Theorem seq3shft2 10842

Description: Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)

Hypotheses

Ref	Expression
seq3shft2.1	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seq3shft2.2	⊢ (𝜑 → 𝐾 ∈ ℤ)
seq3shft2.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
seq3shft2.f	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
seq3shft2.g	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)
seq3shft2.pl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)

Assertion

Ref	Expression
seq3shft2	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(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)

Proof of Theorem seq3shft2

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seq3shft2.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 10365	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2295	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 5669	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
6		fvoveq1 6072	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))
7	5, 6	eqeq12d 2247	. . . . . 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 234	. . . . 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 230	. . . 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 2295	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
11		fveq2 5669	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
12		fvoveq1 6072	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))
13	11, 12	eqeq12d 2247	. . . . . 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 234	. . . . 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 230	. . . 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 2295	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
17		fveq2 5669	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18		fvoveq1 6072	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)))
19	17, 18	eqeq12d 2247	. . . . . 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 234	. . . . 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 230	. . . 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 2295	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
23		fveq2 5669	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
24		fvoveq1 6072	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))
25	23, 24	eqeq12d 2247	. . . . . 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 234	. . . . 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 230	. . . 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 5669	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
29		fvoveq1 6072	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
30	28, 29	eqeq12d 2247	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾))))
31		seq3shft2.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
32	31	ralrimiva 2615	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
33		eluzfz1 10364	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
34	1, 33	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
35	30, 32, 34	rspcdva 2925	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾)))
36		eluzel2 9857	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
37	1, 36	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
38		seq3shft2.f	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
39		seq3shft2.pl	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
40	37, 38, 39	seq3-1 10823	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
41		seq3shft2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
42	37, 41	zaddcld 9703	. . . . . . 7 ⊢ (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
43		seq3shft2.g	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)
44	42, 43, 39	seq3-1 10823	. . . . . 6 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
45	35, 40, 44	3eqtr4d 2275	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))
46	45	a1i13 24	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))))
47		peano2fzr 10370	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
48	47	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
49	48	expr 375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
50	49	imim1d 75	. . . . 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)))))
51		oveq1 6056	. . . . . 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))))
52		simprl 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
53	38	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
54	39	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
55	52, 53, 54	seq3p1 10826	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
56	41	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℤ)
57		eluzadd 9882	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
58	52, 56, 57	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
59	43	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)
60	58, 59, 54	seq3p1 10826	. . . . . . . 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))))
61		eluzelz 9862	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
62	52, 61	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
63	62	zcnd 9700	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℂ)
64		1cnd 8289	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 1 ∈ ℂ)
65	56	zcnd 9700	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℂ)
66	63, 64, 65	add32d 8440	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
67	66	fveq2d 5673	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 𝐾) + 1)))
68		fveq2 5669	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
69		fvoveq1 6072	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘((𝑛 + 1) + 𝐾)))
70	68, 69	eqeq12d 2247	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
71	32	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
72		simprr 533	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
73	70, 71, 72	rspcdva 2925	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾)))
74	66	fveq2d 5673	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐺‘((𝑛 + 1) + 𝐾)) = (𝐺‘((𝑛 + 𝐾) + 1)))
75	73, 74	eqtrd 2265	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 𝐾) + 1)))
76	75	oveq2d 6065	. . . . . . . 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))))
77	60, 67, 76	3eqtr4d 2275	. . . . . . 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))))
78	55, 77	eqeq12d 2247	. . . . . 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)))))
79	51, 78	imbitrrid 156	. . . . 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))))
80	50, 79	animpimp2impd 561	. . . 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))))))
81	9, 15, 21, 27, 46, 80	uzind4 9919	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))))
82	1, 81	mpcom 36	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))))
83	3, 82	mpd 13	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ‘cfv 5351  (class class class)co 6049  1c1 8127   + caddc 8129  ℤcz 9576  ℤ_≥cuz 9852  ...cfz 10341  seqcseq 10808

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-n0 9496  df-z 9577  df-uz 9853  df-fz 10342  df-seqfrec 10809

This theorem is referenced by:  seq3f1olemqsumkj  10872  seq3shft  11519  mulgnndir  13860