Theorem iseqshft2 9452

Description: Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)

Hypotheses

Ref	Expression
iseqshft2.1	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
iseqshft2.2	⊢ (𝜑 → 𝐾 ∈ ℤ)
iseqshft2.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
iseqshft2.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
iseqshft2.f	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
iseqshft2.g	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)
iseqshft2.pl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)

Assertion

Ref	Expression
iseqshft2	⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))

Distinct variable groups:   𝑥, + ,𝑦   𝑘,𝐹,𝑥   𝑦,𝐹   𝑘,𝐺,𝑥   𝑦,𝐺   𝑘,𝐾,𝑥   𝑦,𝐾   𝑘,𝑀,𝑥   𝑦,𝑀   𝑘,𝑁,𝑥   𝑦,𝑁   𝑥,𝑆,𝑦   𝜑,𝑘,𝑥   𝜑,𝑦

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

Proof of Theorem iseqshft2

Dummy variables 𝑛 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqshft2.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 9051	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2141	. . . . . 6 ⊢ (𝑤 = 𝑀 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 5198	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
6		oveq1 5539	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (𝑤 + 𝐾) = (𝑀 + 𝐾))
7	6	fveq2d 5202	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))
8	5, 7	eqeq12d 2095	. . . . . 6 ⊢ (𝑤 = 𝑀 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾))))
9	4, 8	imbi12d 232	. . . . 5 ⊢ (𝑤 = 𝑀 → ((𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾))) ↔ (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))))
10	9	imbi2d 228	. . . 4 ⊢ (𝑤 = 𝑀 → ((𝜑 → (𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾))))))
11		eleq1 2141	. . . . . 6 ⊢ (𝑤 = 𝑛 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
12		fveq2 5198	. . . . . . 7 ⊢ (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
13		oveq1 5539	. . . . . . . 8 ⊢ (𝑤 = 𝑛 → (𝑤 + 𝐾) = (𝑛 + 𝐾))
14	13	fveq2d 5202	. . . . . . 7 ⊢ (𝑤 = 𝑛 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))
15	12, 14	eqeq12d 2095	. . . . . 6 ⊢ (𝑤 = 𝑛 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))))
16	11, 15	imbi12d 232	. . . . 5 ⊢ (𝑤 = 𝑛 → ((𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾))) ↔ (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))))
17	16	imbi2d 228	. . . 4 ⊢ (𝑤 = 𝑛 → ((𝜑 → (𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))))))
18		eleq1 2141	. . . . . 6 ⊢ (𝑤 = (𝑛 + 1) → (𝑤 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
19		fveq2 5198	. . . . . . 7 ⊢ (𝑤 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
20		oveq1 5539	. . . . . . . 8 ⊢ (𝑤 = (𝑛 + 1) → (𝑤 + 𝐾) = ((𝑛 + 1) + 𝐾))
21	20	fveq2d 5202	. . . . . . 7 ⊢ (𝑤 = (𝑛 + 1) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))
22	19, 21	eqeq12d 2095	. . . . . 6 ⊢ (𝑤 = (𝑛 + 1) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) ↔ (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾))))
23	18, 22	imbi12d 232	. . . . 5 ⊢ (𝑤 = (𝑛 + 1) → ((𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾))) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))))
24	23	imbi2d 228	. . . 4 ⊢ (𝑤 = (𝑛 + 1) → ((𝜑 → (𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾))))))
25		eleq1 2141	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
26		fveq2 5198	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
27		oveq1 5539	. . . . . . . 8 ⊢ (𝑤 = 𝑁 → (𝑤 + 𝐾) = (𝑁 + 𝐾))
28	27	fveq2d 5202	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))
29	26, 28	eqeq12d 2095	. . . . . 6 ⊢ (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) ↔ (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾))))
30	25, 29	imbi12d 232	. . . . 5 ⊢ (𝑤 = 𝑁 → ((𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾))) ↔ (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))))
31	30	imbi2d 228	. . . 4 ⊢ (𝑤 = 𝑁 → ((𝜑 → (𝑤 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾))))))
32		eluzfz1 9050	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
33	1, 32	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
34		iseqshft2.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
35	34	ralrimiva 2434	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
36		fveq2 5198	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
37		oveq1 5539	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑀 → (𝑘 + 𝐾) = (𝑀 + 𝐾))
38	37	fveq2d 5202	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
39	36, 38	eqeq12d 2095	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾))))
40	39	rspcv 2697	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) → (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾))))
41	33, 35, 40	sylc 61	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾)))
42		eluzel2 8624	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
43	1, 42	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
44		iseqshft2.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
45		iseqshft2.f	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
46		iseqshft2.pl	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
47	43, 44, 45, 46	iseq1 9442	. . . . . . 7 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀))
48		iseqshft2.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℤ)
49	43, 48	zaddcld 8473	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
50		iseqshft2.g	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)
51	49, 44, 50, 46	iseq1 9442	. . . . . . 7 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
52	41, 47, 51	3eqtr4d 2123	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))
53	52	a1d 22	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾))))
54	53	a1i 9	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))))
55		peano2fzr 9056	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
56	55	adantl 271	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
57	56	expr 367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
58	57	imim1d 74	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))))
59		oveq1 5539	. . . . . . . . . 10 ⊢ ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))))
60		simprl 497	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
61	44	adantr 270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑆 ∈ 𝑉)
62	45	adantlr 460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
63	46	adantlr 460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
64	60, 61, 62, 63	iseqp1 9445	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
65	48	adantr 270	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℤ)
66		eluzadd 8647	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
67	60, 65, 66	syl2anc 403	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
68	50	adantlr 460	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)
69	67, 61, 68, 63	iseqp1 9445	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 𝐾) + 1)) = ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐺‘((𝑛 + 𝐾) + 1))))
70		eluzelz 8628	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
71	60, 70	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
72		zcn 8356	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
73		zcn 8356	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
74		ax-1cn 7069	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
75		add32 7267	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
76	74, 75	mp3an2 1256	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
77	72, 73, 76	syl2an 283	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
78	71, 65, 77	syl2anc 403	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
79	78	fveq2d 5202	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 𝐾) + 1)))
80		simprr 498	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
81	35	adantr 270	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
82		fveq2 5198	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
83		oveq1 5539	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑛 + 1) → (𝑘 + 𝐾) = ((𝑛 + 1) + 𝐾))
84	83	fveq2d 5202	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘((𝑛 + 1) + 𝐾)))
85	82, 84	eqeq12d 2095	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
86	85	rspcv 2697	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
87	80, 81, 86	sylc 61	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾)))
88	78	fveq2d 5202	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐺‘((𝑛 + 1) + 𝐾)) = (𝐺‘((𝑛 + 𝐾) + 1)))
89	87, 88	eqtrd 2113	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 𝐾) + 1)))
90	89	oveq2d 5548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐺‘((𝑛 + 𝐾) + 1))))
91	69, 79, 90	3eqtr4d 2123	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)) = ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))))
92	64, 91	eqeq12d 2095	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)) ↔ ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1)))))
93	59, 92	syl5ibr 154	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾))))
94	93	expr 367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))))
95	94	a2d 26	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))))
96	58, 95	syld 44	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))))
97	96	expcom 114	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝜑 → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾))))))
98	97	a2d 26	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾))))))
99	10, 17, 24, 31, 54, 98	uzind4 8676	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))))
100	1, 99	mpcom 36	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾))))
101	3, 100	mpd 13	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ‘cfv 4922  (class class class)co 5532  ℂcc 6979  1c1 6982   + caddc 6984  ℤcz 8351  ℤ_≥cuz 8619  ...cfz 9029  seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030  df-iseq 9432

This theorem is referenced by: (None)