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Theorem iseqcaopr2 9790
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 9466 . . . . 5 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
87adantl 271 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
95ralrimiva 2442 . . . . . 6 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
109adantr 270 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
11 fveq2 5256 . . . . . . 7 (𝑘 = 𝑥 → (𝐺𝑘) = (𝐺𝑥))
1211eleq1d 2153 . . . . . 6 (𝑘 = 𝑥 → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
1312rspccva 2713 . . . . 5 ((∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
1410, 13sylan 277 . . . 4 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
151adantlr 461 . . . 4 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
168, 14, 15iseqcl 9770 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺, 𝑆)‘𝑛) ∈ 𝑆)
17 fzssuz 9387 . . . . 5 (𝑀...𝑁) ⊆ (ℤ𝑀)
18 fzofzp1 9541 . . . . 5 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
1917, 18sseldi 3010 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
20 fveq2 5256 . . . . . 6 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
2120eleq1d 2153 . . . . 5 (𝑘 = (𝑛 + 1) → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺‘(𝑛 + 1)) ∈ 𝑆))
2221rspccva 2713 . . . 4 ((∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
239, 19, 22syl2an 283 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
244ralrimiva 2442 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆)
25 fveq2 5256 . . . . . . . . 9 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
2625eleq1d 2153 . . . . . . . 8 (𝑘 = 𝑥 → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
2726rspccva 2713 . . . . . . 7 ((∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
2824, 27sylan 277 . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
2928adantlr 461 . . . . 5 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
308, 29, 15iseqcl 9770 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆)
31 fveq2 5256 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
3231eleq1d 2153 . . . . . 6 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
3332rspccva 2713 . . . . 5 ((∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
3424, 19, 33syl2an 283 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
35 iseqcaopr2.3 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
3635anassrs 392 . . . . . . 7 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ (𝑧𝑆𝑤𝑆)) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
3736ralrimivva 2451 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
3837ralrimivva 2451 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
3938adantr 270 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
40 oveq1 5601 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝑥𝑄𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧))
4140oveq1d 5609 . . . . . . 7 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)))
42 oveq1 5601 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦))
4342oveq1d 5609 . . . . . . 7 (𝑥 = (seq𝑀( + , 𝐹, 𝑆)‘𝑛) → ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)))
4441, 43eqeq12d 2099 . . . . . 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))))
45442ralbidv 2398 . . . . 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))))
46 oveq1 5601 . . . . . . . 8 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝑦𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄𝑤))
4746oveq2d 5610 . . . . . . 7 (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
48 oveq2 5602 . . . . . . . 8 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
4948oveq1d 5609 . . . . . . 7 (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
5047, 49eqeq12d 2099 . . . . . 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))))
51502ralbidv 2398 . . . . 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))))
5245, 51rspc2va 2726 . . . 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)))
5330, 34, 39, 52syl21anc 1171 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
54 oveq2 5602 . . . . . 6 (𝑧 = (seq𝑀( + , 𝐺, 𝑆)‘𝑛) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄𝑧) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)))
5554oveq1d 5609 . . . . 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)))
56 oveq1 5601 . . . . . 6 (𝑧 = (seq𝑀( + , 𝐺, 𝑆)‘𝑛) → (𝑧 + 𝑤) = ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + 𝑤))
5756oveq2d 5610 . . . . 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)))
5855, 57eqeq12d 2099 . . . 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))))
59 oveq2 5602 . . . . . 6 (𝑤 = (𝐺‘(𝑛 + 1)) → ((𝐹‘(𝑛 + 1))𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
6059oveq2d 5610 . . . . 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)))))
61 oveq2 5602 . . . . . 6 (𝑤 = (𝐺‘(𝑛 + 1)) → ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + 𝑤) = ((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1))))
6261oveq2d 5610 . . . . 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)))))
6360, 62eqeq12d 2099 . . . 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))))))
6458, 63rspc2va 2726 . . 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)))))
6516, 23, 53, 64syl21anc 1171 . 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)))))
66 iseqcaopr2.s . 2 (𝜑𝑆𝑉)
671, 2, 3, 4, 5, 6, 65, 66iseqcaopr3 9789 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 102   = wceq 1287  wcel 1436  wral 2355  cfv 4972  (class class class)co 5594  1c1 7272   + caddc 7274  cuz 8928  ...cfz 9333  ..^cfzo 9457  seqcseq 9754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-ltadd 7382
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-frec 6091  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-inn 8335  df-n0 8584  df-z 8661  df-uz 8929  df-fz 9334  df-fzo 9458  df-iseq 9755
This theorem is referenced by:  iseqcaopr  9791  isersub  9793
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