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Metamath Proof Explorer

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 ) )