dvfsumlem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvfsumlem3		Structured version Visualization version GIF version

Theorem dvfsumlem3 25952

Description: Lemma for dvfsumrlim 25955. (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 13329	. . . 4 ⊢ (𝑇(,)+∞) ⊆ ℝ
3	1, 2	eqsstri 3984	. . 3 ⊢ 𝑆 ⊆ ℝ
4		dvfsumlem1.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
5	3, 4	sselid 3935	. 2 ⊢ (𝜑 → 𝑌 ∈ ℝ)
6		dvfsumlem1.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
7	3, 6	sselid 3935	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8		reflcl 13719	. . 3 ⊢ (𝑋 ∈ ℝ → (⌊‘𝑋) ∈ ℝ)
9		peano2re 11308	. . 3 ⊢ ((⌊‘𝑋) ∈ ℝ → ((⌊‘𝑋) + 1) ∈ ℝ)
10	7, 8, 9	3syl 18	. 2 ⊢ (𝜑 → ((⌊‘𝑋) + 1) ∈ ℝ)
11		dvfsum.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑀)
12		dvfsum.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑀 ∈ ℤ)
14		dvfsum.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝐷 ∈ ℝ)
16		dvfsum.md	. . . 4 ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1))
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑀 ≤ (𝐷 + 1))
18		dvfsum.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ)
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑇 ∈ ℝ)
20		dvfsum.a	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
21	20	adantlr 715	. . 3 ⊢ (((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
22		dvfsum.b1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
23	22	adantlr 715	. . 3 ⊢ (((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
24		dvfsum.b2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
25	24	adantlr 715	. . 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 480	. . 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 480	. . 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 1178	. . 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 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑋 ∈ 𝑆)
35	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑌 ∈ 𝑆)
36		dvfsumlem1.3	. . . 4 ⊢ (𝜑 → 𝐷 ≤ 𝑋)
37	36	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝐷 ≤ 𝑋)
38		dvfsumlem1.4	. . . 4 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
39	38	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑋 ≤ 𝑌)
40		dvfsumlem1.5	. . . 4 ⊢ (𝜑 → 𝑌 ≤ 𝑈)
41	40	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≤ ((⌊‘𝑋) + 1)) → 𝑌 ≤ 𝑈)
42		simpr 484	. . 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 25950	. 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 3937	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℝ)
46		reflcl 13719	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
47	45, 46	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (⌊‘𝑥) ∈ ℝ)
48	45, 47	resubcld 11567	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 − (⌊‘𝑥)) ∈ ℝ)
49	44, 20, 22, 26	dvmptrecl 25947	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ)
50	48, 49	remulcld 11164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑥 − (⌊‘𝑥)) · 𝐵) ∈ ℝ)
51		fzfid 13899	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑀...(⌊‘𝑥)) ∈ Fin)
52	24	ralrimiva 3121	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 𝐵 ∈ ℝ)
53	52	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∀𝑥 ∈ 𝑍 𝐵 ∈ ℝ)
54		elfzuz 13442	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑥)) → 𝑘 ∈ (ℤ_≥‘𝑀))
55	54, 11	eleqtrrdi 2839	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀...(⌊‘𝑥)) → 𝑘 ∈ 𝑍)
56	28	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ))
57	56	rspccva 3578	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝑍 𝐵 ∈ ℝ ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℝ)
58	53, 55, 57	syl2an 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑘 ∈ (𝑀...(⌊‘𝑥))) → 𝐶 ∈ ℝ)
59	51, 58	fsumrecl 15660	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 ∈ ℝ)
60	59, 20	resubcld 11567	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴) ∈ ℝ)
61	50, 60	readdcld 11163	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) ∈ ℝ)
62	61, 33	fmptd 7052	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑆⟶ℝ)
63	62	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝐻:𝑆⟶ℝ)
64	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ∈ 𝑆)
65	63, 64	ffvelcdmd 7023	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘𝑌) ∈ ℝ)
66	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ∈ ℝ)
67		reflcl 13719	. . . . . . . 8 ⊢ (𝑌 ∈ ℝ → (⌊‘𝑌) ∈ ℝ)
68	66, 67	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ ℝ)
69	18	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑇 ∈ ℝ)
70	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑋 ∈ ℝ)
71	70, 8, 9	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ∈ ℝ)
72	6, 1	eleqtrdi 2838	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (𝑇(,)+∞))
73	18	rexrd 11184	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ℝ^*)
74		elioopnf 13365	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ ℝ^* → (𝑋 ∈ (𝑇(,)+∞) ↔ (𝑋 ∈ ℝ ∧ 𝑇 < 𝑋)))
75	73, 74	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ (𝑇(,)+∞) ↔ (𝑋 ∈ ℝ ∧ 𝑇 < 𝑋)))
76	72, 75	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ℝ ∧ 𝑇 < 𝑋))
77	76	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 < 𝑋)
78		fllep1 13724	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 𝑋 ≤ ((⌊‘𝑋) + 1))
79	7, 78	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≤ ((⌊‘𝑋) + 1))
80	18, 7, 10, 77, 79	ltletrd 11295	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 < ((⌊‘𝑋) + 1))
81	80	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑇 < ((⌊‘𝑋) + 1))
82		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ≤ 𝑌)
83	70	flcld 13721	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑋) ∈ ℤ)
84	83	peano2zd 12602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ∈ ℤ)
85		flge 13728	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ℝ ∧ ((⌊‘𝑋) + 1) ∈ ℤ) → (((⌊‘𝑋) + 1) ≤ 𝑌 ↔ ((⌊‘𝑋) + 1) ≤ (⌊‘𝑌)))
86	66, 84, 85	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (((⌊‘𝑋) + 1) ≤ 𝑌 ↔ ((⌊‘𝑋) + 1) ≤ (⌊‘𝑌)))
87	82, 86	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ≤ (⌊‘𝑌))
88	69, 71, 68, 81, 87	ltletrd 11295	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑇 < (⌊‘𝑌))
89	73	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑇 ∈ ℝ^*)
90		elioopnf 13365	. . . . . . . 8 ⊢ (𝑇 ∈ ℝ^* → ((⌊‘𝑌) ∈ (𝑇(,)+∞) ↔ ((⌊‘𝑌) ∈ ℝ ∧ 𝑇 < (⌊‘𝑌))))
91	89, 90	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑌) ∈ (𝑇(,)+∞) ↔ ((⌊‘𝑌) ∈ ℝ ∧ 𝑇 < (⌊‘𝑌))))
92	68, 88, 91	mpbir2and 713	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ (𝑇(,)+∞))
93	92, 1	eleqtrrdi 2839	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ 𝑆)
94	63, 93	ffvelcdmd 7023	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘(⌊‘𝑌)) ∈ ℝ)
95	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑋 ∈ 𝑆)
96	63, 95	ffvelcdmd 7023	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘𝑋) ∈ ℝ)
97	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑀 ∈ ℤ)
98	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝐷 ∈ ℝ)
99	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑀 ≤ (𝐷 + 1))
100	20	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
101	22	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
102	24	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
103	26	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
104	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑈 ∈ ℝ^*)
105	31	3adant1r 1178	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵)
106	36	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝐷 ≤ 𝑋)
107	70, 78	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑋 ≤ ((⌊‘𝑋) + 1))
108	98, 70, 71, 106, 107	letrd 11292	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝐷 ≤ ((⌊‘𝑋) + 1))
109	98, 71, 68, 108, 87	letrd 11292	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝐷 ≤ (⌊‘𝑌))
110		flle 13722	. . . . . . 7 ⊢ (𝑌 ∈ ℝ → (⌊‘𝑌) ≤ 𝑌)
111	66, 110	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ≤ 𝑌)
112	40	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ≤ 𝑈)
113		fllep1 13724	. . . . . . . 8 ⊢ (𝑌 ∈ ℝ → 𝑌 ≤ ((⌊‘𝑌) + 1))
114	66, 113	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ≤ ((⌊‘𝑌) + 1))
115		flidm 13732	. . . . . . . . 9 ⊢ (𝑌 ∈ ℝ → (⌊‘(⌊‘𝑌)) = (⌊‘𝑌))
116	66, 115	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘(⌊‘𝑌)) = (⌊‘𝑌))
117	116	oveq1d 7368	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘(⌊‘𝑌)) + 1) = ((⌊‘𝑌) + 1))
118	114, 117	breqtrrd 5123	. . . . . 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 25950	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑌) ≤ (𝐻‘(⌊‘𝑌)) ∧ ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵)))
120	119	simpld 494	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘𝑌) ≤ (𝐻‘(⌊‘𝑌)))
121		elioopnf 13365	. . . . . . . . . 10 ⊢ (𝑇 ∈ ℝ^* → (((⌊‘𝑋) + 1) ∈ (𝑇(,)+∞) ↔ (((⌊‘𝑋) + 1) ∈ ℝ ∧ 𝑇 < ((⌊‘𝑋) + 1))))
122	73, 121	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘𝑋) + 1) ∈ (𝑇(,)+∞) ↔ (((⌊‘𝑋) + 1) ∈ ℝ ∧ 𝑇 < ((⌊‘𝑋) + 1))))
123	10, 80, 122	mpbir2and 713	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘𝑋) + 1) ∈ (𝑇(,)+∞))
124	123, 1	eleqtrrdi 2839	. . . . . . 7 ⊢ (𝜑 → ((⌊‘𝑋) + 1) ∈ 𝑆)
125	124	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ∈ 𝑆)
126	63, 125	ffvelcdmd 7023	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘((⌊‘𝑋) + 1)) ∈ ℝ)
127	66	flcld 13721	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ ℤ)
128		eluz2 12760	. . . . . . 7 ⊢ ((⌊‘𝑌) ∈ (ℤ_≥‘((⌊‘𝑋) + 1)) ↔ (((⌊‘𝑋) + 1) ∈ ℤ ∧ (⌊‘𝑌) ∈ ℤ ∧ ((⌊‘𝑋) + 1) ≤ (⌊‘𝑌)))
129	84, 127, 87, 128	syl3anbrc 1344	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ (ℤ_≥‘((⌊‘𝑋) + 1)))
130	63	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝐻:𝑆⟶ℝ)
131		elfzelz 13446	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌)) → 𝑚 ∈ ℤ)
132	131	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑚 ∈ ℤ)
133	132	zred 12599	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑚 ∈ ℝ)
134	69	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑇 ∈ ℝ)
135	71	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → ((⌊‘𝑋) + 1) ∈ ℝ)
136	80	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑇 < ((⌊‘𝑋) + 1))
137		elfzle1 13449	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌)) → ((⌊‘𝑋) + 1) ≤ 𝑚)
138	137	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → ((⌊‘𝑋) + 1) ≤ 𝑚)
139	134, 135, 133, 136, 138	ltletrd 11295	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑇 < 𝑚)
140	73	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑇 ∈ ℝ^*)
141		elioopnf 13365	. . . . . . . . . 10 ⊢ (𝑇 ∈ ℝ^* → (𝑚 ∈ (𝑇(,)+∞) ↔ (𝑚 ∈ ℝ ∧ 𝑇 < 𝑚)))
142	140, 141	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → (𝑚 ∈ (𝑇(,)+∞) ↔ (𝑚 ∈ ℝ ∧ 𝑇 < 𝑚)))
143	133, 139, 142	mpbir2and 713	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑚 ∈ (𝑇(,)+∞))
144	143, 1	eleqtrrdi 2839	. . . . . . 7 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → 𝑚 ∈ 𝑆)
145	130, 144	ffvelcdmd 7023	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → (𝐻‘𝑚) ∈ ℝ)
146	97	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑀 ∈ ℤ)
147	98	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝐷 ∈ ℝ)
148	16	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑀 ≤ (𝐷 + 1))
149	69	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑇 ∈ ℝ)
150	100	adantlr 715	. . . . . . . 8 ⊢ ((((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
151	101	adantlr 715	. . . . . . . 8 ⊢ ((((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)
152	102	adantlr 715	. . . . . . . 8 ⊢ ((((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)
153	103	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))
154	104	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑈 ∈ ℝ^*)
155	105	3adant1r 1178	. . . . . . . 8 ⊢ ((((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵)
156		elfzelz 13446	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1)) → 𝑚 ∈ ℤ)
157	156	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ∈ ℤ)
158	157	zred 12599	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ∈ ℝ)
159	71	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((⌊‘𝑋) + 1) ∈ ℝ)
160	80	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑇 < ((⌊‘𝑋) + 1))
161		elfzle1 13449	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1)) → ((⌊‘𝑋) + 1) ≤ 𝑚)
162	161	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((⌊‘𝑋) + 1) ≤ 𝑚)
163	149, 159, 158, 160, 162	ltletrd 11295	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑇 < 𝑚)
164	149	rexrd 11184	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑇 ∈ ℝ^*)
165	164, 141	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 ∈ (𝑇(,)+∞) ↔ (𝑚 ∈ ℝ ∧ 𝑇 < 𝑚)))
166	158, 163, 165	mpbir2and 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ∈ (𝑇(,)+∞))
167	166, 1	eleqtrrdi 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ∈ 𝑆)
168		peano2re 11308	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℝ → (𝑚 + 1) ∈ ℝ)
169	158, 168	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ∈ ℝ)
170	158	lep1d 12075	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ≤ (𝑚 + 1))
171	149, 158, 169, 163, 170	ltletrd 11295	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑇 < (𝑚 + 1))
172		elioopnf 13365	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ℝ^* → ((𝑚 + 1) ∈ (𝑇(,)+∞) ↔ ((𝑚 + 1) ∈ ℝ ∧ 𝑇 < (𝑚 + 1))))
173	164, 172	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((𝑚 + 1) ∈ (𝑇(,)+∞) ↔ ((𝑚 + 1) ∈ ℝ ∧ 𝑇 < (𝑚 + 1))))
174	169, 171, 173	mpbir2and 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ∈ (𝑇(,)+∞))
175	174, 1	eleqtrrdi 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ∈ 𝑆)
176	108	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝐷 ≤ ((⌊‘𝑋) + 1))
177	147, 159, 158, 176, 162	letrd 11292	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝐷 ≤ 𝑚)
178	169	rexrd 11184	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ∈ ℝ^*)
179	68	rexrd 11184	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ∈ ℝ^*)
180	179	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (⌊‘𝑌) ∈ ℝ^*)
181		elfzle2 13450	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1)) → 𝑚 ≤ ((⌊‘𝑌) − 1))
182	181	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 ≤ ((⌊‘𝑌) − 1))
183		1red 11135	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 1 ∈ ℝ)
184	66	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑌 ∈ ℝ)
185	184, 67	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (⌊‘𝑌) ∈ ℝ)
186		leaddsub 11615	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℝ ∧ 1 ∈ ℝ ∧ (⌊‘𝑌) ∈ ℝ) → ((𝑚 + 1) ≤ (⌊‘𝑌) ↔ 𝑚 ≤ ((⌊‘𝑌) − 1)))
187	158, 183, 185, 186	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((𝑚 + 1) ≤ (⌊‘𝑌) ↔ 𝑚 ≤ ((⌊‘𝑌) − 1)))
188	182, 187	mpbird 257	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ≤ (⌊‘𝑌))
189	66	rexrd 11184	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → 𝑌 ∈ ℝ^*)
190	179, 189, 104, 111, 112	xrletrd 13083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (⌊‘𝑌) ≤ 𝑈)
191	190	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (⌊‘𝑌) ≤ 𝑈)
192	178, 180, 154, 188, 191	xrletrd 13083	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) ≤ 𝑈)
193		flid 13731	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℤ → (⌊‘𝑚) = 𝑚)
194	157, 193	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (⌊‘𝑚) = 𝑚)
195	194	eqcomd 2735	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → 𝑚 = (⌊‘𝑚))
196	195	oveq1d 7368	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝑚 + 1) = ((⌊‘𝑚) + 1))
197	169, 196	eqled 11238	. . . . . . . 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 25950	. . . . . . 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 494	. . . . . 6 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → (𝐻‘(𝑚 + 1)) ≤ (𝐻‘𝑚))
200	129, 145, 199	monoord2 13959	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘(⌊‘𝑌)) ≤ (𝐻‘((⌊‘𝑋) + 1)))
201	71	rexrd 11184	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ∈ ℝ^*)
202	201, 179, 104, 87, 190	xrletrd 13083	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((⌊‘𝑋) + 1) ≤ 𝑈)
203	71	leidd 11705	. . . . . . 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 25950	. . . . . 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 494	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘((⌊‘𝑋) + 1)) ≤ (𝐻‘𝑋))
206	94, 126, 96, 200, 205	letrd 11292	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘(⌊‘𝑌)) ≤ (𝐻‘𝑋))
207	65, 94, 96, 120, 206	letrd 11292	. . 3 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → (𝐻‘𝑌) ≤ (𝐻‘𝑋))
208		csbeq1 3856	. . . . . . 7 ⊢ (𝑚 = 𝑋 → ⦋𝑚 / 𝑥⦌𝐵 = ⦋𝑋 / 𝑥⦌𝐵)
209	208	eleq1d 2813	. . . . . 6 ⊢ (𝑚 = 𝑋 → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋𝑋 / 𝑥⦌𝐵 ∈ ℝ))
210	49	ralrimiva 3121	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
211	210	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ)
212		nfcsb1v 3877	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑚 / 𝑥⦌𝐵
213	212	nfel1 2908	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ
214		csbeq1a 3867	. . . . . . . . . 10 ⊢ (𝑥 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑥⦌𝐵)
215	214	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → (𝐵 ∈ ℝ ↔ ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ))
216	213, 215	rspc 3567	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑆 → (∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ → ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ))
217	211, 216	mpan9 506	. . . . . . 7 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ 𝑆) → ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ)
218	217	ralrimiva 3121	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ∀𝑚 ∈ 𝑆 ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ)
219	209, 218, 95	rspcdva 3580	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ⦋𝑋 / 𝑥⦌𝐵 ∈ ℝ)
220	96, 219	resubcld 11567	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ∈ ℝ)
221		csbeq1 3856	. . . . . . 7 ⊢ (𝑚 = (⌊‘𝑌) → ⦋𝑚 / 𝑥⦌𝐵 = ⦋(⌊‘𝑌) / 𝑥⦌𝐵)
222	221	eleq1d 2813	. . . . . 6 ⊢ (𝑚 = (⌊‘𝑌) → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋(⌊‘𝑌) / 𝑥⦌𝐵 ∈ ℝ))
223	222, 218, 93	rspcdva 3580	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ⦋(⌊‘𝑌) / 𝑥⦌𝐵 ∈ ℝ)
224	94, 223	resubcld 11567	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵) ∈ ℝ)
225		csbeq1 3856	. . . . . . 7 ⊢ (𝑚 = 𝑌 → ⦋𝑚 / 𝑥⦌𝐵 = ⦋𝑌 / 𝑥⦌𝐵)
226	225	eleq1d 2813	. . . . . 6 ⊢ (𝑚 = 𝑌 → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋𝑌 / 𝑥⦌𝐵 ∈ ℝ))
227	226, 218, 64	rspcdva 3580	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ⦋𝑌 / 𝑥⦌𝐵 ∈ ℝ)
228	65, 227	resubcld 11567	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵) ∈ ℝ)
229		csbeq1 3856	. . . . . . . 8 ⊢ (𝑚 = ((⌊‘𝑋) + 1) → ⦋𝑚 / 𝑥⦌𝐵 = ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵)
230	229	eleq1d 2813	. . . . . . 7 ⊢ (𝑚 = ((⌊‘𝑋) + 1) → (⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ ↔ ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵 ∈ ℝ))
231	230, 218, 125	rspcdva 3580	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵 ∈ ℝ)
232	126, 231	resubcld 11567	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵) ∈ ℝ)
233	204	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵))
234		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑚 → (𝐻‘𝑦) = (𝐻‘𝑚))
235		csbeq1 3856	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑚 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑚 / 𝑥⦌𝐵)
236	234, 235	oveq12d 7371	. . . . . . . . . 10 ⊢ (𝑦 = 𝑚 → ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵) = ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵))
237		eqid 2729	. . . . . . . . . 10 ⊢ (𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵)) = (𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))
238		ovex 7386	. . . . . . . . . 10 ⊢ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵) ∈ V
239	236, 237, 238	fvmpt3i 6939	. . . . . . . . 9 ⊢ (𝑚 ∈ V → ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘𝑚) = ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵))
240	239	elv 3443	. . . . . . . 8 ⊢ ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘𝑚) = ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵)
241	144, 217	syldan 591	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → ⦋𝑚 / 𝑥⦌𝐵 ∈ ℝ)
242	145, 241	resubcld 11567	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵) ∈ ℝ)
243	240, 242	eqeltrid 2832	. . . . . . 7 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...(⌊‘𝑌))) → ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘𝑚) ∈ ℝ)
244	198	simprd 495	. . . . . . . 8 ⊢ (((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) ∧ 𝑚 ∈ (((⌊‘𝑋) + 1)...((⌊‘𝑌) − 1))) → ((𝐻‘𝑚) − ⦋𝑚 / 𝑥⦌𝐵) ≤ ((𝐻‘(𝑚 + 1)) − ⦋(𝑚 + 1) / 𝑥⦌𝐵))
245		ovex 7386	. . . . . . . . 9 ⊢ (𝑚 + 1) ∈ V
246		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑚 + 1) → (𝐻‘𝑦) = (𝐻‘(𝑚 + 1)))
247		csbeq1 3856	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑚 + 1) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋(𝑚 + 1) / 𝑥⦌𝐵)
248	246, 247	oveq12d 7371	. . . . . . . . . 10 ⊢ (𝑦 = (𝑚 + 1) → ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵) = ((𝐻‘(𝑚 + 1)) − ⦋(𝑚 + 1) / 𝑥⦌𝐵))
249	248, 237, 238	fvmpt3i 6939	. . . . . . . . 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 5129	. . . . . . 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 13958	. . . . . 6 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘((⌊‘𝑋) + 1)) ≤ ((𝑦 ∈ V ↦ ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵))‘(⌊‘𝑌)))
253		ovex 7386	. . . . . . 7 ⊢ ((⌊‘𝑋) + 1) ∈ V
254		fveq2 6826	. . . . . . . . 9 ⊢ (𝑦 = ((⌊‘𝑋) + 1) → (𝐻‘𝑦) = (𝐻‘((⌊‘𝑋) + 1)))
255		csbeq1 3856	. . . . . . . . 9 ⊢ (𝑦 = ((⌊‘𝑋) + 1) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵)
256	254, 255	oveq12d 7371	. . . . . . . 8 ⊢ (𝑦 = ((⌊‘𝑋) + 1) → ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵) = ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵))
257	256, 237, 238	fvmpt3i 6939	. . . . . . 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 6839	. . . . . . 7 ⊢ (⌊‘𝑌) ∈ V
260		fveq2 6826	. . . . . . . . 9 ⊢ (𝑦 = (⌊‘𝑌) → (𝐻‘𝑦) = (𝐻‘(⌊‘𝑌)))
261		csbeq1 3856	. . . . . . . . 9 ⊢ (𝑦 = (⌊‘𝑌) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋(⌊‘𝑌) / 𝑥⦌𝐵)
262	260, 261	oveq12d 7371	. . . . . . . 8 ⊢ (𝑦 = (⌊‘𝑌) → ((𝐻‘𝑦) − ⦋𝑦 / 𝑥⦌𝐵) = ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵))
263	262, 237, 238	fvmpt3i 6939	. . . . . . 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 5128	. . . . 5 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘((⌊‘𝑋) + 1)) − ⦋((⌊‘𝑋) + 1) / 𝑥⦌𝐵) ≤ ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵))
266	220, 232, 224, 233, 265	letrd 11292	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵))
267	119	simprd 495	. . . 4 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘(⌊‘𝑌)) − ⦋(⌊‘𝑌) / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵))
268	220, 224, 228, 266, 267	letrd 11292	. . 3 ⊢ ((𝜑 ∧ ((⌊‘𝑋) + 1) ≤ 𝑌) → ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵))
269	207, 268	jca 511	. 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 11241	1 ⊢ (𝜑 → ((𝐻‘𝑌) ≤ (𝐻‘𝑋) ∧ ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3438 ⦋csb 3853 ⊆ wss 3905 class class class wbr 5095 ↦ cmpt 5176 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ℝcr 11027 1c1 11029 + caddc 11031 · cmul 11033 +∞cpnf 11165 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 − cmin 11366 ℤcz 12490 ℤ_≥cuz 12754 (,)cioo 13267 ...cfz 13429 ⌊cfl 13713 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: dvfsumlem4 25953 dvfsum2 25958

W3C validator