Theorem seqcaopr2 12699

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 12343	. . . . 5 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
8	7	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀))
9		elfzouz2 12353	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑛))
10	9	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑁 ∈ (ℤ≥‘𝑛))
11		fzss2 12252	. . . . . . 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 3568	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → 𝑥 ∈ (𝑀...𝑁))
14	5	ralrimiva 2949	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐺‘𝑘) ∈ 𝑆)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (𝑀...𝑁)(𝐺‘𝑘) ∈ 𝑆)
16		fveq2 6103	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝐺‘𝑘) = (𝐺‘𝑥))
17	16	eleq1d 2672	. . . . . . 7 ⊢ (𝑘 = 𝑥 → ((𝐺‘𝑘) ∈ 𝑆 ↔ (𝐺‘𝑥) ∈ 𝑆))
18	17	rspccva 3281	. . . . . 6 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)(𝐺‘𝑘) ∈ 𝑆 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝑆)
19	15, 18	sylan 487	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝑆)
20	13, 19	syldan 486	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐺‘𝑥) ∈ 𝑆)
21	1	adantlr 747	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
22	8, 20, 21	seqcl 12683	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
23		fzofzp1 12431	. . . 4 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
24		fveq2 6103	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
25	24	eleq1d 2672	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((𝐺‘𝑘) ∈ 𝑆 ↔ (𝐺‘(𝑛 + 1)) ∈ 𝑆))
26	25	rspccva 3281	. . . 4 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)(𝐺‘𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
27	14, 23, 26	syl2an 493	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
28	4	ralrimiva 2949	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑆)
29		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
30	29	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → ((𝐹‘𝑘) ∈ 𝑆 ↔ (𝐹‘𝑥) ∈ 𝑆))
31	30	rspccva 3281	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑆 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
32	28, 31	sylan 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
33	32	adantlr 747	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
34	13, 33	syldan 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀...𝑛)) → (𝐹‘𝑥) ∈ 𝑆)
35	8, 34, 21	seqcl 12683	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
36		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
37	36	eleq1d 2672	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
38	37	rspccva 3281	. . . . 5 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
39	28, 23, 38	syl2an 493	. . . 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 678	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
42	41	ralrimivva 2954	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
43	42	ralrimivva 2954	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
44	43	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
45		oveq1 6556	. . . . . . . 8 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥𝑄𝑧) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧))
46	45	oveq1d 6564	. . . . . . 7 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)))
47		oveq1 6556	. . . . . . . 8 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
48	47	oveq1d 6564	. . . . . . 7 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)))
49	46, 48	eqeq12d 2625	. . . . . 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 2972	. . . . 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 6556	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (𝑦𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄𝑤))
52	51	oveq2d 6565	. . . . . . 7 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
53		oveq2 6557	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
54	53	oveq1d 6564	. . . . . . 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 2625	. . . . . 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 2972	. . . . 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 3294	. . . 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 1317	. . 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 6557	. . . . . 6 ⊢ (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))
60	59	oveq1d 6564	. . . . 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 6556	. . . . . 6 ⊢ (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → (𝑧 + 𝑤) = ((seq𝑀( + , 𝐺)‘𝑛) + 𝑤))
62	61	oveq2d 6565	. . . . 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 2625	. . . 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 6557	. . . . . 6 ⊢ (𝑤 = (𝐺‘(𝑛 + 1)) → ((𝐹‘(𝑛 + 1))𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
65	64	oveq2d 6565	. . . . 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 6557	. . . . . 6 ⊢ (𝑤 = (𝐺‘(𝑛 + 1)) → ((seq𝑀( + , 𝐺)‘𝑛) + 𝑤) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
67	66	oveq2d 6565	. . . . 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 2625	. . . 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 3294	. . 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 1317	. 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 12698	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334  seqcseq 12663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664

This theorem is referenced by:  seqcaopr  12700  sersub  12706