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Theorem dvfsumlem3 26255

Description: Lemma for dvfsumrlim 26258. (Contributed by Mario Carneiro, 17-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)
dvfsum.h	⊢ 𝐻 = (𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))
dvfsumlem1.1	⊢ (𝜑 → 𝑋 ∈ 𝑆)
dvfsumlem1.2	⊢ (𝜑 → 𝑌 ∈ 𝑆)
dvfsumlem1.3	⊢ (𝜑 → 𝐷 ≤ 𝑋)
dvfsumlem1.4	⊢ (𝜑 → 𝑋 ≤ 𝑌)
dvfsumlem1.5	⊢ (𝜑 → 𝑌 ≤ 𝑈)

**Assertion**
Ref	Expression
dvfsumlem3	⊢ (𝜑 → ((𝐻‘𝑌) ≤ (𝐻‘𝑋) ∧ ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 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) 𝐻(𝑥, 𝑘) 𝑉(𝑥, 𝑘) 𝑍(𝑘)

Proof of Theorem dvfsumlem3 Dummy variables 𝑦 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvfsum.s	. . . 4 ⊢ 𝑆 = (𝑇(,)+∞)
2		ioossre 13460	. . . 4 ⊢ (𝑇(,)+∞) ⊆ ℝ
3	1, 2	eqsstri 3977	. . 3 ⊢ 𝑆 ⊆ ℝ
4		dvfsumlem1.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
5	3, 4	sselid 3929	. 2 ⊢ (𝜑 → 𝑌 ∈ ℝ)
6		dvfsumlem1.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
7	3, 6	sselid 3929	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8		reflcl 13857	. . 3 ⊢ (𝑋 ∈ ℝ → (⌊‘𝑋) ∈ ℝ)
9		peano2re 11407	. . 3 ⊢ ((⌊‘𝑋) ∈ ℝ → ((⌊‘𝑋) + 1) ∈ ℝ)
10	7, 8, 9	3syl 19	. 2 ⊢ (𝜑 → ((⌊‘𝑋) + 1) ∈ ℝ)
11		dvfsum.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑀)
12		dvfsum.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑀 ∈ ℤ)
14		dvfsum.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
15	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝐷 ∈ ℝ)
16		dvfsum.md	. . . 4 ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1))
17	16	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑀 ≤ (𝐷 + 1))
18		dvfsum.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ)
19	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑇 ∈ ℝ)
20		dvfsum.a	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
21	20	adantlr 728	. . 3 ⊢ (((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
22		dvfsum.b1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
23	22	adantlr 728	. . 3 ⊢ (((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
24		dvfsum.b2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
25	24	adantlr 728	. . 3 ⊢ (((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
26		dvfsum.b3	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
27	26	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
28		dvfsum.c	. . 3 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶)
29		dvfsum.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ℝ^*)
30	29	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑈 ∈ ℝ^*)
31		dvfsum.l	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵)
32	31	3adant1r 1196	. . 3 ⊢ (((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵)
33		dvfsum.h	. . 3 ⊢ 𝐻 = (𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))
34	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑋 ∈ 𝑆)
35	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑌 ∈ 𝑆)
36		dvfsumlem1.3	. . . 4 ⊢ (𝜑 → 𝐷 ≤ 𝑋)
37	36	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝐷 ≤ 𝑋)
38		dvfsumlem1.4	. . . 4 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
39	38	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑋 ≤ 𝑌)
40		dvfsumlem1.5	. . . 4 ⊢ (𝜑 → 𝑌 ≤ 𝑈)
41	40	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑌 ≤ 𝑈)
42		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑌 ≤ ((⌊‘𝑋) + 1))
43	1, 11, 13, 15, 17, 19, 21, 23, 25, 27, 28, 30, 32, 33, 34, 35, 37, 39, 41, 42	dvfsumlem2 26254	. 2 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → ((𝐻‘𝑌) ≤ (𝐻‘𝑋) ∧ ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵)))
44	3	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ ℝ)
45	44	sselda 3931	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℝ)
46		reflcl 13857	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
47	45, 46	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (⌊‘𝑥) ∈ ℝ)
48	45, 47	resubcld 11666	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 − (⌊‘𝑥)) ∈ ℝ)
49	44, 20, 22, 26	dvmptrecl 26251	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ)
50	48, 49	remulcld 11263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑥 − (⌊‘𝑥)) · 𝐵) ∈ ℝ)
51		fzfid 14037	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑀...(⌊‘𝑥)) ∈ Fin)
52	24	ralrimiva 3154	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 𝐵 ∈ ℝ)
53	52	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∀𝑥 ∈ 𝑍 𝐵 ∈ ℝ)
54		elfzuz 13574	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑥)) → 𝑘 ∈ (ℤ_≥‘𝑀))
55	54, 11	eleqtrrdi 2871	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑥)) → 𝑘 ∈ 𝑍)
56	28	eleq1d 2845	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ))
57	56	rspccva 3575	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝑍 𝐵 ∈ ℝ ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℝ)
58	53, 55, 57	syl2an 608	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑘 ∈ (𝑀...(⌊‘𝑥))) → 𝐶 ∈ ℝ)
59	51, 58	fsumrecl 15820	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 ∈ ℝ)
60	59, 20	resubcld 11666	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴) ∈ ℝ)
61	50, 60	readdcld 11262	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) ∈ ℝ)
62	61, 33	fmptd 7107	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑆⟶ℝ)
63	62	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝐻:𝑆⟶ℝ)
64	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ∈ 𝑆)
65	63, 64	ffvelcdmd 7078	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘𝑌) ∈ ℝ)
66	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ∈ ℝ)
67		reflcl 13857	. . . . . . . 8 ⊢ (𝑌 ∈ ℝ → (⌊‘𝑌) ∈ ℝ)
68	66, 67	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ ℝ)
69	18	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑇 ∈ ℝ)
70	7	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑋 ∈ ℝ)
71	70, 8, 9	3syl 19	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ∈ ℝ)
72	6, 1	eleqtrdi 2870	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (𝑇(,)+∞))
73	18	rexrd 11283	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ℝ^*)
74		elioopnf 13496	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ ℝ^* → (𝑋 ∈ (𝑇(,)+∞) ↔ (𝑋 ∈ ℝ ∧ 𝑇 < 𝑋)))
75	73, 74	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ (𝑇(,)+∞) ↔ (𝑋 ∈ ℝ ∧ 𝑇 < 𝑋)))
76	72, 75	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ℝ ∧ 𝑇 < 𝑋))
77	76	simprd 501	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 < 𝑋)
78		fllep1 13862	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 𝑋 ≤ ((⌊‘𝑋) + 1))
79	7, 78	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ ((⌊‘𝑋) + 1))
80	18, 7, 10, 77, 79	ltletrd 11394	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 < ((⌊‘𝑋) + 1))
81	80	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑇 < ((⌊‘𝑋) + 1))
82		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ≤ 𝑌)
83	70	flcld 13859	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑋) ∈ ℤ)
84	83	peano2zd 12728	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ∈ ℤ)
85		flge 13866	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ℝ ∧ ((⌊‘𝑋) + 1) ∈ ℤ) → (((⌊‘𝑋) + 1) ≤ 𝑌 ↔ ((⌊‘𝑋) + 1) ≤ (⌊‘𝑌)))
86	66, 84, 85	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (((⌊‘𝑋) + 1) ≤ 𝑌 ↔ ((⌊‘𝑋) + 1) ≤ (⌊‘𝑌)))
87	82, 86	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ≤ (⌊‘𝑌))
88	69, 71, 68, 81, 87	ltletrd 11394	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑇 < (⌊‘𝑌))
89	73	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑇 ∈ ℝ^*)
90		elioopnf 13496	. . . . . . . 8 ⊢ (𝑇 ∈ ℝ^* → ((⌊‘𝑌) ∈ (𝑇(,)+∞) ↔ ((⌊‘𝑌) ∈ ℝ ∧ 𝑇 < (⌊‘𝑌))))
91	89, 90	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑌) ∈ (𝑇(,)+∞) ↔ ((⌊‘𝑌) ∈ ℝ ∧ 𝑇 < (⌊‘𝑌))))
92	68, 88, 91	mpbir2and 726	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ (𝑇(,)+∞))
93	92, 1	eleqtrrdi 2871	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ 𝑆)
94	63, 93	ffvelcdmd 7078	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘(⌊‘𝑌)) ∈ ℝ)
95	6	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑋 ∈ 𝑆)
96	63, 95	ffvelcdmd 7078	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘𝑋) ∈ ℝ)
97	12	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑀 ∈ ℤ)
98	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝐷 ∈ ℝ)
99	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑀 ≤ (𝐷 + 1))
100	20	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
101	22	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
102	24	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
103	26	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
104	29	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑈 ∈ ℝ^*)
105	31	3adant1r 1196	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵)
106	36	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝐷 ≤ 𝑋)
107	70, 78	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑋 ≤ ((⌊‘𝑋) + 1))
108	98, 70, 71, 106, 107	letrd 11391	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝐷 ≤ ((⌊‘𝑋) + 1))
109	98, 71, 68, 108, 87	letrd 11391	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝐷 ≤ (⌊‘𝑌))
110		flle 13860	. . . . . . 7 ⊢ (𝑌 ∈ ℝ → (⌊‘𝑌) ≤ 𝑌)
111	66, 110	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ≤ 𝑌)
112	40	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ≤ 𝑈)
113		fllep1 13862	. . . . . . . 8 ⊢ (𝑌 ∈ ℝ → 𝑌 ≤ ((⌊‘𝑌) + 1))
114	66, 113	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ≤ ((⌊‘𝑌) + 1))
115		flidm 13870	. . . . . . . . 9 ⊢ (𝑌 ∈ ℝ → (⌊‘(⌊‘𝑌)) = (⌊‘𝑌))
116	66, 115	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘(⌊‘𝑌)) = (⌊‘𝑌))
117	116	oveq1d 7428	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘(⌊‘𝑌)) + 1) = ((⌊‘𝑌) + 1))
118	114, 117	breqtrrd 5133	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ≤ ((⌊‘(⌊‘𝑌)) + 1))
119	1, 11, 97, 98, 99, 69, 100, 101, 102, 103, 28, 104, 105, 33, 93, 64, 109, 111, 112, 118	dvfsumlem2 26254	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑌) ≤ (𝐻‘(⌊‘𝑌)) ∧ ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵)))
120	119	simpld 500	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘𝑌) ≤ (𝐻‘(⌊‘𝑌)))
121		elioopnf 13496	. . . . . . . . . 10 ⊢ (𝑇 ∈ ℝ^* → (((⌊‘𝑋) + 1) ∈ (𝑇(,)+∞) ↔ (((⌊‘𝑋) + 1) ∈ ℝ ∧ 𝑇 < ((⌊‘𝑋) + 1))))
122	73, 121	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘𝑋) + 1) ∈ (𝑇(,)+∞) ↔ (((⌊‘𝑋) + 1) ∈ ℝ ∧ 𝑇 < ((⌊‘𝑋) + 1))))
123	10, 80, 122	mpbir2and 726	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘𝑋) + 1) ∈ (𝑇(,)+∞))
124	123, 1	eleqtrrdi 2871	. . . . . . 7 ⊢ (𝜑 → ((⌊‘𝑋) + 1) ∈ 𝑆)
125	124	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ∈ 𝑆)
126	63, 125	ffvelcdmd 7078	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘((⌊‘𝑋) + 1)) ∈ ℝ)
127	66	flcld 13859	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ ℤ)
128		eluz2 12893	. . . . . . 7 ⊢ ((⌊‘𝑌) ∈ (ℤ_≥‘((⌊‘𝑋) + 1)) ↔ (((⌊‘𝑋) + 1) ∈ ℤ ∧ (⌊‘𝑌) ∈ ℤ ∧ ((⌊‘𝑋) + 1) ≤ (⌊‘𝑌)))
129	84, 127, 87, 128	syl3anbrc 1362	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ (ℤ_≥‘((⌊‘𝑋) + 1)))
130	63	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝐻:𝑆⟶ℝ)
131		elfzelz 13578	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌)) → 𝑚 ∈ ℤ)
132	131	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑚 ∈ ℤ)
133	132	zred 12725	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑚 ∈ ℝ)
134	69	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑇 ∈ ℝ)
135	71	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → ((⌊‘𝑋) + 1) ∈ ℝ)
136	80	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑇 < ((⌊‘𝑋) + 1))
137		elfzle1 13581	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌)) → ((⌊‘𝑋) + 1) ≤ 𝑚)
138	137	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → ((⌊‘𝑋) + 1) ≤ 𝑚)
139	134, 135, 133, 136, 138	ltletrd 11394	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑇 < 𝑚)
140	73	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑇 ∈ ℝ^*)
141		elioopnf 13496	. . . . . . . . . 10 ⊢ (𝑇 ∈ ℝ^* → (𝑚 ∈ (𝑇(,)+∞) ↔ (𝑚 ∈ ℝ ∧ 𝑇 < 𝑚)))
142	140, 141	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → (𝑚 ∈ (𝑇(,)+∞) ↔ (𝑚 ∈ ℝ ∧ 𝑇 < 𝑚)))
143	133, 139, 142	mpbir2and 726	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑚 ∈ (𝑇(,)+∞))
144	143, 1	eleqtrrdi 2871	. . . . . . 7 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑚 ∈ 𝑆)
145	130, 144	ffvelcdmd 7078	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → (𝐻‘𝑚) ∈ ℝ)
146	97	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑀 ∈ ℤ)
147	98	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝐷 ∈ ℝ)
148	16	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑀 ≤ (𝐷 + 1))
149	69	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑇 ∈ ℝ)
150	100	adantlr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
151	101	adantlr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
152	102	adantlr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
153	103	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
154	104	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑈 ∈ ℝ^*)
155	105	3adant1r 1196	. . . . . . . 8 ⊢ ((((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵)
156		elfzelz 13578	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1)) → 𝑚 ∈ ℤ)
157	156	adantl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ∈ ℤ)
158	157	zred 12725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ∈ ℝ)
159	71	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((⌊‘𝑋) + 1) ∈ ℝ)
160	80	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑇 < ((⌊‘𝑋) + 1))
161		elfzle1 13581	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1)) → ((⌊‘𝑋) + 1) ≤ 𝑚)
162	161	adantl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((⌊‘𝑋) + 1) ≤ 𝑚)
163	149, 159, 158, 160, 162	ltletrd 11394	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑇 < 𝑚)
164	149	rexrd 11283	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑇 ∈ ℝ^*)
165	164, 141	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 ∈ (𝑇(,)+∞) ↔ (𝑚 ∈ ℝ ∧ 𝑇 < 𝑚)))
166	158, 163, 165	mpbir2and 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ∈ (𝑇(,)+∞))
167	166, 1	eleqtrrdi 2871	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ∈ 𝑆)
168		peano2re 11407	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℝ → (𝑚 + 1) ∈ ℝ)
169	158, 168	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ∈ ℝ)
170	158	lep1d 12170	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ≤ (𝑚 + 1))
171	149, 158, 169, 163, 170	ltletrd 11394	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑇 < (𝑚 + 1))
172		elioopnf 13496	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ℝ^* → ((𝑚 + 1) ∈ (𝑇(,)+∞) ↔ ((𝑚 + 1) ∈ ℝ ∧ 𝑇 < (𝑚 + 1))))
173	164, 172	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((𝑚 + 1) ∈ (𝑇(,)+∞) ↔ ((𝑚 + 1) ∈ ℝ ∧ 𝑇 < (𝑚 + 1))))
174	169, 171, 173	mpbir2and 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ∈ (𝑇(,)+∞))
175	174, 1	eleqtrrdi 2871	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ∈ 𝑆)
176	108	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝐷 ≤ ((⌊‘𝑋) + 1))
177	147, 159, 158, 176, 162	letrd 11391	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝐷 ≤ 𝑚)
178	169	rexrd 11283	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ∈ ℝ^*)
179	68	rexrd 11283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ ℝ^*)
180	179	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (⌊‘𝑌) ∈ ℝ^*)
181		elfzle2 13582	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1)) → 𝑚 ≤ ((⌊‘𝑌) − 1))
182	181	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ≤ ((⌊‘𝑌) − 1))
183		1red 11233	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 1 ∈ ℝ)
184	66	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑌 ∈ ℝ)
185	184, 67	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (⌊‘𝑌) ∈ ℝ)
186		leaddsub 11714	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℝ ∧ 1 ∈ ℝ ∧ (⌊‘𝑌) ∈ ℝ) → ((𝑚 + 1) ≤ (⌊‘𝑌) ↔ 𝑚 ≤ ((⌊‘𝑌) − 1)))
187	158, 183, 185, 186	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((𝑚 + 1) ≤ (⌊‘𝑌) ↔ 𝑚 ≤ ((⌊‘𝑌) − 1)))
188	182, 187	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ≤ (⌊‘𝑌))
189	66	rexrd 11283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ∈ ℝ^*)
190	179, 189, 104, 111, 112	xrletrd 13213	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ≤ 𝑈)
191	190	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (⌊‘𝑌) ≤ 𝑈)
192	178, 180, 154, 188, 191	xrletrd 13213	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ≤ 𝑈)
193		flid 13869	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℤ → (⌊‘𝑚) = 𝑚)
194	157, 193	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (⌊‘𝑚) = 𝑚)
195	194	eqcomd 2766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 = (⌊‘𝑚))
196	195	oveq1d 7428	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) = ((⌊‘𝑚) + 1))
197	169, 196	eqled 11337	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ≤ ((⌊‘𝑚) + 1))
198	1, 11, 146, 147, 148, 149, 150, 151, 152, 153, 28, 154, 155, 33, 167, 175, 177, 170, 192, 197	dvfsumlem2 26254	. . . . . . 7 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((𝐻‘(𝑚 + 1)) ≤ (𝐻‘𝑚) ∧ ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵) ≤ ((𝐻‘(𝑚 + 1)) − ⦋(𝑚 + 1) / 𝑥⦌𝐵)))
199	198	simpld 500	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝐻‘(𝑚 + 1)) ≤ (𝐻‘𝑚))
200	129, 145, 199	monoord2 14097	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘(⌊‘𝑌)) ≤ (𝐻‘((⌊‘𝑋) + 1)))
201	71	rexrd 11283	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ∈ ℝ^*)
202	201, 179, 104, 87, 190	xrletrd 13213	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ≤ 𝑈)
203	71	leidd 11804	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ≤ ((⌊‘𝑋) + 1))
204	1, 11, 97, 98, 99, 69, 100, 101, 102, 103, 28, 104, 105, 33, 95, 125, 106, 107, 202, 203	dvfsumlem2 26254	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘((⌊‘𝑋) + 1)) ≤ (𝐻‘𝑋) ∧ ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵)))
205	204	simpld 500	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘((⌊‘𝑋) + 1)) ≤ (𝐻‘𝑋))
206	94, 126, 96, 200, 205	letrd 11391	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘(⌊‘𝑌)) ≤ (𝐻‘𝑋))
207	65, 94, 96, 120, 206	letrd 11391	. . 3 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘𝑌) ≤ (𝐻‘𝑋))
208		csbeq1 3850	. . . . . . 7 ⊢ (𝑚 = 𝑋 → ⦋𝑚 / 𝑥⦌𝐵 = ⦋𝑋 / 𝑥⦌𝐵)
209	208	eleq1d 2845	. . . . . 6 ⊢ (𝑚 = 𝑋 → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋𝑋 / 𝑥⦌𝐵 ∈ ℝ))
210	49	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
211	210	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
212		nfcsb1v 3871	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑚 / 𝑥⦌𝐵
213	212	nfel1 2938	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ
214		csbeq1a 3861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑥⦌𝐵)
215	214	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → (𝐵 ∈ ℝ ↔ ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ))
216	213, 215	rspc 3564	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑆 → (∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ → ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ))
217	211, 216	mpan9 516	. . . . . . 7 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ 𝑆) → ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ)
218	217	ralrimiva 3154	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ∀𝑚 ∈ 𝑆 ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ)
219	209, 218, 95	rspcdva 3577	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ⦋𝑋 / 𝑥⦌𝐵 ∈ ℝ)
220	96, 219	resubcld 11666	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ∈ ℝ)
221		csbeq1 3850	. . . . . . 7 ⊢ (𝑚 = (⌊‘𝑌) → ⦋𝑚 / 𝑥⦌𝐵 = ⦋(⌊‘𝑌) / 𝑥⦌𝐵)
222	221	eleq1d 2845	. . . . . 6 ⊢ (𝑚 = (⌊‘𝑌) → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋(⌊‘𝑌) / 𝑥⦌𝐵 ∈ ℝ))
223	222, 218, 93	rspcdva 3577	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ⦋(⌊‘𝑌) / 𝑥⦌𝐵 ∈ ℝ)
224	94, 223	resubcld 11666	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵) ∈ ℝ)
225		csbeq1 3850	. . . . . . 7 ⊢ (𝑚 = 𝑌 → ⦋𝑚 / 𝑥⦌𝐵 = ⦋𝑌 / 𝑥⦌𝐵)
226	225	eleq1d 2845	. . . . . 6 ⊢ (𝑚 = 𝑌 → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋𝑌 / 𝑥⦌𝐵 ∈ ℝ))
227	226, 218, 64	rspcdva 3577	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ⦋𝑌 / 𝑥⦌𝐵 ∈ ℝ)
228	65, 227	resubcld 11666	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵) ∈ ℝ)
229		csbeq1 3850	. . . . . . . 8 ⊢ (𝑚 = ((⌊‘𝑋) + 1) → ⦋𝑚 / 𝑥⦌𝐵 = ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵)
230	229	eleq1d 2845	. . . . . . 7 ⊢ (𝑚 = ((⌊‘𝑋) + 1) → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵 ∈ ℝ))
231	230, 218, 125	rspcdva 3577	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵 ∈ ℝ)
232	126, 231	resubcld 11666	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵) ∈ ℝ)
233	204	simprd 501	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵))
234		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑚 → (𝐻‘𝑦) = (𝐻‘𝑚))
235		csbeq1 3850	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑚 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑚 / 𝑥⦌𝐵)
236	234, 235	oveq12d 7431	. . . . . . . . . 10 ⊢ (𝑦 = 𝑚 → ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵) = ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵))
237		eqid 2760	. . . . . . . . . 10 ⊢ (𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵)) = (𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))
238		ovex 7446	. . . . . . . . . 10 ⊢ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵) ∈ V
239	236, 237, 238	fvmpt3i 6992	. . . . . . . . 9 ⊢ (𝑚 ∈ V → ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘𝑚) = ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵))
240	239	elv 3455	. . . . . . . 8 ⊢ ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘𝑚) = ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵)
241	144, 217	syldan 603	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ)
242	145, 241	resubcld 11666	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵) ∈ ℝ)
243	240, 242	eqeltrid 2864	. . . . . . 7 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘𝑚) ∈ ℝ)
244	198	simprd 501	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵) ≤ ((𝐻‘(𝑚 + 1)) − ⦋(𝑚 + 1) / 𝑥⦌𝐵))
245		ovex 7446	. . . . . . . . 9 ⊢ (𝑚 + 1) ∈ V
246		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑚 + 1) → (𝐻‘𝑦) = (𝐻‘(𝑚 + 1)))
247		csbeq1 3850	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑚 + 1) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋(𝑚 + 1) / 𝑥⦌𝐵)
248	246, 247	oveq12d 7431	. . . . . . . . . 10 ⊢ (𝑦 = (𝑚 + 1) → ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵) = ((𝐻‘(𝑚 + 1)) − ⦋(𝑚 + 1) / 𝑥⦌𝐵))
249	248, 237, 238	fvmpt3i 6992	. . . . . . . . 9 ⊢ ((𝑚 + 1) ∈ V → ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘(𝑚 + 1)) = ((𝐻‘(𝑚 + 1)) − ⦋(𝑚 + 1) / 𝑥⦌𝐵))
250	245, 249	ax-mp 5	. . . . . . . 8 ⊢ ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘(𝑚 + 1)) = ((𝐻‘(𝑚 + 1)) − ⦋(𝑚 + 1) / 𝑥⦌𝐵)
251	244, 240, 250	3brtr4g 5139	. . . . . . 7 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘𝑚) ≤ ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘(𝑚 + 1)))
252	129, 243, 251	monoord 14096	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘((⌊‘𝑋) + 1)) ≤ ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘(⌊‘𝑌)))
253		ovex 7446	. . . . . . 7 ⊢ ((⌊‘𝑋) + 1) ∈ V
254		fveq2 6878	. . . . . . . . 9 ⊢ (𝑦 = ((⌊‘𝑋) + 1) → (𝐻‘𝑦) = (𝐻‘((⌊‘𝑋) + 1)))
255		csbeq1 3850	. . . . . . . . 9 ⊢ (𝑦 = ((⌊‘𝑋) + 1) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵)
256	254, 255	oveq12d 7431	. . . . . . . 8 ⊢ (𝑦 = ((⌊‘𝑋) + 1) → ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵) = ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵))
257	256, 237, 238	fvmpt3i 6992	. . . . . . 7 ⊢ (((⌊‘𝑋) + 1) ∈ V → ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘((⌊‘𝑋) + 1)) = ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵))
258	253, 257	ax-mp 5	. . . . . 6 ⊢ ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘((⌊‘𝑋) + 1)) = ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵)
259		fvex 6891	. . . . . . 7 ⊢ (⌊‘𝑌) ∈ V
260		fveq2 6878	. . . . . . . . 9 ⊢ (𝑦 = (⌊‘𝑌) → (𝐻‘𝑦) = (𝐻‘(⌊‘𝑌)))
261		csbeq1 3850	. . . . . . . . 9 ⊢ (𝑦 = (⌊‘𝑌) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋(⌊‘𝑌) / 𝑥⦌𝐵)
262	260, 261	oveq12d 7431	. . . . . . . 8 ⊢ (𝑦 = (⌊‘𝑌) → ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵) = ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵))
263	262, 237, 238	fvmpt3i 6992	. . . . . . 7 ⊢ ((⌊‘𝑌) ∈ V → ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘(⌊‘𝑌)) = ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵))
264	259, 263	ax-mp 5	. . . . . 6 ⊢ ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘(⌊‘𝑌)) = ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵)
265	252, 258, 264	3brtr3g 5138	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵) ≤ ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵))
266	220, 232, 224, 233, 265	letrd 11391	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵))
267	119	simprd 501	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵))
268	220, 224, 228, 266, 267	letrd 11391	. . 3 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵))
269	207, 268	jca 521	. 2 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑌) ≤ (𝐻‘𝑋) ∧ ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵)))
270	5, 10, 43, 269	lecasei 11340	1 ⊢ (𝜑 → ((𝐻‘𝑌) ≤ (𝐻‘𝑋) ∧ ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 Vcvv 3450 ⦋csb 3847 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ℝcr 11123 1c1 11125 + caddc 11127 · cmul 11129 +∞cpnf 11264 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 − cmin 11465 ℤcz 12615 ℤ_≥cuz 12887 (,)cioo 13398 ...cfz 13561 ⌊cfl 13851 Σcsu 15773 D cdv 26090

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575 df-sum 15774 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-cmp 23612 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-limc 26093 df-dv 26094

This theorem is used by: dvfsumlem4 26256 dvfsum2 26261

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