Theorem telgsums 19113

Description: Telescoping finitely supported group sum ranging over nonnegative integers, using explicit substitution. (Contributed by AV, 24-Oct-2019.)

Hypotheses

Ref	Expression
telgsums.b	⊢ 𝐵 = (Base‘𝐺)
telgsums.g	⊢ (𝜑 → 𝐺 ∈ Abel)
telgsums.m	⊢ − = (-g‘𝐺)
telgsums.0	⊢ 0 = (0g‘𝐺)
telgsums.f	⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵)
telgsums.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)
telgsums.u	⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 ))

Assertion

Ref	Expression
telgsums	⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = ⦋0 / 𝑘⦌𝐶)

Distinct variable groups:   𝐵,𝑖,𝑘   𝐶,𝑖   𝑖,𝐺   𝑆,𝑖,𝑘   0 ,𝑖,𝑘   𝜑,𝑖   − ,𝑖

Allowed substitution hints:   𝜑(𝑘)   𝐶(𝑘)   𝐺(𝑘)   − (𝑘)

Proof of Theorem telgsums

Step	Hyp	Ref	Expression
1		telgsums.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		telgsums.0	. . 3 ⊢ 0 = (0g‘𝐺)
3		telgsums.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
4		ablcmn 18913	. . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd)
6		ablgrp 18911	. . . . . . 7 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
7	3, 6	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
8	7	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐺 ∈ Grp)
9		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0)
10		telgsums.f	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵)
11	10	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵)
12		rspcsbela 4387	. . . . . 6 ⊢ ((𝑖 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵)
13	9, 11, 12	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵)
14		peano2nn0 11938	. . . . . 6 ⊢ (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0)
15		rspcsbela 4387	. . . . . 6 ⊢ (((𝑖 + 1) ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
16	14, 10, 15	syl2anr 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
17		telgsums.m	. . . . . 6 ⊢ − = (-g‘𝐺)
18	1, 17	grpsubcl 18179	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) ∈ 𝐵)
19	8, 13, 16, 18	syl3anc 1367	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) ∈ 𝐵)
20	19	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ ℕ0 (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) ∈ 𝐵)
21		telgsums.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
22		telgsums.u	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 ))
23		rspsbca 3863	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 )) → [𝑖 / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ))
24		sbcimg 3820	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ V → ([𝑖 / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ) ↔ ([𝑖 / 𝑘]𝑆 < 𝑘 → [𝑖 / 𝑘]𝐶 = 0 )))
25		sbcbr2g 5124	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ V → ([𝑖 / 𝑘]𝑆 < 𝑘 ↔ 𝑆 < ⦋𝑖 / 𝑘⦌𝑘))
26		csbvarg 4383	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ V → ⦋𝑖 / 𝑘⦌𝑘 = 𝑖)
27	26	breq2d 5078	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ V → (𝑆 < ⦋𝑖 / 𝑘⦌𝑘 ↔ 𝑆 < 𝑖))
28	25, 27	bitrd 281	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ V → ([𝑖 / 𝑘]𝑆 < 𝑘 ↔ 𝑆 < 𝑖))
29		sbceq1g 4366	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ V → ([𝑖 / 𝑘]𝐶 = 0 ↔ ⦋𝑖 / 𝑘⦌𝐶 = 0 ))
30	28, 29	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ V → (([𝑖 / 𝑘]𝑆 < 𝑘 → [𝑖 / 𝑘]𝐶 = 0 ) ↔ (𝑆 < 𝑖 → ⦋𝑖 / 𝑘⦌𝐶 = 0 )))
31	24, 30	bitrd 281	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ V → ([𝑖 / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ) ↔ (𝑆 < 𝑖 → ⦋𝑖 / 𝑘⦌𝐶 = 0 )))
32	31	elv 3499	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ) ↔ (𝑆 < 𝑖 → ⦋𝑖 / 𝑘⦌𝐶 = 0 ))
33	23, 32	sylib 220	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 )) → (𝑆 < 𝑖 → ⦋𝑖 / 𝑘⦌𝐶 = 0 ))
34	33	expcom 416	. . . . . . . . 9 ⊢ (∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 ) → (𝑖 ∈ ℕ0 → (𝑆 < 𝑖 → ⦋𝑖 / 𝑘⦌𝐶 = 0 )))
35	22, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ ℕ0 → (𝑆 < 𝑖 → ⦋𝑖 / 𝑘⦌𝐶 = 0 )))
36	35	imp31 420	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → ⦋𝑖 / 𝑘⦌𝐶 = 0 )
37	21	nn0red 11957	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ ℝ)
38	37	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑆 ∈ ℝ)
39	38	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → 𝑆 ∈ ℝ)
40		nn0re 11907	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ0 → 𝑖 ∈ ℝ)
41	40	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → 𝑖 ∈ ℝ)
42	14	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → (𝑖 + 1) ∈ ℕ0)
43	42	nn0red 11957	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → (𝑖 + 1) ∈ ℝ)
44		simpr 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → 𝑆 < 𝑖)
45	41	ltp1d 11570	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → 𝑖 < (𝑖 + 1))
46	39, 41, 43, 44, 45	lttrd 10801	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → 𝑆 < (𝑖 + 1))
47	46	ex 415	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑆 < 𝑖 → 𝑆 < (𝑖 + 1)))
48		rspsbca 3863	. . . . . . . . . . 11 ⊢ (((𝑖 + 1) ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 )) → [(𝑖 + 1) / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ))
49		ovex 7189	. . . . . . . . . . . 12 ⊢ (𝑖 + 1) ∈ V
50		sbcimg 3820	. . . . . . . . . . . . 13 ⊢ ((𝑖 + 1) ∈ V → ([(𝑖 + 1) / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ) ↔ ([(𝑖 + 1) / 𝑘]𝑆 < 𝑘 → [(𝑖 + 1) / 𝑘]𝐶 = 0 )))
51		sbcbr2g 5124	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 + 1) ∈ V → ([(𝑖 + 1) / 𝑘]𝑆 < 𝑘 ↔ 𝑆 < ⦋(𝑖 + 1) / 𝑘⦌𝑘))
52		csbvarg 4383	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 + 1) ∈ V → ⦋(𝑖 + 1) / 𝑘⦌𝑘 = (𝑖 + 1))
53	52	breq2d 5078	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 + 1) ∈ V → (𝑆 < ⦋(𝑖 + 1) / 𝑘⦌𝑘 ↔ 𝑆 < (𝑖 + 1)))
54	51, 53	bitrd 281	. . . . . . . . . . . . . 14 ⊢ ((𝑖 + 1) ∈ V → ([(𝑖 + 1) / 𝑘]𝑆 < 𝑘 ↔ 𝑆 < (𝑖 + 1)))
55		sbceq1g 4366	. . . . . . . . . . . . . 14 ⊢ ((𝑖 + 1) ∈ V → ([(𝑖 + 1) / 𝑘]𝐶 = 0 ↔ ⦋(𝑖 + 1) / 𝑘⦌𝐶 = 0 ))
56	54, 55	imbi12d 347	. . . . . . . . . . . . 13 ⊢ ((𝑖 + 1) ∈ V → (([(𝑖 + 1) / 𝑘]𝑆 < 𝑘 → [(𝑖 + 1) / 𝑘]𝐶 = 0 ) ↔ (𝑆 < (𝑖 + 1) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = 0 )))
57	50, 56	bitrd 281	. . . . . . . . . . . 12 ⊢ ((𝑖 + 1) ∈ V → ([(𝑖 + 1) / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ) ↔ (𝑆 < (𝑖 + 1) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = 0 )))
58	49, 57	ax-mp 5	. . . . . . . . . . 11 ⊢ ([(𝑖 + 1) / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ) ↔ (𝑆 < (𝑖 + 1) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = 0 ))
59	48, 58	sylib 220	. . . . . . . . . 10 ⊢ (((𝑖 + 1) ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 )) → (𝑆 < (𝑖 + 1) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = 0 ))
60	14, 22, 59	syl2anr 598	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑆 < (𝑖 + 1) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = 0 ))
61	47, 60	syld 47	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑆 < 𝑖 → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = 0 ))
62	61	imp 409	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = 0 )
63	36, 62	oveq12d 7174	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = ( 0 − 0 ))
64	8	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → 𝐺 ∈ Grp)
65	1, 2	grpidcl 18131	. . . . . . 7 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
66	1, 2, 17	grpsubid 18183	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 − 0 ) = 0 )
67	64, 65, 66	syl2anc2 587	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → ( 0 − 0 ) = 0 )
68	63, 67	eqtrd 2856	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑆 < 𝑖) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = 0 )
69	68	ex 415	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑆 < 𝑖 → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = 0 ))
70	69	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ ℕ0 (𝑆 < 𝑖 → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = 0 ))
71	1, 2, 5, 20, 21, 70	gsummptnn0fz 19106	. 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))))
72		fzssuz 12949	. . . . . 6 ⊢ (0...(𝑆 + 1)) ⊆ (ℤ≥‘0)
73	72	a1i 11	. . . . 5 ⊢ (𝜑 → (0...(𝑆 + 1)) ⊆ (ℤ≥‘0))
74		nn0uz 12281	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
75	73, 74	sseqtrrdi 4018	. . . 4 ⊢ (𝜑 → (0...(𝑆 + 1)) ⊆ ℕ0)
76		ssralv 4033	. . . 4 ⊢ ((0...(𝑆 + 1)) ⊆ ℕ0 → (∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵 → ∀𝑘 ∈ (0...(𝑆 + 1))𝐶 ∈ 𝐵))
77	75, 10, 76	sylc 65	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...(𝑆 + 1))𝐶 ∈ 𝐵)
78	1, 3, 17, 21, 77	telgsumfz0s 19111	. 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋0 / 𝑘⦌𝐶 − ⦋(𝑆 + 1) / 𝑘⦌𝐶))
79		peano2nn0 11938	. . . . . 6 ⊢ (𝑆 ∈ ℕ0 → (𝑆 + 1) ∈ ℕ0)
80	21, 79	syl 17	. . . . 5 ⊢ (𝜑 → (𝑆 + 1) ∈ ℕ0)
81	37	ltp1d 11570	. . . . 5 ⊢ (𝜑 → 𝑆 < (𝑆 + 1))
82		rspsbca 3863	. . . . . . 7 ⊢ (((𝑆 + 1) ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 )) → [(𝑆 + 1) / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ))
83		ovex 7189	. . . . . . . 8 ⊢ (𝑆 + 1) ∈ V
84		sbcimg 3820	. . . . . . . . 9 ⊢ ((𝑆 + 1) ∈ V → ([(𝑆 + 1) / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ) ↔ ([(𝑆 + 1) / 𝑘]𝑆 < 𝑘 → [(𝑆 + 1) / 𝑘]𝐶 = 0 )))
85		sbcbr2g 5124	. . . . . . . . . . 11 ⊢ ((𝑆 + 1) ∈ V → ([(𝑆 + 1) / 𝑘]𝑆 < 𝑘 ↔ 𝑆 < ⦋(𝑆 + 1) / 𝑘⦌𝑘))
86		csbvarg 4383	. . . . . . . . . . . 12 ⊢ ((𝑆 + 1) ∈ V → ⦋(𝑆 + 1) / 𝑘⦌𝑘 = (𝑆 + 1))
87	86	breq2d 5078	. . . . . . . . . . 11 ⊢ ((𝑆 + 1) ∈ V → (𝑆 < ⦋(𝑆 + 1) / 𝑘⦌𝑘 ↔ 𝑆 < (𝑆 + 1)))
88	85, 87	bitrd 281	. . . . . . . . . 10 ⊢ ((𝑆 + 1) ∈ V → ([(𝑆 + 1) / 𝑘]𝑆 < 𝑘 ↔ 𝑆 < (𝑆 + 1)))
89		sbceq1g 4366	. . . . . . . . . 10 ⊢ ((𝑆 + 1) ∈ V → ([(𝑆 + 1) / 𝑘]𝐶 = 0 ↔ ⦋(𝑆 + 1) / 𝑘⦌𝐶 = 0 ))
90	88, 89	imbi12d 347	. . . . . . . . 9 ⊢ ((𝑆 + 1) ∈ V → (([(𝑆 + 1) / 𝑘]𝑆 < 𝑘 → [(𝑆 + 1) / 𝑘]𝐶 = 0 ) ↔ (𝑆 < (𝑆 + 1) → ⦋(𝑆 + 1) / 𝑘⦌𝐶 = 0 )))
91	84, 90	bitrd 281	. . . . . . . 8 ⊢ ((𝑆 + 1) ∈ V → ([(𝑆 + 1) / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ) ↔ (𝑆 < (𝑆 + 1) → ⦋(𝑆 + 1) / 𝑘⦌𝐶 = 0 )))
92	83, 91	ax-mp 5	. . . . . . 7 ⊢ ([(𝑆 + 1) / 𝑘](𝑆 < 𝑘 → 𝐶 = 0 ) ↔ (𝑆 < (𝑆 + 1) → ⦋(𝑆 + 1) / 𝑘⦌𝐶 = 0 ))
93	82, 92	sylib 220	. . . . . 6 ⊢ (((𝑆 + 1) ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 )) → (𝑆 < (𝑆 + 1) → ⦋(𝑆 + 1) / 𝑘⦌𝐶 = 0 ))
94	93	ex 415	. . . . 5 ⊢ ((𝑆 + 1) ∈ ℕ0 → (∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 ) → (𝑆 < (𝑆 + 1) → ⦋(𝑆 + 1) / 𝑘⦌𝐶 = 0 )))
95	80, 22, 81, 94	syl3c 66	. . . 4 ⊢ (𝜑 → ⦋(𝑆 + 1) / 𝑘⦌𝐶 = 0 )
96	95	oveq2d 7172	. . 3 ⊢ (𝜑 → (⦋0 / 𝑘⦌𝐶 − ⦋(𝑆 + 1) / 𝑘⦌𝐶) = (⦋0 / 𝑘⦌𝐶 − 0 ))
97		0nn0 11913	. . . . . 6 ⊢ 0 ∈ ℕ0
98	97	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℕ0)
99		rspcsbela 4387	. . . . 5 ⊢ ((0 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) → ⦋0 / 𝑘⦌𝐶 ∈ 𝐵)
100	98, 10, 99	syl2anc 586	. . . 4 ⊢ (𝜑 → ⦋0 / 𝑘⦌𝐶 ∈ 𝐵)
101	1, 2, 17	grpsubid1 18184	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ ⦋0 / 𝑘⦌𝐶 ∈ 𝐵) → (⦋0 / 𝑘⦌𝐶 − 0 ) = ⦋0 / 𝑘⦌𝐶)
102	7, 100, 101	syl2anc 586	. . 3 ⊢ (𝜑 → (⦋0 / 𝑘⦌𝐶 − 0 ) = ⦋0 / 𝑘⦌𝐶)
103	96, 102	eqtrd 2856	. 2 ⊢ (𝜑 → (⦋0 / 𝑘⦌𝐶 − ⦋(𝑆 + 1) / 𝑘⦌𝐶) = ⦋0 / 𝑘⦌𝐶)
104	71, 78, 103	3eqtrd 2860	1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = ⦋0 / 𝑘⦌𝐶)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494  [wsbc 3772  ⦋csb 3883   ⊆ wss 3936   class class class wbr 5066   ↦ cmpt 5146  ‘cfv 6355  (class class class)co 7156  ℝcr 10536  0cc0 10537  1c1 10538   + caddc 10540   < clt 10675  ℕ₀cn0 11898  ℤ_≥cuz 12244  ...cfz 12893  Basecbs 16483  0_gc0g 16713   Σ_g cgsu 16714  Grpcgrp 18103  -_gcsg 18105  CMndccmn 18906  Abelcabl 18907

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 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-oi 8974  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-fzo 13035  df-seq 13371  df-hash 13692  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-0g 16715  df-gsum 16716  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-cntz 18447  df-cmn 18908  df-abl 18909

This theorem is referenced by:  telgsum  19114