Theorem seqfeq3

Description: Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015) (Revised by Mario Carneiro, 25-Apr-2016)

	Ref	Expression
Hypotheses	seqfeq3.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	seqfeq3.f	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 )
	seqfeq3.cl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
	seqfeq3.id	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑥 𝑄 𝑦 ) )
Assertion	seqfeq3	⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) = seq 𝑀 ( 𝑄 , 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		seqfeq3.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
2		seqfeq3.f	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 )
3		seqfeq3.cl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
4		seqfeq3.id	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑥 𝑄 𝑦 ) )
5		seqfn	⊢ ( 𝑀 ∈ ℤ → seq 𝑀 ( + , 𝐹 ) Fn ( ℤ≥ ‘ 𝑀 ) )
6	1 5	syl	⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) Fn ( ℤ≥ ‘ 𝑀 ) )
7		seqfn	⊢ ( 𝑀 ∈ ℤ → seq 𝑀 ( 𝑄 , 𝐹 ) Fn ( ℤ≥ ‘ 𝑀 ) )
8	1 7	syl	⊢ ( 𝜑 → seq 𝑀 ( 𝑄 , 𝐹 ) Fn ( ℤ≥ ‘ 𝑀 ) )
9		simpr	⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑎 ∈ ( ℤ≥ ‘ 𝑀 ) )
10		simpll	⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑎 ) ) → 𝜑 )
11		elfzuz	⊢ ( 𝑥 ∈ ( 𝑀 ... 𝑎 ) → 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) )
12	11	adantl	⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑎 ) ) → 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) )
13	10 12 2	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑎 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 )
14	3	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
15	4	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑥 𝑄 𝑦 ) )
16	9 13 14 15	seqfeq4	⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑎 ) = ( seq 𝑀 ( 𝑄 , 𝐹 ) ‘ 𝑎 ) )
17	6 8 16	eqfnfvd	⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) = seq 𝑀 ( 𝑄 , 𝐹 ) )