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Theorem iseqcaopr2 9241
 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 9008 . . . . 5 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
87adantl 262 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
9 iseqcaopr2.s . . . . 5 (𝜑𝑆𝑉)
109adantr 261 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑆𝑉)
115ralrimiva 2392 . . . . . 6 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
1211adantr 261 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
13 fveq2 5178 . . . . . . 7 (𝑘 = 𝑥 → (𝐺𝑘) = (𝐺𝑥))
1413eleq1d 2106 . . . . . 6 (𝑘 = 𝑥 → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
1514rspccva 2655 . . . . 5 ((∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
1612, 15sylan 267 . . . 4 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
171adantlr 446 . . . 4 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
188, 10, 16, 17iseqcl 9223 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺, 𝑆)‘𝑛) ∈ 𝑆)
19 fzssuz 8928 . . . . 5 (𝑀...𝑁) ⊆ (ℤ𝑀)
20 fzofzp1 9083 . . . . 5 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
2119, 20sseldi 2943 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
22 fveq2 5178 . . . . . 6 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
2322eleq1d 2106 . . . . 5 (𝑘 = (𝑛 + 1) → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺‘(𝑛 + 1)) ∈ 𝑆))
2423rspccva 2655 . . . 4 ((∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
2511, 21, 24syl2an 273 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
264ralrimiva 2392 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆)
27 fveq2 5178 . . . . . . . . 9 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
2827eleq1d 2106 . . . . . . . 8 (𝑘 = 𝑥 → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
2928rspccva 2655 . . . . . . 7 ((∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3026, 29sylan 267 . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3130adantlr 446 . . . . 5 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
328, 10, 31, 17iseqcl 9223 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆)
33 fveq2 5178 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
3433eleq1d 2106 . . . . . 6 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
3534rspccva 2655 . . . . 5 ((∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
3626, 21, 35syl2an 273 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
37 iseqcaopr2.3 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
3837anassrs 380 . . . . . . 7 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ (𝑧𝑆𝑤𝑆)) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
3938ralrimivva 2401 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4039ralrimivva 2401 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4140adantr 261 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
42 oveq1 5519 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝑥𝑄𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧))
4342oveq1d 5527 . . . . . . 7 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)))
44 oveq1 5519 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦))
4544oveq1d 5527 . . . . . . 7 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)))
4643, 45eqeq12d 2054 . . . . . 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 2348 . . . . 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 5519 . . . . . . . 8 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝑦𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄𝑤))
4948oveq2d 5528 . . . . . . 7 (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
50 oveq2 5520 . . . . . . . 8 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
5150oveq1d 5527 . . . . . . 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 2054 . . . . . 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 2348 . . . . 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 2663 . . . 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 1134 . . 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 5520 . . . . . 6 (𝑧 = (seq𝑀( + , 𝐺, 𝑆)‘𝑛) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)))
5756oveq1d 5527 . . . . 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 5519 . . . . . 6 (𝑧 = (seq𝑀( + , 𝐺, 𝑆)‘𝑛) → (𝑧 + 𝑤) = ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + 𝑤))
5958oveq2d 5528 . . . . 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 2054 . . . 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 5520 . . . . . 6 (𝑤 = (𝐺‘(𝑛 + 1)) → ((𝐹‘(𝑛 + 1))𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
6261oveq2d 5528 . . . . 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 5520 . . . . . 6 (𝑤 = (𝐺‘(𝑛 + 1)) → ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + 𝑤) = ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1))))
6463oveq2d 5528 . . . . 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 2054 . . . 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 2663 . . 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 1134 . 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 9240 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 97   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ‘cfv 4902  (class class class)co 5512  1c1 6890   + caddc 6892  ℤ≥cuz 8473  ...cfz 8874  ..^cfzo 8999  seqcseq 9211 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-fz 8875  df-fzo 9000  df-iseq 9212 This theorem is referenced by:  iseqcaopr  9242  isersub  9244
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