Theorem mulg1

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

	Ref	Expression
Hypotheses	mulg1.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mulg1.m	⊢ · = ( .g ‘ 𝐺 )
Assertion	mulg1	⊢ ( 𝑋 ∈ 𝐵 → ( 1 · 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		mulg1.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mulg1.m	⊢ · = ( .g ‘ 𝐺 )
3		1nn	⊢ 1 ∈ ℕ
4		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
5		eqid	⊢ seq 1 ( ( +g ‘ 𝐺 ) , ( ℕ × { 𝑋 } ) ) = seq 1 ( ( +g ‘ 𝐺 ) , ( ℕ × { 𝑋 } ) )
6	1 4 2 5	mulgnn	⊢ ( ( 1 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → ( 1 · 𝑋 ) = ( seq 1 ( ( +g ‘ 𝐺 ) , ( ℕ × { 𝑋 } ) ) ‘ 1 ) )
7	3 6	mpan	⊢ ( 𝑋 ∈ 𝐵 → ( 1 · 𝑋 ) = ( seq 1 ( ( +g ‘ 𝐺 ) , ( ℕ × { 𝑋 } ) ) ‘ 1 ) )
8		1z	⊢ 1 ∈ ℤ
9		fvconst2g	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 1 ∈ ℕ ) → ( ( ℕ × { 𝑋 } ) ‘ 1 ) = 𝑋 )
10	3 9	mpan2	⊢ ( 𝑋 ∈ 𝐵 → ( ( ℕ × { 𝑋 } ) ‘ 1 ) = 𝑋 )
11	8 10	seq1i	⊢ ( 𝑋 ∈ 𝐵 → ( seq 1 ( ( +g ‘ 𝐺 ) , ( ℕ × { 𝑋 } ) ) ‘ 1 ) = 𝑋 )
12	7 11	eqtrd	⊢ ( 𝑋 ∈ 𝐵 → ( 1 · 𝑋 ) = 𝑋 )