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Theorem seq3caopr2 10500
Description: The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
Hypotheses
Ref Expression
seqcaopr2.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqcaopr2.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
seqcaopr2.3 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
seqcaopr2.4 (𝜑𝑁 ∈ (ℤ𝑀))
seq3caopr2.5 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
seq3caopr2.6 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)
seq3caopr2.7 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
Assertion
Ref Expression
seq3caopr2 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘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)

Proof of Theorem seq3caopr2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 seqcaopr2.1 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2 seqcaopr2.2 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
3 seqcaopr2.4 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
4 seq3caopr2.5 . 2 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)
5 seq3caopr2.6 . 2 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)
6 seq3caopr2.7 . 2 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
7 eqid 2189 . . . . 5 (ℤ𝑀) = (ℤ𝑀)
8 eluzel2 9551 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
93, 8syl 14 . . . . . 6 (𝜑𝑀 ∈ ℤ)
109adantr 276 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℤ)
115ralrimiva 2563 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
1211adantr 276 . . . . . 6 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆)
13 fveq2 5530 . . . . . . . 8 (𝑘 = 𝑥 → (𝐺𝑘) = (𝐺𝑥))
1413eleq1d 2258 . . . . . . 7 (𝑘 = 𝑥 → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
1514rspccva 2855 . . . . . 6 ((∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
1612, 15sylan 283 . . . . 5 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
171adantlr 477 . . . . 5 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
187, 10, 16, 17seqf 10479 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → seq𝑀( + , 𝐺):(ℤ𝑀)⟶𝑆)
19 elfzouz 10169 . . . . 5 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
2019adantl 277 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
2118, 20ffvelcdmd 5668 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
22 fzssuz 10083 . . . . 5 (𝑀...𝑁) ⊆ (ℤ𝑀)
23 fzofzp1 10245 . . . . 5 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
2422, 23sselid 3168 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (ℤ𝑀))
25 fveq2 5530 . . . . . 6 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
2625eleq1d 2258 . . . . 5 (𝑘 = (𝑛 + 1) → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺‘(𝑛 + 1)) ∈ 𝑆))
2726rspccva 2855 . . . 4 ((∀𝑘 ∈ (ℤ𝑀)(𝐺𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
2811, 24, 27syl2an 289 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
294ralrimiva 2563 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆)
30 fveq2 5530 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
3130eleq1d 2258 . . . . . . . . 9 (𝑘 = 𝑥 → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
3231rspccva 2855 . . . . . . . 8 ((∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3329, 32sylan 283 . . . . . . 7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
3433adantlr 477 . . . . . 6 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)
357, 10, 34, 17seqf 10479 . . . . 5 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → seq𝑀( + , 𝐹):(ℤ𝑀)⟶𝑆)
3635, 20ffvelcdmd 5668 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
37 fveq2 5530 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
3837eleq1d 2258 . . . . . 6 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
3938rspccva 2855 . . . . 5 ((∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
4029, 24, 39syl2an 289 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
41 seqcaopr2.3 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4241anassrs 400 . . . . . . 7 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ (𝑧𝑆𝑤𝑆)) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4342ralrimivva 2572 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4443ralrimivva 2572 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4544adantr 276 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
46 oveq1 5898 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥𝑄𝑧) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧))
4746oveq1d 5906 . . . . . . 7 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)))
48 oveq1 5898 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
4948oveq1d 5906 . . . . . . 7 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)))
5047, 49eqeq12d 2204 . . . . . 6 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) ↔ (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤))))
51502ralbidv 2514 . . . . 5 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (∀𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) ↔ ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤))))
52 oveq1 5898 . . . . . . . 8 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝑦𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄𝑤))
5352oveq2d 5907 . . . . . . 7 (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
54 oveq2 5899 . . . . . . . 8 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5554oveq1d 5906 . . . . . . 7 (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
5653, 55eqeq12d 2204 . . . . . 6 (𝑦 = (𝐹‘(𝑛 + 1)) → ((((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) ↔ (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤))))
57562ralbidv 2514 . . . . 5 (𝑦 = (𝐹‘(𝑛 + 1)) → (∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) ↔ ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤))))
5851, 57rspc2va 2870 . . . 4 ((((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤))) → ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
5936, 40, 45, 58syl21anc 1248 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
60 oveq2 5899 . . . . . 6 (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))
6160oveq1d 5906 . . . . 5 (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
62 oveq1 5898 . . . . . 6 (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → (𝑧 + 𝑤) = ((seq𝑀( + , 𝐺)‘𝑛) + 𝑤))
6362oveq2d 5907 . . . . 5 (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + 𝑤)))
6461, 63eqeq12d 2204 . . . 4 (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → ((((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)) ↔ (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + 𝑤))))
65 oveq2 5899 . . . . . 6 (𝑤 = (𝐺‘(𝑛 + 1)) → ((𝐹‘(𝑛 + 1))𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
6665oveq2d 5907 . . . . 5 (𝑤 = (𝐺‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
67 oveq2 5899 . . . . . 6 (𝑤 = (𝐺‘(𝑛 + 1)) → ((seq𝑀( + , 𝐺)‘𝑛) + 𝑤) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
6867oveq2d 5907 . . . . 5 (𝑤 = (𝐺‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
6966, 68eqeq12d 2204 . . . 4 (𝑤 = (𝐺‘(𝑛 + 1)) → ((((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + 𝑤)) ↔ (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))))
7064, 69rspc2va 2870 . . 3 ((((seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆 ∧ (𝐺‘(𝑛 + 1)) ∈ 𝑆) ∧ ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤))) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
7121, 28, 59, 70syl21anc 1248 . 2 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
721, 2, 3, 4, 5, 6, 71seq3caopr3 10499 1 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wral 2468  cfv 5231  (class class class)co 5891  1c1 7830   + caddc 7832  cz 9271  cuz 9546  ...cfz 10026  ..^cfzo 10160  seqcseq 10463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-ltadd 7945
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-inn 8938  df-n0 9195  df-z 9272  df-uz 9547  df-fz 10027  df-fzo 10161  df-seqfrec 10464
This theorem is referenced by:  seq3caopr  10501  ser3sub  10524
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