Theorem seqcaopr3 13943

Description: Lemma for seqcaopr2 13944. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
seqcaopr3.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqcaopr3.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
seqcaopr3.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seqcaopr3.4	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)
seqcaopr3.5	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆)
seqcaopr3.6	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
seqcaopr3.7	⊢ ((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)))))

Assertion

Ref	Expression
seqcaopr3	⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))

Distinct variable groups:   𝑘,𝑛,𝑥,𝑦,𝐹   𝑘,𝐻,𝑛   𝑘,𝑁,𝑛,𝑥,𝑦   𝜑,𝑘,𝑛,𝑥,𝑦   𝑘,𝐺,𝑛,𝑥,𝑦   𝑘,𝑀,𝑛,𝑥,𝑦   𝑄,𝑘,𝑛,𝑥,𝑦   + ,𝑛,𝑥,𝑦   𝑆,𝑘,𝑥,𝑦

Allowed substitution hints:   + (𝑘)   𝑆(𝑛)   𝐻(𝑥,𝑦)

Proof of Theorem seqcaopr3

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqcaopr3.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 13449	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 6842	. . . . 5 ⊢ (𝑧 = 𝑀 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑀))
5		fveq2 6842	. . . . . 6 ⊢ (𝑧 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑀))
6		fveq2 6842	. . . . . 6 ⊢ (𝑧 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑀))
7	5, 6	oveq12d 7375	. . . . 5 ⊢ (𝑧 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))
8	4, 7	eqeq12d 2752	. . . 4 ⊢ (𝑧 = 𝑀 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀))))
9	8	imbi2d 340	. . 3 ⊢ (𝑧 = 𝑀 → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))))
10		fveq2 6842	. . . . 5 ⊢ (𝑧 = 𝑛 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑛))
11		fveq2 6842	. . . . . 6 ⊢ (𝑧 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑛))
12		fveq2 6842	. . . . . 6 ⊢ (𝑧 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑛))
13	11, 12	oveq12d 7375	. . . . 5 ⊢ (𝑧 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))
14	10, 13	eqeq12d 2752	. . . 4 ⊢ (𝑧 = 𝑛 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛))))
15	14	imbi2d 340	. . 3 ⊢ (𝑧 = 𝑛 → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))))
16		fveq2 6842	. . . . 5 ⊢ (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘(𝑛 + 1)))
17		fveq2 6842	. . . . . 6 ⊢ (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18		fveq2 6842	. . . . . 6 ⊢ (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
19	17, 18	oveq12d 7375	. . . . 5 ⊢ (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))
20	16, 19	eqeq12d 2752	. . . 4 ⊢ (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1)))))
21	20	imbi2d 340	. . 3 ⊢ (𝑧 = (𝑛 + 1) → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))))
22		fveq2 6842	. . . . 5 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑁))
23		fveq2 6842	. . . . . 6 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑁))
24		fveq2 6842	. . . . . 6 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
25	23, 24	oveq12d 7375	. . . . 5 ⊢ (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
26	22, 25	eqeq12d 2752	. . . 4 ⊢ (𝑧 = 𝑁 → ((seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) ↔ (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁))))
27	26	imbi2d 340	. . 3 ⊢ (𝑧 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑧) = ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧))) ↔ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))))
28		fveq2 6842	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐻‘𝑘) = (𝐻‘𝑀))
29		fveq2 6842	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
30		fveq2 6842	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀))
31	29, 30	oveq12d 7375	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀)))
32	28, 31	eqeq12d 2752	. . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ↔ (𝐻‘𝑀) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀))))
33		seqcaopr3.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
34	33	ralrimiva 3143	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
35		eluzfz1 13448	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
36	1, 35	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
37	32, 34, 36	rspcdva 3582	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑀) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀)))
38		eluzel2 12768	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
39	1, 38	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
40		seq1 13919	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐻)‘𝑀) = (𝐻‘𝑀))
41	39, 40	syl 17	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = (𝐻‘𝑀))
42		seq1 13919	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
43		seq1 13919	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺‘𝑀))
44	42, 43	oveq12d 7375	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀)))
45	39, 44	syl 17	. . . . 5 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀)))
46	37, 41, 45	3eqtr4d 2786	. . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))
47	46	a1i 11	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀))))
48		oveq1 7364	. . . . . 6 ⊢ ((seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) → ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1))))
49		elfzouz 13576	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
50	49	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀))
51		seqp1 13921	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))))
52	50, 51	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))))
53		seqcaopr3.7	. . . . . . . 8 ⊢ ((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)))))
54		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → (𝐻‘𝑘) = (𝐻‘(𝑛 + 1)))
55		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
56		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
57	55, 56	oveq12d 7375	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
58	54, 57	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → ((𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ↔ (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
59	34	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (𝑀...𝑁)(𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
60		fzofzp1 13669	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
61	60	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
62	58, 59, 61	rspcdva 3582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
63	62	oveq2d 7373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
64		seqp1 13921	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
65		seqp1 13921	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
66	64, 65	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
67	50, 66	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))
68	53, 63, 67	3eqtr4rd 2787	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1))))
69	52, 68	eqeq12d 2752	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) ↔ ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1)))))
70	48, 69	syl5ibr 245	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1)))))
71	70	expcom 414	. . . 4 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))))
72	71	a2d 29	. . 3 ⊢ (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( + , 𝐻)‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛))) → (𝜑 → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))))
73	9, 15, 21, 27, 47, 72	fzind2 13690	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁))))
74	3, 73	mpcom 38	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ‘cfv 6496  (class class class)co 7357  1c1 11052   + caddc 11054  ℤcz 12499  ℤ_≥cuz 12763  ...cfz 13424  ..^cfzo 13567  seqcseq 13906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907

This theorem is referenced by:  seqcaopr2  13944  gsumzaddlem  19698