Theorem seqid2 13419

Description: The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for +) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
seqid2.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑥)
seqid2.2	⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
seqid2.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
seqid2.4	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)
seqid2.5	⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) = 𝑍)

Assertion

Ref	Expression
seqid2	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐾   𝑥,𝑀   𝑥,𝑁   𝜑,𝑥   𝑥,𝑆   𝑥, +   𝑥,𝑍

Proof of Theorem seqid2

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqid2.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
2		eluzfz2 12918	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (𝐾...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾...𝑁))
4		eleq1 2902	. . . . . 6 ⊢ (𝑥 = 𝐾 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5		fveq2 6672	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝐾))
6	5	eqeq2d 2834	. . . . . 6 ⊢ (𝑥 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾)))
7	4, 6	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝐾 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))))
8	7	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝐾 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾)))))
9		eleq1 2902	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑛 ∈ (𝐾...𝑁)))
10		fveq2 6672	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
11	10	eqeq2d 2834	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)))
12	9, 11	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))))
13	12	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)))))
14		eleq1 2902	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝐾...𝑁) ↔ (𝑛 + 1) ∈ (𝐾...𝑁)))
15		fveq2 6672	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
16	15	eqeq2d 2834	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))
17	14, 16	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))))
18	17	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
19		eleq1 2902	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
20		fveq2 6672	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
21	20	eqeq2d 2834	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))
22	19, 21	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))))
23	22	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))))
24		eqidd 2824	. . . . 5 ⊢ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))
25	24	2a1i 12	. . . 4 ⊢ (𝐾 ∈ ℤ → (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))))
26		peano2fzr 12923	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁)) → 𝑛 ∈ (𝐾...𝑁))
27	26	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (𝐾...𝑁))
28	27	expr 459	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑛 ∈ (𝐾...𝑁)))
29	28	imim1d 82	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → ((𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))))
30		oveq1 7165	. . . . . 6 ⊢ ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
31		fveqeq2 6681	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑥) = 𝑍 ↔ (𝐹‘(𝑛 + 1)) = 𝑍))
32		seqid2.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) = 𝑍)
33	32	ralrimiva 3184	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑥) = 𝑍)
34	33	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑥) = 𝑍)
35		eluzp1p1 12273	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → (𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
36	35	ad2antrl 726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
37		elfzuz3 12908	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
38	37	ad2antll 727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
39		elfzuzb 12905	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))))
40	36, 38, 39	sylanbrc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ ((𝐾 + 1)...𝑁))
41	31, 34, 40	rspcdva 3627	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑛 + 1)) = 𝑍)
42	41	oveq2d 7174	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍))
43		oveq1 7165	. . . . . . . . . . 11 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍))
44		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → 𝑥 = (seq𝑀( + , 𝐹)‘𝐾))
45	43, 44	eqeq12d 2839	. . . . . . . . . 10 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → ((𝑥 + 𝑍) = 𝑥 ↔ ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾)))
46		seqid2.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑥)
47	46	ralrimiva 3184	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥 + 𝑍) = 𝑥)
48		seqid2.4	. . . . . . . . . 10 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)
49	45, 47, 48	rspcdva 3627	. . . . . . . . 9 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾))
50	49	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾))
51	42, 50	eqtr2d 2859	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))))
52		simprl 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ≥‘𝐾))
53		seqid2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
54	53	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ≥‘𝑀))
55		uztrn 12264	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
56	52, 54, 55	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
57		seqp1 13387	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
58	56, 57	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
59	51, 58	eqeq12d 2839	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ↔ ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
60	30, 59	syl5ibr 248	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))
61	29, 60	animpimp2impd 842	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → ((𝜑 → (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
62	8, 13, 18, 23, 25, 61	uzind4 12309	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))))
63	1, 62	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))
64	3, 63	mpd 15	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ‘cfv 6357  (class class class)co 7158  1c1 10540   + caddc 10542  ℤcz 11984  ℤ_≥cuz 12246  ...cfz 12895  seqcseq 13372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896  df-seq 13373

This theorem is referenced by:  seqcoll  13825  seqcoll2  13826  fsumcvg  15071  fprodcvg  15286  ovolicc1  24119  lgsdilem2  25911