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Theorem dvfsumlem4 25953

Description: Lemma for dvfsumrlim 25955. (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 13899	. . . . . . 7 ⊢ (𝜑 → (𝑀...(⌊‘𝑌)) ∈ Fin)
3		dvfsum.b2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	ralrimiva 3121	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 𝐵 ∈ ℝ)
5		elfzuz 13442	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑌)) → 𝑘 ∈ (ℤ_≥‘𝑀))
6		dvfsum.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
7	5, 6	eleqtrrdi 2839	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑌)) → 𝑘 ∈ 𝑍)
8		dvfsum.c	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶)
9	8	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ))
10	9	rspccva 3578	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝑍 𝐵 ∈ ℝ ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℝ)
11	4, 7, 10	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(⌊‘𝑌))) → 𝐶 ∈ ℝ)
12	2, 11	fsumrecl 15660	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 ∈ ℝ)
13		dvfsum.a	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
14	13	ralrimiva 3121	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ)
15		nfcsb1v 3877	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑌 / 𝑥⦌𝐴
16	15	nfel1 2908	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑌 / 𝑥⦌𝐴 ∈ ℝ
17		csbeq1a 3867	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → 𝐴 = ⦋𝑌 / 𝑥⦌𝐴)
18	17	eleq1d 2813	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝐴 ∈ ℝ ↔ ⦋𝑌 / 𝑥⦌𝐴 ∈ ℝ))
19	16, 18	rspc 3567	. . . . . . 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 11567	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ∈ ℝ)
22		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥𝑌
23		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶
24		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥 −
25	23, 24, 15	nfov 7383	. . . . . 6 ⊢ Ⅎ𝑥(Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)
26		fveq2 6826	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → (⌊‘𝑥) = (⌊‘𝑌))
27	26	oveq2d 7369	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝑀...(⌊‘𝑥)) = (𝑀...(⌊‘𝑌)))
28	27	sumeq1d 15626	. . . . . . 7 ⊢ (𝑥 = 𝑌 → Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 = Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶)
29	28, 17	oveq12d 7371	. . . . . 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 6955	. . . . 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 584	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) = (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))
33		dvfsumlem4.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
34		fzfid 13899	. . . . . . 7 ⊢ (𝜑 → (𝑀...(⌊‘𝑋)) ∈ Fin)
35		elfzuz 13442	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑋)) → 𝑘 ∈ (ℤ_≥‘𝑀))
36	35, 6	eleqtrrdi 2839	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑋)) → 𝑘 ∈ 𝑍)
37	4, 36, 10	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(⌊‘𝑋))) → 𝐶 ∈ ℝ)
38	34, 37	fsumrecl 15660	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 ∈ ℝ)
39		nfcsb1v 3877	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑋 / 𝑥⦌𝐴
40	39	nfel1 2908	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑋 / 𝑥⦌𝐴 ∈ ℝ
41		csbeq1a 3867	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → 𝐴 = ⦋𝑋 / 𝑥⦌𝐴)
42	41	eleq1d 2813	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ ℝ ↔ ⦋𝑋 / 𝑥⦌𝐴 ∈ ℝ))
43	40, 42	rspc 3567	. . . . . . 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 11567	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) ∈ ℝ)
46		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥𝑋
47		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶
48	47, 24, 39	nfov 7383	. . . . . 6 ⊢ Ⅎ𝑥(Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)
49		fveq2 6826	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (⌊‘𝑥) = (⌊‘𝑋))
50	49	oveq2d 7369	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑀...(⌊‘𝑥)) = (𝑀...(⌊‘𝑋)))
51	50	sumeq1d 15626	. . . . . . 7 ⊢ (𝑥 = 𝑋 → Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 = Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶)
52	51, 41	oveq12d 7371	. . . . . 6 ⊢ (𝑥 = 𝑋 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴) = (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))
53	46, 48, 52, 30	fvmptf 6955	. . . . 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 584	. . . 4 ⊢ (𝜑 → (𝐺‘𝑋) = (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))
55	32, 54	oveq12d 7371	. . 3 ⊢ (𝜑 → ((𝐺‘𝑌) − (𝐺‘𝑋)) = ((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) − (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
56	55	fveq2d 6830	. 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 13329	. . . . . . . . . . 11 ⊢ (𝑇(,)+∞) ⊆ ℝ
59	57, 58	eqsstri 3984	. . . . . . . . . 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 25947	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ)
64	63	ralrimiva 3121	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
65		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑚 𝐵 ∈ ℝ
66		nfcsb1v 3877	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑚 / 𝑥⦌𝐵
67	66	nfel1 2908	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ
68		csbeq1a 3867	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑥⦌𝐵)
69	68	eleq1d 2813	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝐵 ∈ ℝ ↔ ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ))
70	65, 67, 69	cbvralw 3272	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ ↔ ∀𝑚 ∈ 𝑆 ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ)
71	64, 70	sylib 218	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ 𝑆 ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ)
72		csbeq1 3856	. . . . . . . 8 ⊢ (𝑚 = 𝑋 → ⦋𝑚 / 𝑥⦌𝐵 = ⦋𝑋 / 𝑥⦌𝐵)
73	72	eleq1d 2813	. . . . . . 7 ⊢ (𝑚 = 𝑋 → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋𝑋 / 𝑥⦌𝐵 ∈ ℝ))
74	73	rspcv 3575	. . . . . 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 11567	. . . 4 ⊢ (𝜑 → ((Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) − ⦋𝑋 / 𝑥⦌𝐵) ∈ ℝ)
77	59, 33	sselid 3935	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
78		reflcl 13719	. . . . . . . . 9 ⊢ (𝑋 ∈ ℝ → (⌊‘𝑋) ∈ ℝ)
79	77, 78	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⌊‘𝑋) ∈ ℝ)
80	77, 79	resubcld 11567	. . . . . . 7 ⊢ (𝜑 → (𝑋 − (⌊‘𝑋)) ∈ ℝ)
81	80, 75	remulcld 11164	. . . . . 6 ⊢ (𝜑 → ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) ∈ ℝ)
82	81, 45	readdcld 11163	. . . . 5 ⊢ (𝜑 → (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) ∈ ℝ)
83	82, 75	resubcld 11567	. . . 4 ⊢ (𝜑 → ((((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) − ⦋𝑋 / 𝑥⦌𝐵) ∈ ℝ)
84		fracge0 13727	. . . . . . . 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 11184	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
88	59, 1	sselid 3935	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℝ)
89	88	rexrd 11184	. . . . . . . . . 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 13083	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≤ 𝑈)
94	33, 86, 93	3jca 1128	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈))
95		simpr1 1195	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈)) → 𝑋 ∈ 𝑆)
96		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈))
97		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥0
98		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 ≤
99		nfcsb1v 3877	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑋 / 𝑥⦌𝐵
100	97, 98, 99	nfbr 5142	. . . . . . . . . . 11 ⊢ Ⅎ𝑥0 ≤ ⦋𝑋 / 𝑥⦌𝐵
101	96, 100	nfim 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈)) → 0 ≤ ⦋𝑋 / 𝑥⦌𝐵)
102		eleq1 2816	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑆 ↔ 𝑋 ∈ 𝑆))
103		breq2 5099	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝐷 ≤ 𝑥 ↔ 𝐷 ≤ 𝑋))
104		breq1 5098	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑈 ↔ 𝑋 ≤ 𝑈))
105	102, 103, 104	3anbi123d 1438	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈) ↔ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈)))
106	105	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈)) ↔ (𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋 ∧ 𝑋 ≤ 𝑈))))
107		csbeq1a 3867	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → 𝐵 = ⦋𝑋 / 𝑥⦌𝐵)
108	107	breq2d 5107	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (0 ≤ 𝐵 ↔ 0 ≤ ⦋𝑋 / 𝑥⦌𝐵))
109	106, 108	imbi12d 344	. . . . . . . . . 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 3527	. . . . . . . . 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 687	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ⦋𝑋 / 𝑥⦌𝐵)
114	80, 75, 85, 113	mulge0d 11716	. . . . . 6 ⊢ (𝜑 → 0 ≤ ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵))
115	45, 81	addge02d 11728	. . . . . 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 232	. . . . 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 11755	. . . 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 13719	. . . . . . . . . 10 ⊢ (𝑌 ∈ ℝ → (⌊‘𝑌) ∈ ℝ)
119	88, 118	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘𝑌) ∈ ℝ)
120	88, 119	resubcld 11567	. . . . . . . 8 ⊢ (𝜑 → (𝑌 − (⌊‘𝑌)) ∈ ℝ)
121		csbeq1 3856	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑌 → ⦋𝑚 / 𝑥⦌𝐵 = ⦋𝑌 / 𝑥⦌𝐵)
122	121	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑚 = 𝑌 → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋𝑌 / 𝑥⦌𝐵 ∈ ℝ))
123	122	rspcv 3575	. . . . . . . . 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 11164	. . . . . . 7 ⊢ (𝜑 → ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ∈ ℝ)
126	125, 21	readdcld 11163	. . . . . 6 ⊢ (𝜑 → (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)) ∈ ℝ)
127	126, 124	resubcld 11567	. . . . 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 2729	. . . . . . . 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 25952	. . . . . . 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 495	. . . . . 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 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑋 − (⌊‘𝑋))
137		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 ·
138	136, 137, 99	nfov 7383	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵)
139		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥 +
140	138, 139, 48	nfov 7383	. . . . . . . . 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 7371	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 − (⌊‘𝑥)) = (𝑋 − (⌊‘𝑋)))
143	142, 107	oveq12d 7371	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝑥 − (⌊‘𝑥)) · 𝐵) = ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵))
144	143, 52	oveq12d 7371	. . . . . . . . 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 6955	. . . . . . . 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 584	. . . . . . 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 7368	. . . . . 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 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑌 − (⌊‘𝑌))
149		nfcsb1v 3877	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑌 / 𝑥⦌𝐵
150	148, 137, 149	nfov 7383	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)
151	150, 139, 25	nfov 7383	. . . . . . . . 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 7371	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → (𝑥 − (⌊‘𝑥)) = (𝑌 − (⌊‘𝑌)))
154		csbeq1a 3867	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑌 → 𝐵 = ⦋𝑌 / 𝑥⦌𝐵)
155	153, 154	oveq12d 7371	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → ((𝑥 − (⌊‘𝑥)) · 𝐵) = ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵))
156	155, 29	oveq12d 7371	. . . . . . . . 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 6955	. . . . . . . 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 584	. . . . . . 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 7368	. . . . . 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 5126	. . . . 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 11162	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ∈ ℂ)
162	124	recnd 11162	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑌 / 𝑥⦌𝐵 ∈ ℂ)
163	125	recnd 11162	. . . . . . . 8 ⊢ (𝜑 → ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ∈ ℂ)
164	161, 162, 163	subsub3d 11524	. . . . . . 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 11337	. . . . . . . 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 7368	. . . . . . 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 2764	. . . . . 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 11135	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
169	129, 77, 88, 86, 91	letrd 11292	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ≤ 𝑌)
170	1, 169, 92	3jca 1128	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈))
171		simpr1 1195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈)) → 𝑌 ∈ 𝑆)
172		nfv 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈))
173	97, 98, 149	nfbr 5142	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥0 ≤ ⦋𝑌 / 𝑥⦌𝐵
174	172, 173	nfim 1896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈)) → 0 ≤ ⦋𝑌 / 𝑥⦌𝐵)
175		eleq1 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ 𝑆 ↔ 𝑌 ∈ 𝑆))
176		breq2 5099	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑌 → (𝐷 ≤ 𝑥 ↔ 𝐷 ≤ 𝑌))
177		breq1 5098	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑌 → (𝑥 ≤ 𝑈 ↔ 𝑌 ≤ 𝑈))
178	175, 176, 177	3anbi123d 1438	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑌 → ((𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈) ↔ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈)))
179	178	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑌 → ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈)) ↔ (𝜑 ∧ (𝑌 ∈ 𝑆 ∧ 𝐷 ≤ 𝑌 ∧ 𝑌 ≤ 𝑈))))
180	154	breq2d 5107	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑌 → (0 ≤ 𝐵 ↔ 0 ≤ ⦋𝑌 / 𝑥⦌𝐵))
181	179, 180	imbi12d 344	. . . . . . . . . . . . 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 3527	. . . . . . . . . . . 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 687	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ ⦋𝑌 / 𝑥⦌𝐵)
185		fracle1 13726	. . . . . . . . . . 11 ⊢ (𝑌 ∈ ℝ → (𝑌 − (⌊‘𝑌)) ≤ 1)
186	88, 185	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 − (⌊‘𝑌)) ≤ 1)
187	120, 168, 124, 184, 186	lemul1ad 12083	. . . . . . . . 9 ⊢ (𝜑 → ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ≤ (1 · ⦋𝑌 / 𝑥⦌𝐵))
188	162	mullidd 11152	. . . . . . . . 9 ⊢ (𝜑 → (1 · ⦋𝑌 / 𝑥⦌𝐵) = ⦋𝑌 / 𝑥⦌𝐵)
189	187, 188	breqtrd 5121	. . . . . . . 8 ⊢ (𝜑 → ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ≤ ⦋𝑌 / 𝑥⦌𝐵)
190	124, 125	subge0d 11729	. . . . . . . 8 ⊢ (𝜑 → (0 ≤ (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)) ↔ ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) ≤ ⦋𝑌 / 𝑥⦌𝐵))
191	189, 190	mpbird 257	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)))
192	124, 125	resubcld 11567	. . . . . . . 8 ⊢ (𝜑 → (⦋𝑌 / 𝑥⦌𝐵 − ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵)) ∈ ℝ)
193	21, 192	subge02d 11731	. . . . . . 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 232	. . . . . 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 5119	. . . . 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 11292	. . . 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 11292	. . 3 ⊢ (𝜑 → ((Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) − ⦋𝑋 / 𝑥⦌𝐵) ≤ (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴))
198	75, 45	readdcld 11163	. . . . 5 ⊢ (𝜑 → (⦋𝑋 / 𝑥⦌𝐵 + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)) ∈ ℝ)
199		fracge0 13727	. . . . . . . . 9 ⊢ (𝑌 ∈ ℝ → 0 ≤ (𝑌 − (⌊‘𝑌)))
200	88, 199	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝑌 − (⌊‘𝑌)))
201	120, 124, 200, 184	mulge0d 11716	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵))
202	21, 125	addge02d 11728	. . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ≤ (((𝑌 − (⌊‘𝑌)) · ⦋𝑌 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴)))
204	134	simpld 494	. . . . . . 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 5126	. . . . . 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 11292	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ≤ (((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
207		fracle1 13726	. . . . . . . . 9 ⊢ (𝑋 ∈ ℝ → (𝑋 − (⌊‘𝑋)) ≤ 1)
208	77, 207	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑋 − (⌊‘𝑋)) ≤ 1)
209	80, 168, 75, 113, 208	lemul1ad 12083	. . . . . . 7 ⊢ (𝜑 → ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) ≤ (1 · ⦋𝑋 / 𝑥⦌𝐵))
210	75	recnd 11162	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌𝐵 ∈ ℂ)
211	210	mullidd 11152	. . . . . . 7 ⊢ (𝜑 → (1 · ⦋𝑋 / 𝑥⦌𝐵) = ⦋𝑋 / 𝑥⦌𝐵)
212	209, 211	breqtrd 5121	. . . . . 6 ⊢ (𝜑 → ((𝑋 − (⌊‘𝑋)) · ⦋𝑋 / 𝑥⦌𝐵) ≤ ⦋𝑋 / 𝑥⦌𝐵)
213	81, 75, 45, 212	leadd1dd 11753	. . . . 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 11292	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ≤ (⦋𝑋 / 𝑥⦌𝐵 + (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴)))
215	45	recnd 11162	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) ∈ ℂ)
216	210, 215	addcomd 11337	. . . 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 5121	. . 3 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) ≤ ((Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴) + ⦋𝑋 / 𝑥⦌𝐵))
218	21, 45, 75	absdifled 15363	. . 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 713	. 2 ⊢ (𝜑 → (abs‘((Σ𝑘 ∈ (𝑀...(⌊‘𝑌))𝐶 − ⦋𝑌 / 𝑥⦌𝐴) − (Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶 − ⦋𝑋 / 𝑥⦌𝐴))) ≤ ⦋𝑋 / 𝑥⦌𝐵)
220	56, 219	eqbrtrd 5117	1 ⊢ (𝜑 → (abs‘((𝐺‘𝑌) − (𝐺‘𝑋))) ≤ ⦋𝑋 / 𝑥⦌𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⦋csb 3853 ⊆ wss 3905 class class class wbr 5095 ↦ cmpt 5176 ‘cfv 6486 (class class class)co 7353 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 +∞cpnf 11165 ℝ^*cxr 11167 ≤ cle 11169 − cmin 11366 ℤcz 12490 ℤ_≥cuz 12754 (,)cioo 13267 ...cfz 13429 ⌊cfl 13713 abscabs 15160 Σcsu 15612 D cdv 25781

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12611 df-uz 12755 df-q 12869 df-rp 12913 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13271 df-ico 13273 df-icc 13274 df-fz 13430 df-fzo 13577 df-fl 13715 df-seq 13928 df-exp 13988 df-hash 14257 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-clim 15414 df-sum 15613 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-starv 17195 df-sca 17196 df-vsca 17197 df-ip 17198 df-tset 17199 df-ple 17200 df-ds 17202 df-unif 17203 df-hom 17204 df-cco 17205 df-rest 17345 df-topn 17346 df-0g 17364 df-gsum 17365 df-topgen 17366 df-pt 17367 df-prds 17370 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-submnd 18677 df-mulg 18966 df-cntz 19215 df-cmn 19680 df-psmet 21272 df-xmet 21273 df-met 21274 df-bl 21275 df-mopn 21276 df-fbas 21277 df-fg 21278 df-cnfld 21281 df-top 22798 df-topon 22815 df-topsp 22837 df-bases 22850 df-cld 22923 df-ntr 22924 df-cls 22925 df-nei 23002 df-lp 23040 df-perf 23041 df-cn 23131 df-cnp 23132 df-haus 23219 df-cmp 23291 df-tx 23466 df-hmeo 23659 df-fil 23750 df-fm 23842 df-flim 23843 df-flf 23844 df-xms 24225 df-ms 24226 df-tms 24227 df-cncf 24788 df-limc 25784 df-dv 25785

This theorem is referenced by: dvfsumrlim 25955 dvfsumrlim2 25956 logexprlim 27153

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