Theorem seqcaopr2 13759

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 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqcaopr2.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
seqcaopr2.3	⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
seqcaopr2.4	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seqcaopr2.5	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)
seqcaopr2.6	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆)
seqcaopr2.7	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))

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.

Step	Hyp	Ref	Expression
1		seqcaopr2.1	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2		seqcaopr2.2	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
3		seqcaopr2.4	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
4		seqcaopr2.5	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)
5		seqcaopr2.6	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆)
6		seqcaopr2.7	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
7		elfzouz 13391	. . . . 5 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
8	7	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀))
9		elfzouz2 13402	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑛))
10	9	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ≥‘𝑛))
11		fzss2 13296	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
12	10, 11	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
13	12	sselda 3921	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ (𝑀...𝑁))
14	5	ralrimiva 3103	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐺‘𝑘) ∈ 𝑆)
15	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (𝑀...𝑁)(𝐺‘𝑘) ∈ 𝑆)
16		fveq2 6774	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝐺‘𝑘) = (𝐺‘𝑥))
17	16	eleq1d 2823	. . . . . . 7 ⊢ (𝑘 = 𝑥 → ((𝐺‘𝑘) ∈ 𝑆 ↔ (𝐺‘𝑥) ∈ 𝑆))
18	17	rspccva 3560	. . . . . 6 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)(𝐺‘𝑘) ∈ 𝑆 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝑆)
19	15, 18	sylan 580	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝑆)
20	13, 19	syldan 591	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺‘𝑥) ∈ 𝑆)
21	1	adantlr 712	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
22	8, 20, 21	seqcl 13743	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
23		fzofzp1 13484	. . . 4 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
24		fveq2 6774	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
25	24	eleq1d 2823	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((𝐺‘𝑘) ∈ 𝑆 ↔ (𝐺‘(𝑛 + 1)) ∈ 𝑆))
26	25	rspccva 3560	. . . 4 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)(𝐺‘𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
27	14, 23, 26	syl2an 596	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
28	4	ralrimiva 3103	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑆)
29		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
30	29	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → ((𝐹‘𝑘) ∈ 𝑆 ↔ (𝐹‘𝑥) ∈ 𝑆))
31	30	rspccva 3560	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑆 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
32	28, 31	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
33	32	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
34	13, 33	syldan 591	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐹‘𝑥) ∈ 𝑆)
35	8, 34, 21	seqcl 13743	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
36		fveq2 6774	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
37	36	eleq1d 2823	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
38	37	rspccva 3560	. . . . 5 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
39	28, 23, 38	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
40		seqcaopr2.3	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
41	40	anassrs 468	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
42	41	ralrimivva 3123	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
43	42	ralrimivva 3123	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
44	43	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
45		oveq1 7282	. . . . . . . 8 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥𝑄𝑧) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧))
46	45	oveq1d 7290	. . . . . . 7 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)))
47		oveq1 7282	. . . . . . . 8 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
48	47	oveq1d 7290	. . . . . . 7 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)))
49	46, 48	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) ↔ (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤))))
50	49	2ralbidv 3129	. . . . 5 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) ↔ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤))))
51		oveq1 7282	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (𝑦𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄𝑤))
52	51	oveq2d 7291	. . . . . . 7 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
53		oveq2 7283	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
54	53	oveq1d 7290	. . . . . . 7 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
55	52, 54	eqeq12d 2754	. . . . . 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))))
56	55	2ralbidv 3129	. . . . 5 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) ↔ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤))))
57	50, 56	rspc2va 3571	. . . 4 ⊢ ((((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤))) → ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
58	35, 39, 44, 57	syl21anc 835	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
59		oveq2 7283	. . . . . 6 ⊢ (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))
60	59	oveq1d 7290	. . . . 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 7282	. . . . . 6 ⊢ (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → (𝑧 + 𝑤) = ((seq𝑀( + , 𝐺)‘𝑛) + 𝑤))
62	61	oveq2d 7291	. . . . 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)))
63	60, 62	eqeq12d 2754	. . . 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 7283	. . . . . 6 ⊢ (𝑤 = (𝐺‘(𝑛 + 1)) → ((𝐹‘(𝑛 + 1))𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
65	64	oveq2d 7291	. . . . 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 7283	. . . . . 6 ⊢ (𝑤 = (𝐺‘(𝑛 + 1)) → ((seq𝑀( + , 𝐺)‘𝑛) + 𝑤) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
67	66	oveq2d 7291	. . . . 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)))))
68	65, 67	eqeq12d 2754	. . . 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))))))
69	63, 68	rspc2va 3571	. . 3 ⊢ ((((seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆 ∧ (𝐺‘(𝑛 + 1)) ∈ 𝑆) ∧ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 (((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)))))
70	22, 27, 58, 69	syl21anc 835	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (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)))))
71	1, 2, 3, 4, 5, 6, 70	seqcaopr3 13758	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  ‘cfv 6433  (class class class)co 7275  1c1 10872   + caddc 10874  ℤ_≥cuz 12582  ...cfz 13239  ..^cfzo 13382  seqcseq 13721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722

This theorem is referenced by:  seqcaopr  13760  sersub  13766