Theorem seqshft2 12827

Description: Shifting the index set of a sequence. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
seqshft2.1	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seqshft2.2	⊢ (𝜑 → 𝐾 ∈ ℤ)
seqshft2.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))

Assertion

Ref	Expression
seqshft2	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))

Distinct variable groups:   𝑘,𝐹   𝑘,𝐺   𝑘,𝐾   𝑘,𝑀   𝜑,𝑘   𝑘,𝑁

Allowed substitution hint:   + (𝑘)

Proof of Theorem seqshft2

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

Step	Hyp	Ref	Expression
1		seqshft2.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 12349	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2689	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
6		oveq1 6657	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝑥 + 𝐾) = (𝑀 + 𝐾))
7	6	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))
8	5, 7	eqeq12d 2637	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾))))
9	4, 8	imbi12d 334	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾))) ↔ (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))))
10	9	imbi2d 330	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾))))))
11		eleq1 2689	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
12		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
13		oveq1 6657	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑥 + 𝐾) = (𝑛 + 𝐾))
14	13	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))
15	12, 14	eqeq12d 2637	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾))))
16	11, 15	imbi12d 334	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾))) ↔ (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))))
17	16	imbi2d 330	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾))))))
18		eleq1 2689	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
19		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
20		oveq1 6657	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 + 𝐾) = ((𝑛 + 1) + 𝐾))
21	20	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)))
22	19, 21	eqeq12d 2637	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾))))
23	18, 22	imbi12d 334	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾))) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)))))
24	23	imbi2d 330	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾))))))
25		eleq1 2689	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
26		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
27		oveq1 6657	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑥 + 𝐾) = (𝑁 + 𝐾))
28	27	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))
29	26, 28	eqeq12d 2637	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))))
30	25, 29	imbi12d 334	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾))) ↔ (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))))
31	30	imbi2d 330	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))))))
32		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
33		oveq1 6657	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝑘 + 𝐾) = (𝑀 + 𝐾))
34	33	fveq2d 6195	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
35	32, 34	eqeq12d 2637	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾))))
36		seqshft2.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
37	36	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
38		eluzfz1 12348	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
39	1, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
40	35, 37, 39	rspcdva 3316	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾)))
41		eluzel2 11692	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
42	1, 41	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
43		seq1 12814	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
44	42, 43	syl 17	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
45		seqshft2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
46	42, 45	zaddcld 11486	. . . . . . 7 ⊢ (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
47		seq1 12814	. . . . . . 7 ⊢ ((𝑀 + 𝐾) ∈ ℤ → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
48	46, 47	syl 17	. . . . . 6 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
49	40, 44, 48	3eqtr4d 2666	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))
50	49	a1i13 27	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))))
51		peano2fzr 12354	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
52	51	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
53	52	expr 643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
54	53	imim1d 82	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))))
55		oveq1 6657	. . . . . . . . . 10 ⊢ ((seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))))
56		simprl 794	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
57		seqp1 12816	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
58	56, 57	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
59	45	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℤ)
60		eluzadd 11716	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
61	56, 59, 60	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
62		seqp1 12816	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 𝐾) + 1)) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐺‘((𝑛 + 𝐾) + 1))))
63	61, 62	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 𝐾) + 1)) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐺‘((𝑛 + 𝐾) + 1))))
64		eluzelz 11697	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
65	56, 64	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
66		zcn 11382	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
67		zcn 11382	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
68		ax-1cn 9994	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
69		add32 10254	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
70	68, 69	mp3an2 1412	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
71	66, 67, 70	syl2an 494	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
72	65, 59, 71	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
73	72	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 𝐾) + 1)))
74		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
75		oveq1 6657	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛 + 1) → (𝑘 + 𝐾) = ((𝑛 + 1) + 𝐾))
76	75	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘((𝑛 + 1) + 𝐾)))
77	74, 76	eqeq12d 2637	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
78	37	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
79		simprr 796	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
80	77, 78, 79	rspcdva 3316	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾)))
81	72	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐺‘((𝑛 + 1) + 𝐾)) = (𝐺‘((𝑛 + 𝐾) + 1)))
82	80, 81	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 𝐾) + 1)))
83	82	oveq2d 6666	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐺‘((𝑛 + 𝐾) + 1))))
84	63, 73, 83	3eqtr4d 2666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))))
85	58, 84	eqeq12d 2637	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)) ↔ ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1)))))
86	55, 85	syl5ibr 236	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾))))
87	86	expr 643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)))))
88	87	a2d 29	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)))))
89	54, 88	syld 47	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)))))
90	89	expcom 451	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝜑 → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾))))))
91	90	a2d 29	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾))))))
92	10, 17, 24, 31, 50, 91	uzind4 11746	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))))
93	1, 92	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))))
94	3, 93	mpd 15	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  1c1 9937   + caddc 9939  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  seqf1olem2  12841  seqshft  13825  isercoll2  14399  fprodser  14679  gsumccat  17378  mulgnndir  17569  mulgnndirOLD  17570