Theorem seq3shft2 10277

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 9843	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2203	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 5429	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
6		fvoveq1 5805	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)))
7	5, 6	eqeq12d 2155	. . . . . 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 233	. . . . 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 229	. . . 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 2203	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
11		fveq2 5429	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
12		fvoveq1 5805	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑛 + 𝐾)))
13	11, 12	eqeq12d 2155	. . . . . 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 233	. . . . 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 229	. . . 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 2203	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
17		fveq2 5429	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18		fvoveq1 5805	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘((𝑛 + 1) + 𝐾)))
19	17, 18	eqeq12d 2155	. . . . . 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 233	. . . . 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 229	. . . 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 2203	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
23		fveq2 5429	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
24		fvoveq1 5805	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑥 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))
25	23, 24	eqeq12d 2155	. . . . . 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 233	. . . . 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 229	. . . 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 5429	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
29		fvoveq1 5805	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
30	28, 29	eqeq12d 2155	. . . . . . 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 2508	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
33		eluzfz1 9842	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
34	1, 33	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
35	30, 32, 34	rspcdva 2798	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘(𝑀 + 𝐾)))
36		eluzel2 9355	. . . . . . . 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 10264	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
41		seq3shft2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
42	37, 41	zaddcld 9201	. . . . . . 7 ⊢ (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
43		seq3shft2.g	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)
44	42, 43, 39	seq3-1 10264	. . . . . 6 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑀 + 𝐾)) = (𝐺‘(𝑀 + 𝐾)))
45	35, 40, 44	3eqtr4d 2183	. . . . 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 9848	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
48	47	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
49	48	expr 373	. . . . . 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 5789	. . . . . 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 521	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
53	38	adantlr 469	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
54	39	adantlr 469	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
55	52, 53, 54	seq3p1 10266	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
56	41	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℤ)
57		eluzadd 9378	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
58	52, 56, 57	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)))
59	43	adantlr 469	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)
60	58, 59, 54	seq3p1 10266	. . . . . . . 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 9359	. . . . . . . . . . . 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 9198	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℂ)
64		1cnd 7806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 1 ∈ ℂ)
65	56	zcnd 9198	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝐾 ∈ ℂ)
66	63, 64, 65	add32d 7954	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝑛 + 1) + 𝐾) = ((𝑛 + 𝐾) + 1))
67	66	fveq2d 5433	. . . . . . . 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 5429	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
69		fvoveq1 5805	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘(𝑘 + 𝐾)) = (𝐺‘((𝑛 + 1) + 𝐾)))
70	68, 69	eqeq12d 2155	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾))))
71	32	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))
72		simprr 522	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
73	70, 71, 72	rspcdva 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 1) + 𝐾)))
74	66	fveq2d 5433	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐺‘((𝑛 + 1) + 𝐾)) = (𝐺‘((𝑛 + 𝐾) + 1)))
75	73, 74	eqtrd 2173	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘((𝑛 + 𝐾) + 1)))
76	75	oveq2d 5798	. . . . . . . 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 2183	. . . . . . 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 2155	. . . . . 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	syl5ibr 155	. . . . 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 549	. . . 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 9410	. . 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 103   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ‘cfv 5131  (class class class)co 5782  1c1 7645   + caddc 7647  ℤcz 9078  ℤ_≥cuz 9350  ...cfz 9821  seqcseq 10249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-fz 9822  df-seqfrec 10250

This theorem is referenced by:  seq3f1olemqsumkj  10302  seq3shft  10642