Theorem seqshft2 13677

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 13193	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2826	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
6		fvoveq1 7278	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))
7	5, 6	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾))))
8	4, 7	imbi12d 344	. . . . 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)))))
9	8	imbi2d 340	. . . 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))))))
10		eleq1 2826	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
11		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
12		fvoveq1 7278	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))
13	11, 12	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾))))
14	10, 13	imbi12d 344	. . . . 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)))))
15	14	imbi2d 340	. . . 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))))))
16		eleq1 2826	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
17		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18		fvoveq1 7278	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)))
19	17, 18	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾))))
20	16, 19	imbi12d 344	. . . . 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)))))
21	20	imbi2d 340	. . . 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))))))
22		eleq1 2826	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
23		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
24		fvoveq1 7278	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))
25	23, 24	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))))
26	22, 25	imbi12d 344	. . . . 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)))))
27	26	imbi2d 340	. . . 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))))))
28		fveq2 6756	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
29		fvoveq1 7278	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
30	28, 29	eqeq12d 2754	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾))))
31		seqshft2.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
32	31	ralrimiva 3107	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
33		eluzfz1 13192	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
34	1, 33	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
35	30, 32, 34	rspcdva 3554	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾)))
36		eluzel2 12516	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
37	1, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
38		seq1 13662	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
39	37, 38	syl 17	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
40		seqshft2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
41	37, 40	zaddcld 12359	. . . . . . 7 ⊢ (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
42		seq1 13662	. . . . . . 7 ⊢ ((𝑀 + 𝐾) ∈ ℤ → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
43	41, 42	syl 17	. . . . . 6 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
44	35, 39, 43	3eqtr4d 2788	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))
45	44	a1i13 27	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))))
46		peano2fzr 13198	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
47	46	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
48	47	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
49	48	imim1d 82	. . . . 5 ⊢ ((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)))))
50		oveq1 7262	. . . . . 6 ⊢ ((seq𝑀( + , 𝐹)‘𝑛) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))))
51		simprl 767	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
52		seqp1 13664	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
53	51, 52	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
54	40	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℤ)
55		eluzadd 12542	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
56	51, 54, 55	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
57		seqp1 13664	. . . . . . . . 9 ⊢ ((𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 𝐾) + 1)) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐺‘((𝑛 + 𝐾) + 1))))
58	56, 57	syl 17	. . . . . . . 8 ⊢ ((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))))
59		eluzelz 12521	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
60	51, 59	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
61		zcn 12254	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
62		zcn 12254	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
63		ax-1cn 10860	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
64		add32 11123	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
65	63, 64	mp3an2 1447	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
66	61, 62, 65	syl2an 595	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
67	60, 54, 66	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
68	67	fveq2d 6760	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 𝐾) + 1)))
69		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
70		fvoveq1 7278	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘((𝑛 + 1) + 𝐾)))
71	69, 70	eqeq12d 2754	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
72	32	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
73		simprr 769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
74	71, 72, 73	rspcdva 3554	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾)))
75	67	fveq2d 6760	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐺‘((𝑛 + 1) + 𝐾)) = (𝐺‘((𝑛 + 𝐾) + 1)))
76	74, 75	eqtrd 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 𝐾) + 1)))
77	76	oveq2d 7271	. . . . . . . 8 ⊢ ((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))))
78	58, 68, 77	3eqtr4d 2788	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)) = ((seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)) + (𝐹‘(𝑛 + 1))))
79	53, 78	eqeq12d 2754	. . . . . 6 ⊢ ((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)))))
80	50, 79	syl5ibr 245	. . . . 5 ⊢ ((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))))
81	49, 80	animpimp2impd 842	. . . 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))))))
82	9, 15, 21, 27, 45, 81	uzind4 12575	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))))
83	1, 82	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))))
84	3, 83	mpd 15	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  1c1 10803   + caddc 10805  ℤcz 12249  ℤ_≥cuz 12511  ...cfz 13168  seqcseq 13649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-seq 13650

This theorem is referenced by:  seqf1olem2  13691  seqshft  14724  isercoll2  15308  fprodser  15587  gsumsgrpccat  18393  gsumccatOLD  18394  mulgnndir  18647