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Theorem iseqcaopr2 9405
Description: The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)
Hypotheses
Ref Expression
iseqcaopr2.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqcaopr2.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
iseqcaopr2.3 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
iseqcaopr2.4 (𝜑𝑁 ∈ (ℤ𝑀))
iseqcaopr2.5 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
iseqcaopr2.6 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)
iseqcaopr2.7 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
iseqcaopr2.s (𝜑𝑆𝑉)
Assertion
Ref Expression
iseqcaopr2 (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))
Distinct variable groups:   𝑤, + ,𝑥,𝑦,𝑧   𝑘,𝐹,𝑤,𝑥,𝑦,𝑧   𝑘,𝐺,𝑤,𝑥,𝑦,𝑧   𝑘,𝐻,𝑥,𝑦,𝑧   𝑘,𝑀,𝑤,𝑥,𝑦,𝑧   𝑘,𝑁,𝑥,𝑦,𝑧   𝑄,𝑘,𝑤,𝑥,𝑦,𝑧   𝑆,𝑘,𝑤,𝑥,𝑦,𝑧   𝜑,𝑘,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   + (𝑘)   𝐻(𝑤)   𝑁(𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤,𝑘)

Proof of Theorem iseqcaopr2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 iseqcaopr2.1 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2 iseqcaopr2.2 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
3 iseqcaopr2.4 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
4 iseqcaopr2.5 . 2 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
5 iseqcaopr2.6 . 2 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)
6 iseqcaopr2.7 . 2 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
7 elfzouz 9110 . . . . 5 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
87adantl 266 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
9 iseqcaopr2.s . . . . 5 (𝜑𝑆𝑉)
109adantr 265 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑆𝑉)
115ralrimiva 2409 . . . . . 6 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
1211adantr 265 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
13 fveq2 5206 . . . . . . 7 (𝑘 = 𝑥 → (𝐺𝑘) = (𝐺𝑥))
1413eleq1d 2122 . . . . . 6 (𝑘 = 𝑥 → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
1514rspccva 2672 . . . . 5 ((∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
1612, 15sylan 271 . . . 4 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
171adantlr 454 . . . 4 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
188, 10, 16, 17iseqcl 9387 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺, 𝑆)‘𝑛) ∈ 𝑆)
19 fzssuz 9030 . . . . 5 (𝑀...𝑁) ⊆ (ℤ𝑀)
20 fzofzp1 9185 . . . . 5 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
2119, 20sseldi 2971 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
22 fveq2 5206 . . . . . 6 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
2322eleq1d 2122 . . . . 5 (𝑘 = (𝑛 + 1) → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺‘(𝑛 + 1)) ∈ 𝑆))
2423rspccva 2672 . . . 4 ((∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
2511, 21, 24syl2an 277 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
264ralrimiva 2409 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆)
27 fveq2 5206 . . . . . . . . 9 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
2827eleq1d 2122 . . . . . . . 8 (𝑘 = 𝑥 → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
2928rspccva 2672 . . . . . . 7 ((∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3026, 29sylan 271 . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3130adantlr 454 . . . . 5 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
328, 10, 31, 17iseqcl 9387 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆)
33 fveq2 5206 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
3433eleq1d 2122 . . . . . 6 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
3534rspccva 2672 . . . . 5 ((∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
3626, 21, 35syl2an 277 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
37 iseqcaopr2.3 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
3837anassrs 386 . . . . . . 7 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ (𝑧𝑆𝑤𝑆)) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
3938ralrimivva 2418 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4039ralrimivva 2418 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4140adantr 265 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
42 oveq1 5547 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝑥𝑄𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧))
4342oveq1d 5555 . . . . . . 7 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)))
44 oveq1 5547 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦))
4544oveq1d 5555 . . . . . . 7 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)))
4643, 45eqeq12d 2070 . . . . . 6 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) ↔ (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤))))
47462ralbidv 2365 . . . . 5 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (∀𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) ↔ ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤))))
48 oveq1 5547 . . . . . . . 8 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝑦𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄𝑤))
4948oveq2d 5556 . . . . . . 7 (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
50 oveq2 5548 . . . . . . . 8 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
5150oveq1d 5555 . . . . . . 7 (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
5249, 51eqeq12d 2070 . . . . . 6 (𝑦 = (𝐹‘(𝑛 + 1)) → ((((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) ↔ (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤))))
53522ralbidv 2365 . . . . 5 (𝑦 = (𝐹‘(𝑛 + 1)) → (∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) ↔ ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤))))
5447, 53rspc2va 2686 . . . 4 ((((seq𝑀( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤))) → ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
5532, 36, 41, 54syl21anc 1145 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
56 oveq2 5548 . . . . . 6 (𝑧 = (seq𝑀( + , 𝐺, 𝑆)‘𝑛) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)))
5756oveq1d 5555 . . . . 5 (𝑧 = (seq𝑀( + , 𝐺, 𝑆)‘𝑛) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
58 oveq1 5547 . . . . . 6 (𝑧 = (seq𝑀( + , 𝐺, 𝑆)‘𝑛) → (𝑧 + 𝑤) = ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + 𝑤))
5958oveq2d 5556 . . . . 5 (𝑧 = (seq𝑀( + , 𝐺, 𝑆)‘𝑛) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + 𝑤)))
6057, 59eqeq12d 2070 . . . 4 (𝑧 = (seq𝑀( + , 𝐺, 𝑆)‘𝑛) → ((((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)) ↔ (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + 𝑤))))
61 oveq2 5548 . . . . . 6 (𝑤 = (𝐺‘(𝑛 + 1)) → ((𝐹‘(𝑛 + 1))𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
6261oveq2d 5556 . . . . 5 (𝑤 = (𝐺‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
63 oveq2 5548 . . . . . 6 (𝑤 = (𝐺‘(𝑛 + 1)) → ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + 𝑤) = ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1))))
6463oveq2d 5556 . . . . 5 (𝑤 = (𝐺‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + 𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1)))))
6562, 64eqeq12d 2070 . . . 4 (𝑤 = (𝐺‘(𝑛 + 1)) → ((((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + 𝑤)) ↔ (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1))))))
6660, 65rspc2va 2686 . . 3 ((((seq𝑀( + , 𝐺, 𝑆)‘𝑛) ∈ 𝑆 ∧ (𝐺‘(𝑛 + 1)) ∈ 𝑆) ∧ ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤))) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1)))))
6718, 25, 55, 66syl21anc 1145 . 2 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1)))))
681, 2, 3, 4, 5, 6, 67, 9iseqcaopr3 9404 1 (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  wral 2323  cfv 4930  (class class class)co 5540  1c1 6948   + caddc 6950  cuz 8569  ...cfz 8976  ..^cfzo 9101  seqcseq 9375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-ltadd 7058
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-fz 8977  df-fzo 9102  df-iseq 9376
This theorem is referenced by:  iseqcaopr  9406  isersub  9408
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