Theorem iseqshft2 9232

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 8896	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2100	. . . . . 6 ⊢ (𝑤 = 𝑀 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 5178	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑀))
6		oveq1 5519	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (𝑤 + 𝐾) = (𝑀 + 𝐾))
7	6	fveq2d 5182	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)))
8	5, 7	eqeq12d 2054	. . . . . 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 223	. . . . 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 219	. . . 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 2100	. . . . . 6 ⊢ (𝑤 = 𝑛 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
12		fveq2 5178	. . . . . . 7 ⊢ (𝑤 = 𝑛 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑛))
13		oveq1 5519	. . . . . . . 8 ⊢ (𝑤 = 𝑛 → (𝑤 + 𝐾) = (𝑛 + 𝐾))
14	13	fveq2d 5182	. . . . . . 7 ⊢ (𝑤 = 𝑛 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑛 + 𝐾)))
15	12, 14	eqeq12d 2054	. . . . . 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 223	. . . . 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 219	. . . 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 2100	. . . . . 6 ⊢ (𝑤 = (𝑛 + 1) → (𝑤 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
19		fveq2 5178	. . . . . . 7 ⊢ (𝑤 = (𝑛 + 1) → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)))
20		oveq1 5519	. . . . . . . 8 ⊢ (𝑤 = (𝑛 + 1) → (𝑤 + 𝐾) = ((𝑛 + 1) + 𝐾))
21	20	fveq2d 5182	. . . . . . 7 ⊢ (𝑤 = (𝑛 + 1) → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘((𝑛 + 1) + 𝐾)))
22	19, 21	eqeq12d 2054	. . . . . 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 223	. . . . 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 219	. . . 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 2100	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝑤 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
26		fveq2 5178	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (seq𝑀( + , 𝐹, 𝑆)‘𝑤) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))
27		oveq1 5519	. . . . . . . 8 ⊢ (𝑤 = 𝑁 → (𝑤 + 𝐾) = (𝑁 + 𝐾))
28	27	fveq2d 5182	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑤 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))
29	26, 28	eqeq12d 2054	. . . . . 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 223	. . . . 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 219	. . . 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 8895	. . . . . . . . 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 2392	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
36		fveq2 5178	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
37		oveq1 5519	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑀 → (𝑘 + 𝐾) = (𝑀 + 𝐾))
38	37	fveq2d 5182	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
39	36, 38	eqeq12d 2054	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾))))
40	39	rspcv 2652	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) → (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾))))
41	33, 35, 40	sylc 56	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾)))
42		eluzel2 8478	. . . . . . . . 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 9222	. . . . . . 7 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀))
48		iseqshft2.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℤ)
49	43, 48	zaddcld 8364	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
50		iseqshft2.g	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)
51	49, 44, 50, 46	iseq1 9222	. . . . . . 7 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
52	41, 47, 51	3eqtr4d 2082	. . . . . 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 8901	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
56	55	adantl 262	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
57	56	expr 357	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
58	57	imim1d 69	. . . . . . 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 5519	. . . . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
61	44	adantr 261	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑆 ∈ 𝑉)
62	45	adantlr 446	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
63	46	adantlr 446	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
64	60, 61, 62, 63	iseqp1 9225	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
65	48	adantr 261	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℤ)
66		eluzadd 8501	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
67	60, 65, 66	syl2anc 391	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
68	50	adantlr 446	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)
69	67, 61, 68, 63	iseqp1 9225	. . . . . . . . . . . 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 8482	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
71	60, 70	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
72		zcn 8250	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
73		zcn 8250	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
74		ax-1cn 6977	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
75		add32 7170	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
76	74, 75	mp3an2 1220	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
77	72, 73, 76	syl2an 273	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
78	71, 65, 77	syl2anc 391	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
79	78	fveq2d 5182	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
81	35	adantr 261	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
82		fveq2 5178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
83		oveq1 5519	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑛 + 1) → (𝑘 + 𝐾) = ((𝑛 + 1) + 𝐾))
84	83	fveq2d 5182	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘((𝑛 + 1) + 𝐾)))
85	82, 84	eqeq12d 2054	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
86	85	rspcv 2652	. . . . . . . . . . . . . . 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 56	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾)))
88	78	fveq2d 5182	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐺‘((𝑛 + 1) + 𝐾)) = (𝐺‘((𝑛 + 𝐾) + 1)))
89	87, 88	eqtrd 2072	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 𝐾) + 1)))
90	89	oveq2d 5528	. . . . . . . . . . . 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 2082	. . . . . . . . . . 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 2054	. . . . . . . . . 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 145	. . . . . . . . 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 357	. . . . . . . 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 23	. . . . . . 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 40	. . . . . 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 109	. . . . 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 23	. . . 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 8531	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))))
100	1, 99	mpcom 32	. 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 97   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ‘cfv 4902  (class class class)co 5512  ℂcc 6887  1c1 6890   + caddc 6892  ℤcz 8245  ℤ_≥cuz 8473  ...cfz 8874  seqcseq 9211

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-fz 8875  df-iseq 9212

This theorem is referenced by: (None)