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Theorem seqcaopr2 14035
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
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))
seqcaopr2.5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)
seqcaopr2.6 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ∈ 𝑆)
seqcaopr2.7 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
Assertion
Ref Expression
seqcaopr2 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
Distinct variable groups:   𝑤,𝑘,𝑥,𝑦,𝑧,𝐹   𝑘,𝐻,𝑧   𝑘,𝑁,𝑥,𝑦,𝑧   𝜑,𝑘,𝑤,𝑥,𝑦,𝑧   𝑘,𝐺,𝑤,𝑥,𝑦,𝑧   𝑘,𝑀,𝑤,𝑥,𝑦,𝑧   𝑄,𝑘,𝑤,𝑥,𝑦,𝑧   𝑤, + ,𝑥,𝑦,𝑧   𝑆,𝑘,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   + (𝑘)   𝐻(𝑥,𝑦,𝑤)   𝑁(𝑤)

Proof of Theorem seqcaopr2
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 seqcaopr2.5 . 2 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)
5 seqcaopr2.6 . 2 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ∈ 𝑆)
6 seqcaopr2.7 . 2 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))
7 elfzouz 13668 . . . . 5 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
87adantl 480 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ𝑀))
9 elfzouz2 13679 . . . . . . . 8 (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝑛))
109adantl 480 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ𝑛))
11 fzss2 13573 . . . . . . 7 (𝑁 ∈ (ℤ𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
1210, 11syl 17 . . . . . 6 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
1312sselda 3972 . . . . 5 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ (𝑀...𝑁))
145ralrimiva 3136 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐺𝑘) ∈ 𝑆)
1514adantr 479 . . . . . 6 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (𝑀...𝑁)(𝐺𝑘) ∈ 𝑆)
16 fveq2 6892 . . . . . . . 8 (𝑘 = 𝑥 → (𝐺𝑘) = (𝐺𝑥))
1716eleq1d 2810 . . . . . . 7 (𝑘 = 𝑥 → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
1817rspccva 3600 . . . . . 6 ((∀𝑘 ∈ (𝑀...𝑁)(𝐺𝑘) ∈ 𝑆𝑥 ∈ (𝑀...𝑁)) → (𝐺𝑥) ∈ 𝑆)
1915, 18sylan 578 . . . . 5 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺𝑥) ∈ 𝑆)
2013, 19syldan 589 . . . 4 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺𝑥) ∈ 𝑆)
211adantlr 713 . . . 4 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
228, 20, 21seqcl 14019 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
23 fzofzp1 13761 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
24 fveq2 6892 . . . . . 6 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
2524eleq1d 2810 . . . . 5 (𝑘 = (𝑛 + 1) → ((𝐺𝑘) ∈ 𝑆 ↔ (𝐺‘(𝑛 + 1)) ∈ 𝑆))
2625rspccva 3600 . . . 4 ((∀𝑘 ∈ (𝑀...𝑁)(𝐺𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
2714, 23, 26syl2an 594 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
284ralrimiva 3136 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ 𝑆)
29 fveq2 6892 . . . . . . . . . 10 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
3029eleq1d 2810 . . . . . . . . 9 (𝑘 = 𝑥 → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
3130rspccva 3600 . . . . . . . 8 ((∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ 𝑆𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)
3228, 31sylan 578 . . . . . . 7 ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)
3332adantlr 713 . . . . . 6 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)
3413, 33syldan 589 . . . . 5 (((𝜑𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐹𝑥) ∈ 𝑆)
358, 34, 21seqcl 14019 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
36 fveq2 6892 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
3736eleq1d 2810 . . . . . 6 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
3837rspccva 3600 . . . . 5 ((∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
3928, 23, 38syl2an 594 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
40 seqcaopr2.3 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4140anassrs 466 . . . . . . 7 (((𝜑 ∧ (𝑥𝑆𝑦𝑆)) ∧ (𝑧𝑆𝑤𝑆)) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4241ralrimivva 3191 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4342ralrimivva 3191 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
4443adantr 479 . . . 4 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑥𝑆𝑦𝑆𝑧𝑆𝑤𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
45 oveq1 7423 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥𝑄𝑧) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧))
4645oveq1d 7431 . . . . . . 7 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)))
47 oveq1 7423 . . . . . . . 8 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
4847oveq1d 7431 . . . . . . 7 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)))
4946, 48eqeq12d 2741 . . . . . 6 (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) ↔ (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤))))
50492ralbidv 3209 . . . . 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))))
51 oveq1 7423 . . . . . . . 8 (𝑦 = (𝐹‘(𝑛 + 1)) → (𝑦𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄𝑤))
5251oveq2d 7432 . . . . . . 7 (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
53 oveq2 7424 . . . . . . . 8 (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
5453oveq1d 7431 . . . . . . 7 (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
5552, 54eqeq12d 2741 . . . . . 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))))
56552ralbidv 3209 . . . . 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))))
5750, 56rspc2va 3613 . . . 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)))
5835, 39, 44, 57syl21anc 836 . . 3 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → ∀𝑧𝑆𝑤𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
59 oveq2 7424 . . . . . 6 (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))
6059oveq1d 7431 . . . . 5 (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
61 oveq1 7423 . . . . . 6 (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → (𝑧 + 𝑤) = ((seq𝑀( + , 𝐺)‘𝑛) + 𝑤))
6261oveq2d 7432 . . . . 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)))
6360, 62eqeq12d 2741 . . . 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))))
64 oveq2 7424 . . . . . 6 (𝑤 = (𝐺‘(𝑛 + 1)) → ((𝐹‘(𝑛 + 1))𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
6564oveq2d 7432 . . . . 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)))))
66 oveq2 7424 . . . . . 6 (𝑤 = (𝐺‘(𝑛 + 1)) → ((seq𝑀( + , 𝐺)‘𝑛) + 𝑤) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
6766oveq2d 7432 . . . . 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)))))
6865, 67eqeq12d 2741 . . . 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))))))
6963, 68rspc2va 3613 . . 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)))))
7022, 27, 58, 69syl21anc 836 . 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)))))
711, 2, 3, 4, 5, 6, 70seqcaopr3 14034 1 (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3051  wss 3939  cfv 6543  (class class class)co 7416  1c1 11139   + caddc 11141  cuz 12852  ...cfz 13516  ..^cfzo 13659  seqcseq 13998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-fzo 13660  df-seq 13999
This theorem is referenced by:  seqcaopr  14036  sersub  14042
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