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Metamath Proof Explorer

Theorem seqcaopr2

Description: The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014)

Ref Expression
Hypotheses seqcaopr2.1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
seqcaopr2.2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝑄 𝑦 ) ∈ 𝑆 )
seqcaopr2.3 ⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) ) ) → ( ( 𝑥 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) )
seqcaopr2.4 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
seqcaopr2.5 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 )
seqcaopr2.6 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ 𝑆 )
seqcaopr2.7 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) 𝑄 ( 𝐺 ‘ 𝑘 ) ) )
Assertion seqcaopr2 ( 𝜑 → ( seq 𝑀 ( + , 𝐻 ) ‘ 𝑁 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 seqcaopr2.1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
2 seqcaopr2.2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝑄 𝑦 ) ∈ 𝑆 )
3 seqcaopr2.3 ⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) ) ) → ( ( 𝑥 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) )
4 seqcaopr2.4 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
5 seqcaopr2.5 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 )
6 seqcaopr2.6 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ 𝑆 )
7 seqcaopr2.7 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) 𝑄 ( 𝐺 ‘ 𝑘 ) ) )
8 elfzouz ⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) )
9 8 adantl ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) )
10 elfzouz2 ⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) )
11 10 adantl ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) )
12 fzss2 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) → ( 𝑀 ... 𝑛 ) ⊆ ( 𝑀 ... 𝑁 ) )
13 11 12 syl ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑀 ... 𝑛 ) ⊆ ( 𝑀 ... 𝑁 ) )
14 13 sselda ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑛 ) ) → 𝑥 ∈ ( 𝑀 ... 𝑁 ) )
15 6 ralrimiva ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐺 ‘ 𝑘 ) ∈ 𝑆 )
16 15 adantr ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐺 ‘ 𝑘 ) ∈ 𝑆 )
17 fveq2 ⊢ ( 𝑘 = 𝑥 → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑥 ) )
18 17 eleq1d ⊢ ( 𝑘 = 𝑥 → ( ( 𝐺 ‘ 𝑘 ) ∈ 𝑆 ↔ ( 𝐺 ‘ 𝑥 ) ∈ 𝑆 ) )
19 18 rspccva ⊢ ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐺 ‘ 𝑘 ) ∈ 𝑆 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝑆 )
20 16 19 sylan ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝑆 )
21 14 20 syldan ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑛 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝑆 )
22 1 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
23 9 21 22 seqcl ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ 𝑆 )
24 fzofzp1 ⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) )
25 fveq2 ⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ ( 𝑛 + 1 ) ) )
26 25 eleq1d ⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( ( 𝐺 ‘ 𝑘 ) ∈ 𝑆 ↔ ( 𝐺 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 ) )
27 26 rspccva ⊢ ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐺 ‘ 𝑘 ) ∈ 𝑆 ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐺 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 )
28 15 24 27 syl2an ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐺 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 )
29 5 ralrimiva ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 )
30 fveq2 ⊢ ( 𝑘 = 𝑥 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑥 ) )
31 30 eleq1d ⊢ ( 𝑘 = 𝑥 → ( ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 ↔ ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) )
32 31 rspccva ⊢ ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 )
33 29 32 sylan ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 )
34 33 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 )
35 14 34 syldan ⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ( 𝑀 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 )
36 9 35 22 seqcl ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ 𝑆 )
37 fveq2 ⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ ( 𝑛 + 1 ) ) )
38 37 eleq1d ⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 ↔ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 ) )
39 38 rspccva ⊢ ( ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 ∧ ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 )
40 29 24 39 syl2an ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 )
41 3 anassrs ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) ∧ ( 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) ) → ( ( 𝑥 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) )
42 41 ralrimivva ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( 𝑥 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) )
43 42 ralrimivva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( 𝑥 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) )
44 43 adantr ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( 𝑥 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) )
45 oveq1 ⊢ ( 𝑥 = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) → ( 𝑥 𝑄 𝑧 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) )
46 45 oveq1d ⊢ ( 𝑥 = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) → ( ( 𝑥 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) )
47 oveq1 ⊢ ( 𝑥 = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) → ( 𝑥 + 𝑦 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + 𝑦 ) )
48 47 oveq1d ⊢ ( 𝑥 = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) → ( ( 𝑥 + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) )
49 46 48 eqeq12d ⊢ ( 𝑥 = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) → ( ( ( 𝑥 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) ↔ ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) ) )
50 49 2ralbidv ⊢ ( 𝑥 = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) → ( ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( 𝑥 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) ↔ ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) ) )
51 oveq1 ⊢ ( 𝑦 = ( 𝐹 ‘ ( 𝑛 + 1 ) ) → ( 𝑦 𝑄 𝑤 ) = ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) )
52 51 oveq2d ⊢ ( 𝑦 = ( 𝐹 ‘ ( 𝑛 + 1 ) ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) )
53 oveq2 ⊢ ( 𝑦 = ( 𝐹 ‘ ( 𝑛 + 1 ) ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + 𝑦 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) )
54 53 oveq1d ⊢ ( 𝑦 = ( 𝐹 ‘ ( 𝑛 + 1 ) ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( 𝑧 + 𝑤 ) ) )
55 52 54 eqeq12d ⊢ ( 𝑦 = ( 𝐹 ‘ ( 𝑛 + 1 ) ) → ( ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) ↔ ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( 𝑧 + 𝑤 ) ) ) )
56 55 2ralbidv ⊢ ( 𝑦 = ( 𝐹 ‘ ( 𝑛 + 1 ) ) → ( ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) ↔ ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( 𝑧 + 𝑤 ) ) ) )
57 50 56 rspc2va ⊢ ( ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ∈ 𝑆 ∧ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( 𝑥 𝑄 𝑧 ) + ( 𝑦 𝑄 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) 𝑄 ( 𝑧 + 𝑤 ) ) ) → ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( 𝑧 + 𝑤 ) ) )
58 36 40 44 57 syl21anc ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( 𝑧 + 𝑤 ) ) )
59 oveq2 ⊢ ( 𝑧 = ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) → ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) )
60 59 oveq1d ⊢ ( 𝑧 = ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) )
61 oveq1 ⊢ ( 𝑧 = ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) → ( 𝑧 + 𝑤 ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + 𝑤 ) )
62 61 oveq2d ⊢ ( 𝑧 = ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( 𝑧 + 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + 𝑤 ) ) )
63 60 62 eqeq12d ⊢ ( 𝑧 = ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) → ( ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( 𝑧 + 𝑤 ) ) ↔ ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + 𝑤 ) ) ) )
64 oveq2 ⊢ ( 𝑤 = ( 𝐺 ‘ ( 𝑛 + 1 ) ) → ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) = ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) )
65 64 oveq2d ⊢ ( 𝑤 = ( 𝐺 ‘ ( 𝑛 + 1 ) ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) )
66 oveq2 ⊢ ( 𝑤 = ( 𝐺 ‘ ( 𝑛 + 1 ) ) → ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + 𝑤 ) = ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) )
67 66 oveq2d ⊢ ( 𝑤 = ( 𝐺 ‘ ( 𝑛 + 1 ) ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) )
68 65 67 eqeq12d ⊢ ( 𝑤 = ( 𝐺 ‘ ( 𝑛 + 1 ) ) → ( ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + 𝑤 ) ) ↔ ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) ) )
69 63 68 rspc2va ⊢ ( ( ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ∈ 𝑆 ∧ ( 𝐺 ‘ ( 𝑛 + 1 ) ) ∈ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝑆 ∀ 𝑤 ∈ 𝑆 ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 𝑧 ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 𝑤 ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( 𝑧 + 𝑤 ) ) ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) )
70 23 28 58 69 syl21anc ⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) ) + ( ( 𝐹 ‘ ( 𝑛 + 1 ) ) 𝑄 ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) = ( ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) + ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) 𝑄 ( ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑛 ) + ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) )
71 1 2 4 5 6 7 70 seqcaopr3 ⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐻 ) ‘ 𝑁 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) 𝑄 ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) )