Theorem seq3caopr2 10567

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
seq3caopr2.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seq3caopr2.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
seq3caopr2.3	⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
seq3caopr2.4	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seq3caopr2.5	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)
seq3caopr2.6	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆)
seq3caopr2.7	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))

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.

Step	Hyp	Ref	Expression
1		seq3caopr2.1	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2		seq3caopr2.2	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
3		seq3caopr2.4	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
4		seq3caopr2.5	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)
5		seq3caopr2.6	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆)
6		seq3caopr2.7	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
7		eqid 2193	. . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
8		eluzel2 9600	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
9	3, 8	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
10	9	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℤ)
11	5	ralrimiva 2567	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐺‘𝑘) ∈ 𝑆)
12	11	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐺‘𝑘) ∈ 𝑆)
13		fveq2 5555	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (𝐺‘𝑘) = (𝐺‘𝑥))
14	13	eleq1d 2262	. . . . . . 7 ⊢ (𝑘 = 𝑥 → ((𝐺‘𝑘) ∈ 𝑆 ↔ (𝐺‘𝑥) ∈ 𝑆))
15	14	rspccva 2864	. . . . . 6 ⊢ ((∀𝑘 ∈ (ℤ≥‘𝑀)(𝐺‘𝑘) ∈ 𝑆 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
16	12, 15	sylan 283	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
17	1	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
18	7, 10, 16, 17	seqf 10538	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → seq𝑀( + , 𝐺):(ℤ≥‘𝑀)⟶𝑆)
19		elfzouz 10220	. . . . 5 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
20	19	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀))
21	18, 20	ffvelcdmd 5695	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ 𝑆)
22		fzssuz 10134	. . . . 5 ⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀)
23		fzofzp1 10297	. . . . 5 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
24	22, 23	sselid 3178	. . . 4 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
25		fveq2 5555	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
26	25	eleq1d 2262	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((𝐺‘𝑘) ∈ 𝑆 ↔ (𝐺‘(𝑛 + 1)) ∈ 𝑆))
27	26	rspccva 2864	. . . 4 ⊢ ((∀𝑘 ∈ (ℤ≥‘𝑀)(𝐺‘𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (ℤ≥‘𝑀)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
28	11, 24, 27	syl2an 289	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) ∈ 𝑆)
29	4	ralrimiva 2567	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ 𝑆)
30		fveq2 5555	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
31	30	eleq1d 2262	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → ((𝐹‘𝑘) ∈ 𝑆 ↔ (𝐹‘𝑥) ∈ 𝑆))
32	31	rspccva 2864	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ 𝑆 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
33	29, 32	sylan 283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
34	33	adantlr 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
35	7, 10, 34, 17	seqf 10538	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝑆)
36	35, 20	ffvelcdmd 5695	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝑆)
37		fveq2 5555	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
38	37	eleq1d 2262	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
39	38	rspccva 2864	. . . . 5 ⊢ ((∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ 𝑆 ∧ (𝑛 + 1) ∈ (ℤ≥‘𝑀)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
40	29, 24, 39	syl2an 289	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
41		seq3caopr2.3	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
42	41	anassrs 400	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
43	42	ralrimivva 2576	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
44	43	ralrimivva 2576	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
45	44	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))
46		oveq1 5926	. . . . . . . 8 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥𝑄𝑧) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧))
47	46	oveq1d 5934	. . . . . . 7 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)))
48		oveq1 5926	. . . . . . . 8 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (𝑥 + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦))
49	48	oveq1d 5934	. . . . . . 7 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)))
50	47, 49	eqeq12d 2208	. . . . . 6 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) → (((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)) ↔ (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤))))
51	50	2ralbidv 2518	. . . . 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))))
52		oveq1 5926	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (𝑦𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄𝑤))
53	52	oveq2d 5935	. . . . . . 7 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + (𝑦𝑄𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) + ((𝐹‘(𝑛 + 1))𝑄𝑤)))
54		oveq2 5927	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → ((seq𝑀( + , 𝐹)‘𝑛) + 𝑦) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
55	54	oveq1d 5934	. . . . . . 7 ⊢ (𝑦 = (𝐹‘(𝑛 + 1)) → (((seq𝑀( + , 𝐹)‘𝑛) + 𝑦)𝑄(𝑧 + 𝑤)) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄(𝑧 + 𝑤)))
56	53, 55	eqeq12d 2208	. . . . . 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))))
57	56	2ralbidv 2518	. . . . 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))))
58	51, 57	rspc2va 2879	. . . 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)))
59	36, 40, 45, 58	syl21anc 1248	. . 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)))
60		oveq2 5927	. . . . . 6 ⊢ (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝑛)𝑄𝑧) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))
61	60	oveq1d 5934	. . . . 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 5926	. . . . . 6 ⊢ (𝑧 = (seq𝑀( + , 𝐺)‘𝑛) → (𝑧 + 𝑤) = ((seq𝑀( + , 𝐺)‘𝑛) + 𝑤))
63	62	oveq2d 5935	. . . . 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)))
64	61, 63	eqeq12d 2208	. . . 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 5927	. . . . . 6 ⊢ (𝑤 = (𝐺‘(𝑛 + 1)) → ((𝐹‘(𝑛 + 1))𝑄𝑤) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
66	65	oveq2d 5935	. . . . 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 5927	. . . . . 6 ⊢ (𝑤 = (𝐺‘(𝑛 + 1)) → ((seq𝑀( + , 𝐺)‘𝑛) + 𝑤) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
68	67	oveq2d 5935	. . . . 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)))))
69	66, 68	eqeq12d 2208	. . . 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))))))
70	64, 69	rspc2va 2879	. . 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)))))
71	21, 28, 59, 70	syl21anc 1248	. 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)))))
72	1, 2, 3, 4, 5, 6, 71	seq3caopr3 10565	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ‘cfv 5255  (class class class)co 5919  1c1 7875   + caddc 7877  ℤcz 9320  ℤ_≥cuz 9595  ...cfz 10077  ..^cfzo 10211  seqcseq 10521

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078  df-fzo 10212  df-seqfrec 10522

This theorem is referenced by:  seq3caopr  10569  ser3sub  10597