Theorem seq3caopr3 10721

Description: Lemma for seq3caopr2 10723. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)

Hypotheses

Ref	Expression
seq3caopr3.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seq3caopr3.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
seq3caopr3.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seq3caopr3.4	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)
seq3caopr3.5	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆)
seq3caopr3.6	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
seq3caopr3.7	⊢ ((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
seq3caopr3	⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))

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

Allowed substitution hint:   + (𝑘)

Proof of Theorem seq3caopr3

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seq3caopr3.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 10236	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5629	. . . . 5 ⊢ (𝑧 = 𝑀 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑀))
5		fveq2 5629	. . . . . 6 ⊢ (𝑧 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑀))
6		fveq2 5629	. . . . . 6 ⊢ (𝑧 = 𝑀 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑀))
7	5, 6	oveq12d 6025	. . . . 5 ⊢ (𝑧 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))
8	4, 7	eqeq12d 2244	. . . 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 230	. . 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 5629	. . . . 5 ⊢ (𝑧 = 𝑛 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑛))
11		fveq2 5629	. . . . . 6 ⊢ (𝑧 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑛))
12		fveq2 5629	. . . . . 6 ⊢ (𝑧 = 𝑛 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑛))
13	11, 12	oveq12d 6025	. . . . 5 ⊢ (𝑧 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)))
14	10, 13	eqeq12d 2244	. . . 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 230	. . 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 5629	. . . . 5 ⊢ (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘(𝑛 + 1)))
17		fveq2 5629	. . . . . 6 ⊢ (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18		fveq2 5629	. . . . . 6 ⊢ (𝑧 = (𝑛 + 1) → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘(𝑛 + 1)))
19	17, 18	oveq12d 6025	. . . . 5 ⊢ (𝑧 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))))
20	16, 19	eqeq12d 2244	. . . 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 230	. . 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 5629	. . . . 5 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐻)‘𝑧) = (seq𝑀( + , 𝐻)‘𝑁))
23		fveq2 5629	. . . . . 6 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑧) = (seq𝑀( + , 𝐹)‘𝑁))
24		fveq2 5629	. . . . . 6 ⊢ (𝑧 = 𝑁 → (seq𝑀( + , 𝐺)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
25	23, 24	oveq12d 6025	. . . . 5 ⊢ (𝑧 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑧)𝑄(seq𝑀( + , 𝐺)‘𝑧)) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
26	22, 25	eqeq12d 2244	. . . 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 230	. . 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 5629	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐻‘𝑘) = (𝐻‘𝑀))
29		fveq2 5629	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
30		fveq2 5629	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀))
31	29, 30	oveq12d 6025	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀)))
32	28, 31	eqeq12d 2244	. . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ↔ (𝐻‘𝑀) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀))))
33		seq3caopr3.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
34	33	ralrimiva 2603	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
35		eluzel2 9735	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
36	1, 35	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
37		uzid 9744	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
38	36, 37	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
39	32, 34, 38	rspcdva 2912	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑀) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀)))
40		seq3caopr3.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)
41	40	ralrimivva 2612	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑄𝑦) ∈ 𝑆)
42	41	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑄𝑦) ∈ 𝑆)
43		seq3caopr3.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)
44		seq3caopr3.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆)
45		oveq1 6014	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥𝑄𝑦) = ((𝐹‘𝑘)𝑄𝑦))
46	45	eleq1d 2298	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐹‘𝑘) → ((𝑥𝑄𝑦) ∈ 𝑆 ↔ ((𝐹‘𝑘)𝑄𝑦) ∈ 𝑆))
47		oveq2 6015	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐺‘𝑘) → ((𝐹‘𝑘)𝑄𝑦) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
48	47	eleq1d 2298	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘𝑘) → (((𝐹‘𝑘)𝑄𝑦) ∈ 𝑆 ↔ ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ∈ 𝑆))
49	46, 48	rspc2v 2920	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘) ∈ 𝑆 ∧ (𝐺‘𝑘) ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ∈ 𝑆))
50	43, 44, 49	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑄𝑦) ∈ 𝑆 → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ∈ 𝑆))
51	42, 50	mpd 13	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ∈ 𝑆)
52	33, 51	eqeltrd 2306	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) ∈ 𝑆)
53	52	ralrimiva 2603	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐻‘𝑘) ∈ 𝑆)
54		fveq2 5629	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (𝐻‘𝑘) = (𝐻‘𝑥))
55	54	eleq1d 2298	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → ((𝐻‘𝑘) ∈ 𝑆 ↔ (𝐻‘𝑥) ∈ 𝑆))
56	55	rspcv 2903	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑀)(𝐻‘𝑘) ∈ 𝑆 → (𝐻‘𝑥) ∈ 𝑆))
57	53, 56	mpan9 281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑥) ∈ 𝑆)
58		seq3caopr3.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
59	36, 57, 58	seq3-1 10692	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = (𝐻‘𝑀))
60	43	ralrimiva 2603	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ 𝑆)
61		fveq2 5629	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
62	61	eleq1d 2298	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → ((𝐹‘𝑘) ∈ 𝑆 ↔ (𝐹‘𝑥) ∈ 𝑆))
63	62	rspcv 2903	. . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ 𝑆 → (𝐹‘𝑥) ∈ 𝑆))
64	60, 63	mpan9 281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
65	36, 64, 58	seq3-1 10692	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
66	44	ralrimiva 2603	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐺‘𝑘) ∈ 𝑆)
67		fveq2 5629	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → (𝐺‘𝑘) = (𝐺‘𝑥))
68	67	eleq1d 2298	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → ((𝐺‘𝑘) ∈ 𝑆 ↔ (𝐺‘𝑥) ∈ 𝑆))
69	68	rspcv 2903	. . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑘 ∈ (ℤ≥‘𝑀)(𝐺‘𝑘) ∈ 𝑆 → (𝐺‘𝑥) ∈ 𝑆))
70	66, 69	mpan9 281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
71	36, 70, 58	seq3-1 10692	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐺)‘𝑀) = (𝐺‘𝑀))
72	65, 71	oveq12d 6025	. . . . 5 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)) = ((𝐹‘𝑀)𝑄(𝐺‘𝑀)))
73	39, 59, 72	3eqtr4d 2272	. . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀)))
74	73	a1i 9	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐻)‘𝑀) = ((seq𝑀( + , 𝐹)‘𝑀)𝑄(seq𝑀( + , 𝐺)‘𝑀))))
75		oveq1 6014	. . . . . 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))))
76		elfzouz 10355	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
77	76	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀))
78	57	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑥) ∈ 𝑆)
79	58	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
80	77, 78, 79	seq3p1 10695	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐻)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐻)‘𝑛) + (𝐻‘(𝑛 + 1))))
81		seq3caopr3.7	. . . . . . . 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)))))
82		fveq2 5629	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → (𝐻‘𝑘) = (𝐻‘(𝑛 + 1)))
83		fveq2 5629	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
84		fveq2 5629	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
85	83, 84	oveq12d 6025	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
86	82, 85	eqeq12d 2244	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → ((𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)) ↔ (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))))
87	34	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))
88		fzofzp1 10441	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
89		elfzuz 10225	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
90	88, 89	syl 14	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
91	90	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
92	86, 87, 91	rspcdva 2912	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐻‘(𝑛 + 1)) = ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1))))
93	92	oveq2d 6023	. . . . . . . 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)))))
94	64	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
95	77, 94, 79	seq3p1 10695	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
96	70	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
97	77, 96, 79	seq3p1 10695	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( + , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
98	95, 97	oveq12d 6025	. . . . . . . 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)))))
99	81, 93, 98	3eqtr4rd 2273	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1))𝑄(seq𝑀( + , 𝐺)‘(𝑛 + 1))) = (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + (𝐻‘(𝑛 + 1))))
100	80, 99	eqeq12d 2244	. . . . . 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)))))
101	75, 100	imbitrrid 156	. . . . 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)))))
102	101	expcom 116	. . . 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))))))
103	102	a2d 26	. . 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))))))
104	9, 15, 21, 27, 74, 103	fzind2 10453	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁))))
105	3, 104	mpcom 36	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ‘cfv 5318  (class class class)co 6007  1c1 8008   + caddc 8010  ℤcz 9454  ℤ_≥cuz 9730  ...cfz 10212  ..^cfzo 10346  seqcseq 10677

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-inn 9119  df-n0 9378  df-z 9455  df-uz 9731  df-fz 10213  df-fzo 10347  df-seqfrec 10678

This theorem is referenced by:  seq3caopr2  10723