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Theorem dvfsumlem4 23792

Description: Lemma for dvfsumrlim 23794. (Contributed by Mario Carneiro, 18-May-2016.)

**Hypotheses**
Ref	Expression
dvfsum.s	⊢ 𝑆 = (𝑇(,)+∞)
dvfsum.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
dvfsum.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
dvfsum.d	⊢ (𝜑 → 𝐷 ∈ ℝ)
dvfsum.md	⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1))
dvfsum.t	⊢ (𝜑 → 𝑇 ∈ ℝ)
dvfsum.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
dvfsum.b1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
dvfsum.b2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
dvfsum.b3	⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
dvfsum.c	⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶)
dvfsum.u	⊢ (𝜑 → 𝑈 ∈ ℝ^*)
dvfsum.l	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵)
dvfsumlem4.g	⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴))
dvfsumlem4.0	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈)) → 0 ≤ 𝐵)
dvfsumlem4.1	⊢ (𝜑 → 𝑋 ∈ 𝑆)
dvfsumlem4.2	⊢ (𝜑 → 𝑌 ∈ 𝑆)
dvfsumlem4.3	⊢ (𝜑 → 𝐷 ≤ 𝑋)
dvfsumlem4.4	⊢ (𝜑 → 𝑋 ≤ 𝑌)
dvfsumlem4.5	⊢ (𝜑 → 𝑌 ≤ 𝑈)

**Assertion**
Ref	Expression
dvfsumlem4	⊢ (𝜑 → (abs‘((𝐺‘𝑌) − (𝐺‘𝑋))) ≤ ⦋𝑋 / 𝑥⦌𝐵)

Distinct variable groups: 𝐵,𝑘 𝑥,𝐶 𝑥,𝑘,𝐷 𝜑,𝑘,𝑥 𝑆,𝑘,𝑥 𝑘,𝑀,𝑥 𝑥,𝑇 𝑘,𝑌,𝑥 𝑥,𝑍 𝑈,𝑘,𝑥 𝑘,𝑋,𝑥 Allowed substitution hints: 𝐴(𝑥,𝑘) 𝐵(𝑥) 𝐶(𝑘) 𝑇(𝑘) 𝐺(𝑥,𝑘) 𝑉(𝑥,𝑘) 𝑍(𝑘)

Proof of Theorem dvfsumlem4 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvfsumlem4.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
2		fzfid 12772	. . . . . . 7 ⊢ (𝜑 → (𝑀...(⌊‘𝑌)) ∈ Fin)
3		dvfsum.b2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	ralrimiva 2966	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 𝐵 ∈ ℝ)
5		elfzuz 12338	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑌)) → 𝑘 ∈ (ℤ_≥‘𝑀))
6		dvfsum.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
7	5, 6	syl6eleqr 2712	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑌)) → 𝑘 ∈ 𝑍)
8		dvfsum.c	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶)
9	8	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ))
10	9	rspccva 3308	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝑍 𝐵 ∈ ℝ ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℝ)
11	4, 7, 10	syl2an 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(⌊‘𝑌))) → 𝐶 ∈ ℝ)
12	2, 11	fsumrecl 14465	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 ∈ ℝ)
13		dvfsum.a	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
14	13	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ)
15		nfcsb1v 3549	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑌 / 𝑥⦌𝐴
16	15	nfel1 2779	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑌 / 𝑥⦌𝐴 ∈ ℝ
17		csbeq1a 3542	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → 𝐴 = ⦋𝑌 / 𝑥⦌𝐴)
18	17	eleq1d 2686	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝐴 ∈ ℝ ↔ ⦋𝑌 / 𝑥⦌𝐴 ∈ ℝ))
19	16, 18	rspc 3303	. . . . . . 7 ⊢ (𝑌 ∈ 𝑆 → (∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ → ⦋𝑌 / 𝑥⦌𝐴 ∈ ℝ))
20	1, 14, 19	sylc 65	. . . . . 6 ⊢ (𝜑 → ⦋𝑌 / 𝑥⦌𝐴 ∈ ℝ)
21	12, 20	resubcld 10458	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ∈ ℝ)
22		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥𝑌
23		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶
24		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥 −
25	23, 24, 15	nfov 6676	. . . . . 6 ⊢ Ⅎ𝑥(Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)
26		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → (⌊‘𝑥) = (⌊‘𝑌))
27	26	oveq2d 6666	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝑀...(⌊‘𝑥)) = (𝑀...(⌊‘𝑌)))
28	27	sumeq1d 14431	. . . . . . 7 ⊢ (𝑥 = 𝑌 → Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 = Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶)
29	28, 17	oveq12d 6668	. . . . . 6 ⊢ (𝑥 = 𝑌 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴) = (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))
30		dvfsumlem4.g	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴))
31	22, 25, 29, 30	fvmptf 6301	. . . . 5 ⊢ ((𝑌 ∈ 𝑆 ∧ (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ∈ ℝ) → (𝐺‘𝑌) = (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))
32	1, 21, 31	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) = (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))
33		dvfsumlem4.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
34		fzfid 12772	. . . . . . 7 ⊢ (𝜑 → (𝑀...(⌊‘𝑋)) ∈ Fin)
35		elfzuz 12338	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑋)) → 𝑘 ∈ (ℤ_≥‘𝑀))
36	35, 6	syl6eleqr 2712	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑋)) → 𝑘 ∈ 𝑍)
37	4, 36, 10	syl2an 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(⌊‘𝑋))) → 𝐶 ∈ ℝ)
38	34, 37	fsumrecl 14465	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 ∈ ℝ)
39		nfcsb1v 3549	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑋 / 𝑥⦌𝐴
40	39	nfel1 2779	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑋 / 𝑥⦌𝐴 ∈ ℝ
41		csbeq1a 3542	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → 𝐴 = ⦋𝑋 / 𝑥⦌𝐴)
42	41	eleq1d 2686	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ ℝ ↔ ⦋𝑋 / 𝑥⦌𝐴 ∈ ℝ))
43	40, 42	rspc 3303	. . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → (∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ → ⦋𝑋 / 𝑥⦌𝐴 ∈ ℝ))
44	33, 14, 43	sylc 65	. . . . . 6 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌𝐴 ∈ ℝ)
45	38, 44	resubcld 10458	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) ∈ ℝ)
46		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥𝑋
47		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶
48	47, 24, 39	nfov 6676	. . . . . 6 ⊢ Ⅎ𝑥(Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)
49		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (⌊‘𝑥) = (⌊‘𝑋))
50	49	oveq2d 6666	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑀...(⌊‘𝑥)) = (𝑀...(⌊‘𝑋)))
51	50	sumeq1d 14431	. . . . . . 7 ⊢ (𝑥 = 𝑋 → Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 = Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶)
52	51, 41	oveq12d 6668	. . . . . 6 ⊢ (𝑥 = 𝑋 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴) = (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))
53	46, 48, 52, 30	fvmptf 6301	. . . . 5 ⊢ ((𝑋 ∈ 𝑆 ∧ (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) ∈ ℝ) → (𝐺‘𝑋) = (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))
54	33, 45, 53	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐺‘𝑋) = (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))
55	32, 54	oveq12d 6668	. . 3 ⊢ (𝜑 → ((𝐺‘𝑌) − (𝐺‘𝑋)) = ((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) − (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
56	55	fveq2d 6195	. 2 ⊢ (𝜑 → (abs‘((𝐺‘𝑌) − (𝐺‘𝑋))) = (abs‘((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) − (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))))
57		dvfsum.s	. . . . . . . . . . 11 ⊢ 𝑆 = (𝑇(,)+∞)
58		ioossre 12235	. . . . . . . . . . 11 ⊢ (𝑇(,)+∞) ⊆ ℝ
59	57, 58	eqsstri 3635	. . . . . . . . . 10 ⊢ 𝑆 ⊆ ℝ
60	59	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℝ)
61		dvfsum.b1	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
62		dvfsum.b3	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
63	60, 13, 61, 62	dvmptrecl 23787	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ)
64	63	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
65		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑚 𝐵 ∈ ℝ
66		nfcsb1v 3549	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑚 / 𝑥⦌𝐵
67	66	nfel1 2779	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ
68		csbeq1a 3542	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑥⦌𝐵)
69	68	eleq1d 2686	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝐵 ∈ ℝ ↔ ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ))
70	65, 67, 69	cbvral 3167	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ ↔ ∀𝑚 ∈ 𝑆 ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ)
71	64, 70	sylib 208	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ 𝑆 ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ)
72		csbeq1 3536	. . . . . . . 8 ⊢ (𝑚 = 𝑋 → ⦋𝑚 / 𝑥⦌𝐵 = ⦋𝑋 / 𝑥⦌𝐵)
73	72	eleq1d 2686	. . . . . . 7 ⊢ (𝑚 = 𝑋 → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋𝑋 / 𝑥⦌𝐵 ∈ ℝ))
74	73	rspcv 3305	. . . . . 6 ⊢ (𝑋 ∈ 𝑆 → (∀𝑚 ∈ 𝑆 ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ → ⦋𝑋 / 𝑥⦌𝐵 ∈ ℝ))
75	33, 71, 74	sylc 65	. . . . 5 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌𝐵 ∈ ℝ)
76	45, 75	resubcld 10458	. . . 4 ⊢ (𝜑 → ((Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) − ⦋𝑋 / 𝑥⦌𝐵) ∈ ℝ)
77	59, 33	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
78		reflcl 12597	. . . . . . . . 9 ⊢ (𝑋 ∈ ℝ → (⌊‘𝑋) ∈ ℝ)
79	77, 78	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⌊‘𝑋) ∈ ℝ)
80	77, 79	resubcld 10458	. . . . . . 7 ⊢ (𝜑 → (𝑋 − (⌊‘𝑋)) ∈ ℝ)
81	80, 75	remulcld 10070	. . . . . 6 ⊢ (𝜑 → ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) ∈ ℝ)
82	81, 45	readdcld 10069	. . . . 5 ⊢ (𝜑 → (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) ∈ ℝ)
83	82, 75	resubcld 10458	. . . 4 ⊢ (𝜑 → ((((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) − ⦋𝑋 / 𝑥⦌𝐵) ∈ ℝ)
84		fracge0 12605	. . . . . . . 8 ⊢ (𝑋 ∈ ℝ → 0 ≤ (𝑋 − (⌊‘𝑋)))
85	77, 84	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑋 − (⌊‘𝑋)))
86		dvfsumlem4.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≤ 𝑋)
87	77	rexrd 10089	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
88	59, 1	sseldi 3601	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℝ)
89	88	rexrd 10089	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
90		dvfsum.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℝ^*)
91		dvfsumlem4.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
92		dvfsumlem4.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ≤ 𝑈)
93	87, 89, 90, 91, 92	xrletrd 11993	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≤ 𝑈)
94	33, 86, 93	3jca 1242	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈))
95		simpr1 1067	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈)) → 𝑋 ∈ 𝑆)
96		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈))
97		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥0
98		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 ≤
99		nfcsb1v 3549	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑋 / 𝑥⦌𝐵
100	97, 98, 99	nfbr 4699	. . . . . . . . . . 11 ⊢ Ⅎ𝑥0 ≤ ⦋𝑋 / 𝑥⦌𝐵
101	96, 100	nfim 1825	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈)) → 0 ≤ ⦋𝑋 / 𝑥⦌𝐵)
102		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑆 ↔ 𝑋 ∈ 𝑆))
103		breq2 4657	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝐷 ≤ 𝑥 ↔ 𝐷 ≤ 𝑋))
104		breq1 4656	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑈 ↔ 𝑋 ≤ 𝑈))
105	102, 103, 104	3anbi123d 1399	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈) ↔ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈)))
106	105	anbi2d 740	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈)) ↔ (𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈))))
107		csbeq1a 3542	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → 𝐵 = ⦋𝑋 / 𝑥⦌𝐵)
108	107	breq2d 4665	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (0 ≤ 𝐵 ↔ 0 ≤ ⦋𝑋 / 𝑥⦌𝐵))
109	106, 108	imbi12d 334	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈)) → 0 ≤ 𝐵) ↔ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈)) → 0 ≤ ⦋𝑋 / 𝑥⦌𝐵)))
110		dvfsumlem4.0	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈)) → 0 ≤ 𝐵)
111	101, 109, 110	vtoclg1f 3265	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑆 → ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈)) → 0 ≤ ⦋𝑋 / 𝑥⦌𝐵))
112	95, 111	mpcom 38	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈)) → 0 ≤ ⦋𝑋 / 𝑥⦌𝐵)
113	94, 112	mpdan 702	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ⦋𝑋 / 𝑥⦌𝐵)
114	80, 75, 85, 113	mulge0d 10604	. . . . . 6 ⊢ (𝜑 → 0 ≤ ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵))
115	45, 81	addge02d 10616	. . . . . 6 ⊢ (𝜑 → (0 ≤ ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) ↔ (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) ≤ (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))))
116	114, 115	mpbid 222	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) ≤ (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
117	45, 82, 75, 116	lesub1dd 10643	. . . 4 ⊢ (𝜑 → ((Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) − ⦋𝑋 / 𝑥⦌𝐵))
118		reflcl 12597	. . . . . . . . . 10 ⊢ (𝑌 ∈ ℝ → (⌊‘𝑌) ∈ ℝ)
119	88, 118	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘𝑌) ∈ ℝ)
120	88, 119	resubcld 10458	. . . . . . . 8 ⊢ (𝜑 → (𝑌 − (⌊‘𝑌)) ∈ ℝ)
121		csbeq1 3536	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑌 → ⦋𝑚 / 𝑥⦌𝐵 = ⦋𝑌 / 𝑥⦌𝐵)
122	121	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑚 = 𝑌 → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋𝑌 / 𝑥⦌𝐵 ∈ ℝ))
123	122	rspcv 3305	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑆 → (∀𝑚 ∈ 𝑆 ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ → ⦋𝑌 / 𝑥⦌𝐵 ∈ ℝ))
124	1, 71, 123	sylc 65	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑌 / 𝑥⦌𝐵 ∈ ℝ)
125	120, 124	remulcld 10070	. . . . . . 7 ⊢ (𝜑 → ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ∈ ℝ)
126	125, 21	readdcld 10069	. . . . . 6 ⊢ (𝜑 → (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)) ∈ ℝ)
127	126, 124	resubcld 10458	. . . . 5 ⊢ (𝜑 → ((((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)) − ⦋𝑌 / 𝑥⦌𝐵) ∈ ℝ)
128		dvfsum.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
129		dvfsum.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
130		dvfsum.md	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1))
131		dvfsum.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℝ)
132		dvfsum.l	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵)
133		eqid 2622	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴))) = (𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))
134	57, 6, 128, 129, 130, 131, 13, 61, 3, 62, 8, 90, 132, 133, 33, 1, 86, 91, 92	dvfsumlem3 23791	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑌) ≤ ((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑋) ∧ (((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ (((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵)))
135	134	simprd 479	. . . . . 6 ⊢ (𝜑 → (((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ (((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵))
136		nfcv 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑋 − (⌊‘𝑋))
137		nfcv 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 ·
138	136, 137, 99	nfov 6676	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵)
139		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑥 +
140	138, 139, 48	nfov 6676	. . . . . . . . 9 ⊢ Ⅎ𝑥(((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))
141		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
142	141, 49	oveq12d 6668	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 − (⌊‘𝑥)) = (𝑋 − (⌊‘𝑋)))
143	142, 107	oveq12d 6668	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝑥 − (⌊‘𝑥)) · 𝐵) = ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵))
144	143, 52	oveq12d 6668	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) = (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
145	46, 140, 144, 133	fvmptf 6301	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑆 ∧ (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) ∈ ℝ) → ((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑋) = (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
146	33, 82, 145	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑋) = (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
147	146	oveq1d 6665	. . . . . 6 ⊢ (𝜑 → (((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) = ((((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) − ⦋𝑋 / 𝑥⦌𝐵))
148		nfcv 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑌 − (⌊‘𝑌))
149		nfcsb1v 3549	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑌 / 𝑥⦌𝐵
150	148, 137, 149	nfov 6676	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)
151	150, 139, 25	nfov 6676	. . . . . . . . 9 ⊢ Ⅎ𝑥(((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))
152		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → 𝑥 = 𝑌)
153	152, 26	oveq12d 6668	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → (𝑥 − (⌊‘𝑥)) = (𝑌 − (⌊‘𝑌)))
154		csbeq1a 3542	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → 𝐵 = ⦋𝑌 / 𝑥⦌𝐵)
155	153, 154	oveq12d 6668	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ((𝑥 − (⌊‘𝑥)) · 𝐵) = ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵))
156	155, 29	oveq12d 6668	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) = (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)))
157	22, 151, 156, 133	fvmptf 6301	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝑆 ∧ (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)) ∈ ℝ) → ((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑌) = (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)))
158	1, 126, 157	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑌) = (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)))
159	158	oveq1d 6665	. . . . . 6 ⊢ (𝜑 → (((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵) = ((((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)) − ⦋𝑌 / 𝑥⦌𝐵))
160	135, 147, 159	3brtr3d 4684	. . . . 5 ⊢ (𝜑 → ((((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)) − ⦋𝑌 / 𝑥⦌𝐵))
161	21	recnd 10068	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ∈ ℂ)
162	124	recnd 10068	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑌 / 𝑥⦌𝐵 ∈ ℂ)
163	125	recnd 10068	. . . . . . . 8 ⊢ (𝜑 → ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ∈ ℂ)
164	161, 162, 163	subsub3d 10422	. . . . . . 7 ⊢ (𝜑 → ((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) − (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵))) = (((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) + ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)) − ⦋𝑌 / 𝑥⦌𝐵))
165	161, 163	addcomd 10238	. . . . . . . 8 ⊢ (𝜑 → ((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) + ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)) = (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)))
166	165	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → (((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) + ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)) − ⦋𝑌 / 𝑥⦌𝐵) = ((((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)) − ⦋𝑌 / 𝑥⦌𝐵))
167	164, 166	eqtrd 2656	. . . . . 6 ⊢ (𝜑 → ((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) − (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵))) = ((((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)) − ⦋𝑌 / 𝑥⦌𝐵))
168		1red 10055	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
169	129, 77, 88, 86, 91	letrd 10194	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ≤ 𝑌)
170	1, 169, 92	3jca 1242	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈))
171		simpr1 1067	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈)) → 𝑌 ∈ 𝑆)
172		nfv 1843	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈))
173	97, 98, 149	nfbr 4699	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥0 ≤ ⦋𝑌 / 𝑥⦌𝐵
174	172, 173	nfim 1825	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈)) → 0 ≤ ⦋𝑌 / 𝑥⦌𝐵)
175		eleq1 2689	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ 𝑆 ↔ 𝑌 ∈ 𝑆))
176		breq2 4657	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑌 → (𝐷 ≤ 𝑥 ↔ 𝐷 ≤ 𝑌))
177		breq1 4656	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑌 → (𝑥 ≤ 𝑈 ↔ 𝑌 ≤ 𝑈))
178	175, 176, 177	3anbi123d 1399	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑌 → ((𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈) ↔ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈)))
179	178	anbi2d 740	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑌 → ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈)) ↔ (𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈))))
180	154	breq2d 4665	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑌 → (0 ≤ 𝐵 ↔ 0 ≤ ⦋𝑌 / 𝑥⦌𝐵))
181	179, 180	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑌 → (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈)) → 0 ≤ 𝐵) ↔ ((𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈)) → 0 ≤ ⦋𝑌 / 𝑥⦌𝐵)))
182	174, 181, 110	vtoclg1f 3265	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ 𝑆 → ((𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈)) → 0 ≤ ⦋𝑌 / 𝑥⦌𝐵))
183	171, 182	mpcom 38	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈)) → 0 ≤ ⦋𝑌 / 𝑥⦌𝐵)
184	170, 183	mpdan 702	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ ⦋𝑌 / 𝑥⦌𝐵)
185		fracle1 12604	. . . . . . . . . . 11 ⊢ (𝑌 ∈ ℝ → (𝑌 − (⌊‘𝑌)) ≤ 1)
186	88, 185	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 − (⌊‘𝑌)) ≤ 1)
187	120, 168, 124, 184, 186	lemul1ad 10963	. . . . . . . . 9 ⊢ (𝜑 → ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ≤ (1 · ⦋𝑌 / 𝑥⦌𝐵))
188	162	mulid2d 10058	. . . . . . . . 9 ⊢ (𝜑 → (1 · ⦋𝑌 / 𝑥⦌𝐵) = ⦋𝑌 / 𝑥⦌𝐵)
189	187, 188	breqtrd 4679	. . . . . . . 8 ⊢ (𝜑 → ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ≤ ⦋𝑌 / 𝑥⦌𝐵)
190	124, 125	subge0d 10617	. . . . . . . 8 ⊢ (𝜑 → (0 ≤ (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)) ↔ ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ≤ ⦋𝑌 / 𝑥⦌𝐵))
191	189, 190	mpbird 247	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)))
192	124, 125	resubcld 10458	. . . . . . . 8 ⊢ (𝜑 → (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)) ∈ ℝ)
193	21, 192	subge02d 10619	. . . . . . 7 ⊢ (𝜑 → (0 ≤ (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)) ↔ ((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) − (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵))) ≤ (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)))
194	191, 193	mpbid 222	. . . . . 6 ⊢ (𝜑 → ((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) − (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵))) ≤ (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))
195	167, 194	eqbrtrrd 4677	. . . . 5 ⊢ (𝜑 → ((((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)) − ⦋𝑌 / 𝑥⦌𝐵) ≤ (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))
196	83, 127, 21, 160, 195	letrd 10194	. . . 4 ⊢ (𝜑 → ((((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) − ⦋𝑋 / 𝑥⦌𝐵) ≤ (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))
197	76, 83, 21, 117, 196	letrd 10194	. . 3 ⊢ (𝜑 → ((Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) − ⦋𝑋 / 𝑥⦌𝐵) ≤ (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))
198	75, 45	readdcld 10069	. . . . 5 ⊢ (𝜑 → (⦋𝑋 / 𝑥⦌𝐵 + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) ∈ ℝ)
199		fracge0 12605	. . . . . . . . 9 ⊢ (𝑌 ∈ ℝ → 0 ≤ (𝑌 − (⌊‘𝑌)))
200	88, 199	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝑌 − (⌊‘𝑌)))
201	120, 124, 200, 184	mulge0d 10604	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵))
202	21, 125	addge02d 10616	. . . . . . 7 ⊢ (𝜑 → (0 ≤ ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ↔ (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ≤ (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))))
203	201, 202	mpbid 222	. . . . . 6 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ≤ (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)))
204	134	simpld 475	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑌) ≤ ((𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))‘𝑋))
205	204, 158, 146	3brtr3d 4684	. . . . . 6 ⊢ (𝜑 → (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)) ≤ (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
206	21, 126, 82, 203, 205	letrd 10194	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ≤ (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
207		fracle1 12604	. . . . . . . . 9 ⊢ (𝑋 ∈ ℝ → (𝑋 − (⌊‘𝑋)) ≤ 1)
208	77, 207	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑋 − (⌊‘𝑋)) ≤ 1)
209	80, 168, 75, 113, 208	lemul1ad 10963	. . . . . . 7 ⊢ (𝜑 → ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) ≤ (1 · ⦋𝑋 / 𝑥⦌𝐵))
210	75	recnd 10068	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌𝐵 ∈ ℂ)
211	210	mulid2d 10058	. . . . . . 7 ⊢ (𝜑 → (1 · ⦋𝑋 / 𝑥⦌𝐵) = ⦋𝑋 / 𝑥⦌𝐵)
212	209, 211	breqtrd 4679	. . . . . 6 ⊢ (𝜑 → ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) ≤ ⦋𝑋 / 𝑥⦌𝐵)
213	81, 75, 45, 212	leadd1dd 10641	. . . . 5 ⊢ (𝜑 → (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) ≤ (⦋𝑋 / 𝑥⦌𝐵 + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
214	21, 82, 198, 206, 213	letrd 10194	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ≤ (⦋𝑋 / 𝑥⦌𝐵 + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
215	45	recnd 10068	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) ∈ ℂ)
216	210, 215	addcomd 10238	. . . 4 ⊢ (𝜑 → (⦋𝑋 / 𝑥⦌𝐵 + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) = ((Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) + ⦋𝑋 / 𝑥⦌𝐵))
217	214, 216	breqtrd 4679	. . 3 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ≤ ((Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) + ⦋𝑋 / 𝑥⦌𝐵))
218	21, 45, 75	absdifled 14173	. . 3 ⊢ (𝜑 → ((abs‘((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) − (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))) ≤ ⦋𝑋 / 𝑥⦌𝐵 ↔ (((Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) − ⦋𝑋 / 𝑥⦌𝐵) ≤ (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ∧ (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ≤ ((Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) + ⦋𝑋 / 𝑥⦌𝐵))))
219	197, 217, 218	mpbir2and 957	. 2 ⊢ (𝜑 → (abs‘((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) − (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))) ≤ ⦋𝑋 / 𝑥⦌𝐵)
220	56, 219	eqbrtrd 4675	1 ⊢ (𝜑 → (abs‘((𝐺‘𝑌) − (𝐺‘𝑋))) ≤ ⦋𝑋 / 𝑥⦌𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⦋csb 3533 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 ℝ^*cxr 10073 ≤ cle 10075 − cmin 10266 ℤcz 11377 ℤ_≥cuz 11687 (,)cioo 12175 ...cfz 12326 ⌊cfl 12591 abscabs 13974 Σcsu 14416 D cdv 23627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631

This theorem is referenced by: dvfsumrlim 23794 dvfsumrlim2 23795 logexprlim 24950

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