = ( 𝑆 ‘ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		mulgnn.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mulgnn.p	⊢ + = ( +g ‘ 𝐺 )
3		mulgnn.t	⊢ · = ( .g ‘ 𝐺 )
4		mulgnn.s	⊢ 𝑆 = seq 1 ( + , ( ℕ × { 𝑋 } ) )
5		nnz	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ )
6		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
7		eqid	⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
8	1 2 6 7 3 4	mulgval	⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) = if ( 𝑁 = 0 , ( 0g ‘ 𝐺 ) , if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) ) )
9	5 8	sylan	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) = if ( 𝑁 = 0 , ( 0g ‘ 𝐺 ) , if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) ) )
10		nnne0	⊢ ( 𝑁 ∈ ℕ → 𝑁 ≠ 0 )
11	10	neneqd	⊢ ( 𝑁 ∈ ℕ → ¬ 𝑁 = 0 )
12	11	iffalsed	⊢ ( 𝑁 ∈ ℕ → if ( 𝑁 = 0 , ( 0g ‘ 𝐺 ) , if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) ) = if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) )
13		nngt0	⊢ ( 𝑁 ∈ ℕ → 0 < 𝑁 )
14	13	iftrued	⊢ ( 𝑁 ∈ ℕ → if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) = ( 𝑆 ‘ 𝑁 ) )
15	12 14	eqtrd	⊢ ( 𝑁 ∈ ℕ → if ( 𝑁 = 0 , ( 0g ‘ 𝐺 ) , if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) ) = ( 𝑆 ‘ 𝑁 ) )
16	15	adantr	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → if ( 𝑁 = 0 , ( 0g ‘ 𝐺 ) , if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) ) = ( 𝑆 ‘ 𝑁 ) )
17	9 16	eqtrd	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) = ( 𝑆 ‘ 𝑁 ) )