monotuz - Mathbox for Stefan O'Rear

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Theorem monotuz 39882

Description: A function defined on an upper set of integers which increases at every adjacent pair is globally strictly monotonic by induction. (Contributed by Stefan O'Rear, 24-Sep-2014.)

**Hypotheses**
Ref	Expression
monotuz.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐻) → 𝐹 < 𝐺)
monotuz.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝐶 ∈ ℝ)
monotuz.3	⊢ 𝐻 = (ℤ_≥‘𝐼)
monotuz.4	⊢ (𝑥 = (𝑦 + 1) → 𝐶 = 𝐺)
monotuz.5	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐹)
monotuz.6	⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
monotuz.7	⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)

**Assertion**
Ref	Expression
monotuz	⊢ ((𝜑 ∧ (𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻)) → (𝐴 < 𝐵 ↔ 𝐷 < 𝐸))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦 𝑦,𝐶 𝑥,𝐷,𝑦 𝑥,𝐸,𝑦 𝑥,𝐹 𝑥,𝐺 𝑥,𝐻,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝐶(𝑥) 𝐹(𝑦) 𝐺(𝑦) 𝐼(𝑥,𝑦)

Proof of Theorem monotuz Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3831	. . 3 ⊢ (𝑎 = 𝑏 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶)
2		csbeq1 3831	. . 3 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
3		csbeq1 3831	. . 3 ⊢ (𝑎 = 𝐵 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)
4		monotuz.3	. . . 4 ⊢ 𝐻 = (ℤ_≥‘𝐼)
5		uzssz 12252	. . . . 5 ⊢ (ℤ_≥‘𝐼) ⊆ ℤ
6		zssre 11976	. . . . 5 ⊢ ℤ ⊆ ℝ
7	5, 6	sstri 3924	. . . 4 ⊢ (ℤ_≥‘𝐼) ⊆ ℝ
8	4, 7	eqsstri 3949	. . 3 ⊢ 𝐻 ⊆ ℝ
9		nfv 1915	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ 𝐻)
10		nfcsb1v 3852	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
11	10	nfel1 2971	. . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 ∈ ℝ
12	9, 11	nfim 1897	. . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 ∈ ℝ)
13		eleq1 2877	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐻 ↔ 𝑎 ∈ 𝐻))
14	13	anbi2d 631	. . . . 5 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑥 ∈ 𝐻) ↔ (𝜑 ∧ 𝑎 ∈ 𝐻)))
15		csbeq1a 3842	. . . . . 6 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
16	15	eleq1d 2874	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝐶 ∈ ℝ ↔ ⦋𝑎 / 𝑥⦌𝐶 ∈ ℝ))
17	14, 16	imbi12d 348	. . . 4 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝐶 ∈ ℝ) ↔ ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 ∈ ℝ)))
18		monotuz.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝐶 ∈ ℝ)
19	12, 17, 18	chvarfv 2240	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 ∈ ℝ)
20		simpl 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐻) ∧ 𝑎 < 𝑏) → (𝜑 ∧ 𝑎 ∈ 𝐻))
21	20	adantlrr 720	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) ∧ 𝑎 < 𝑏) → (𝜑 ∧ 𝑎 ∈ 𝐻))
22	4, 5	eqsstri 3949	. . . . . . 7 ⊢ 𝐻 ⊆ ℤ
23		simplrl 776	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) ∧ 𝑎 < 𝑏) → 𝑎 ∈ 𝐻)
24	22, 23	sseldi 3913	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) ∧ 𝑎 < 𝑏) → 𝑎 ∈ ℤ)
25		simplrr 777	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) ∧ 𝑎 < 𝑏) → 𝑏 ∈ 𝐻)
26	22, 25	sseldi 3913	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) ∧ 𝑎 < 𝑏) → 𝑏 ∈ ℤ)
27		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) ∧ 𝑎 < 𝑏) → 𝑎 < 𝑏)
28		csbeq1 3831	. . . . . . . . 9 ⊢ (𝑐 = (𝑎 + 1) → ⦋𝑐 / 𝑥⦌𝐶 = ⦋(𝑎 + 1) / 𝑥⦌𝐶)
29	28	breq2d 5042	. . . . . . . 8 ⊢ (𝑐 = (𝑎 + 1) → (⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑐 / 𝑥⦌𝐶 ↔ ⦋𝑎 / 𝑥⦌𝐶 < ⦋(𝑎 + 1) / 𝑥⦌𝐶))
30	29	imbi2d 344	. . . . . . 7 ⊢ (𝑐 = (𝑎 + 1) → (((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑐 / 𝑥⦌𝐶) ↔ ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋(𝑎 + 1) / 𝑥⦌𝐶)))
31		csbeq1 3831	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ⦋𝑐 / 𝑥⦌𝐶 = ⦋𝑑 / 𝑥⦌𝐶)
32	31	breq2d 5042	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑐 / 𝑥⦌𝐶 ↔ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶))
33	32	imbi2d 344	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑐 / 𝑥⦌𝐶) ↔ ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶)))
34		csbeq1 3831	. . . . . . . . 9 ⊢ (𝑐 = (𝑑 + 1) → ⦋𝑐 / 𝑥⦌𝐶 = ⦋(𝑑 + 1) / 𝑥⦌𝐶)
35	34	breq2d 5042	. . . . . . . 8 ⊢ (𝑐 = (𝑑 + 1) → (⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑐 / 𝑥⦌𝐶 ↔ ⦋𝑎 / 𝑥⦌𝐶 < ⦋(𝑑 + 1) / 𝑥⦌𝐶))
36	35	imbi2d 344	. . . . . . 7 ⊢ (𝑐 = (𝑑 + 1) → (((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑐 / 𝑥⦌𝐶) ↔ ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋(𝑑 + 1) / 𝑥⦌𝐶)))
37		csbeq1 3831	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → ⦋𝑐 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶)
38	37	breq2d 5042	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑐 / 𝑥⦌𝐶 ↔ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑏 / 𝑥⦌𝐶))
39	38	imbi2d 344	. . . . . . 7 ⊢ (𝑐 = 𝑏 → (((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑐 / 𝑥⦌𝐶) ↔ ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑏 / 𝑥⦌𝐶)))
40		eleq1 2877	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ 𝐻 ↔ 𝑎 ∈ 𝐻))
41	40	anbi2d 631	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → ((𝜑 ∧ 𝑦 ∈ 𝐻) ↔ (𝜑 ∧ 𝑎 ∈ 𝐻)))
42		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
43		monotuz.5	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐹)
44	42, 43	csbie 3863	. . . . . . . . . . 11 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐹
45		csbeq1 3831	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
46	44, 45	syl5eqr 2847	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → 𝐹 = ⦋𝑎 / 𝑥⦌𝐶)
47		ovex 7168	. . . . . . . . . . . 12 ⊢ (𝑦 + 1) ∈ V
48		monotuz.4	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 + 1) → 𝐶 = 𝐺)
49	47, 48	csbie 3863	. . . . . . . . . . 11 ⊢ ⦋(𝑦 + 1) / 𝑥⦌𝐶 = 𝐺
50		oveq1 7142	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑎 → (𝑦 + 1) = (𝑎 + 1))
51	50	csbeq1d 3832	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → ⦋(𝑦 + 1) / 𝑥⦌𝐶 = ⦋(𝑎 + 1) / 𝑥⦌𝐶)
52	49, 51	syl5eqr 2847	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → 𝐺 = ⦋(𝑎 + 1) / 𝑥⦌𝐶)
53	46, 52	breq12d 5043	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → (𝐹 < 𝐺 ↔ ⦋𝑎 / 𝑥⦌𝐶 < ⦋(𝑎 + 1) / 𝑥⦌𝐶))
54	41, 53	imbi12d 348	. . . . . . . 8 ⊢ (𝑦 = 𝑎 → (((𝜑 ∧ 𝑦 ∈ 𝐻) → 𝐹 < 𝐺) ↔ ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋(𝑎 + 1) / 𝑥⦌𝐶)))
55		monotuz.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐻) → 𝐹 < 𝐺)
56	54, 55	vtoclg 3515	. . . . . . 7 ⊢ (𝑎 ∈ ℤ → ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋(𝑎 + 1) / 𝑥⦌𝐶))
57	19	3ad2ant2 1131	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → ⦋𝑎 / 𝑥⦌𝐶 ∈ ℝ)
58		simp2l 1196	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → 𝜑)
59		zre 11973	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ ℤ → 𝑎 ∈ ℝ)
60	59	3ad2ant1 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) → 𝑎 ∈ ℝ)
61		zre 11973	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ ℤ → 𝑑 ∈ ℝ)
62	61	3ad2ant2 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) → 𝑑 ∈ ℝ)
63		simp3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) → 𝑎 < 𝑑)
64	60, 62, 63	ltled 10777	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) → 𝑎 ≤ 𝑑)
65	64	3ad2ant1 1130	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → 𝑎 ≤ 𝑑)
66		simp11 1200	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → 𝑎 ∈ ℤ)
67		simp12 1201	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → 𝑑 ∈ ℤ)
68		eluz 12245	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ) → (𝑑 ∈ (ℤ_≥‘𝑎) ↔ 𝑎 ≤ 𝑑))
69	66, 67, 68	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → (𝑑 ∈ (ℤ_≥‘𝑎) ↔ 𝑎 ≤ 𝑑))
70	65, 69	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → 𝑑 ∈ (ℤ_≥‘𝑎))
71		simp2r 1197	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → 𝑎 ∈ 𝐻)
72	71, 4	eleqtrdi 2900	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → 𝑎 ∈ (ℤ_≥‘𝐼))
73		uztrn 12249	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ (ℤ_≥‘𝑎) ∧ 𝑎 ∈ (ℤ_≥‘𝐼)) → 𝑑 ∈ (ℤ_≥‘𝐼))
74	70, 72, 73	syl2anc 587	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → 𝑑 ∈ (ℤ_≥‘𝐼))
75	74, 4	eleqtrrdi 2901	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → 𝑑 ∈ 𝐻)
76		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑑 ∈ 𝐻)
77		nfcsb1v 3852	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑑 / 𝑥⦌𝐶
78	77	nfel1 2971	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥⦋𝑑 / 𝑥⦌𝐶 ∈ ℝ
79	76, 78	nfim 1897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑑 ∈ 𝐻) → ⦋𝑑 / 𝑥⦌𝐶 ∈ ℝ)
80		eleq1 2877	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑑 → (𝑥 ∈ 𝐻 ↔ 𝑑 ∈ 𝐻))
81	80	anbi2d 631	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑑 → ((𝜑 ∧ 𝑥 ∈ 𝐻) ↔ (𝜑 ∧ 𝑑 ∈ 𝐻)))
82		csbeq1a 3842	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑑 → 𝐶 = ⦋𝑑 / 𝑥⦌𝐶)
83	82	eleq1d 2874	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑑 → (𝐶 ∈ ℝ ↔ ⦋𝑑 / 𝑥⦌𝐶 ∈ ℝ))
84	81, 83	imbi12d 348	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑑 → (((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝐶 ∈ ℝ) ↔ ((𝜑 ∧ 𝑑 ∈ 𝐻) → ⦋𝑑 / 𝑥⦌𝐶 ∈ ℝ)))
85	79, 84, 18	chvarfv 2240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐻) → ⦋𝑑 / 𝑥⦌𝐶 ∈ ℝ)
86	58, 75, 85	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → ⦋𝑑 / 𝑥⦌𝐶 ∈ ℝ)
87		peano2uz 12289	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (ℤ_≥‘𝐼) → (𝑑 + 1) ∈ (ℤ_≥‘𝐼))
88	74, 87	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → (𝑑 + 1) ∈ (ℤ_≥‘𝐼))
89	88, 4	eleqtrrdi 2901	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → (𝑑 + 1) ∈ 𝐻)
90		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑑 + 1) ∈ 𝐻)
91		nfcsb1v 3852	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋(𝑑 + 1) / 𝑥⦌𝐶
92	91	nfel1 2971	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥⦋(𝑑 + 1) / 𝑥⦌𝐶 ∈ ℝ
93	90, 92	nfim 1897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝜑 ∧ (𝑑 + 1) ∈ 𝐻) → ⦋(𝑑 + 1) / 𝑥⦌𝐶 ∈ ℝ)
94		ovex 7168	. . . . . . . . . . . 12 ⊢ (𝑑 + 1) ∈ V
95		eleq1 2877	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑑 + 1) → (𝑥 ∈ 𝐻 ↔ (𝑑 + 1) ∈ 𝐻))
96	95	anbi2d 631	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑑 + 1) → ((𝜑 ∧ 𝑥 ∈ 𝐻) ↔ (𝜑 ∧ (𝑑 + 1) ∈ 𝐻)))
97		csbeq1a 3842	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑑 + 1) → 𝐶 = ⦋(𝑑 + 1) / 𝑥⦌𝐶)
98	97	eleq1d 2874	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑑 + 1) → (𝐶 ∈ ℝ ↔ ⦋(𝑑 + 1) / 𝑥⦌𝐶 ∈ ℝ))
99	96, 98	imbi12d 348	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑑 + 1) → (((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝐶 ∈ ℝ) ↔ ((𝜑 ∧ (𝑑 + 1) ∈ 𝐻) → ⦋(𝑑 + 1) / 𝑥⦌𝐶 ∈ ℝ)))
100	93, 94, 99, 18	vtoclf 3506	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 + 1) ∈ 𝐻) → ⦋(𝑑 + 1) / 𝑥⦌𝐶 ∈ ℝ)
101	58, 89, 100	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → ⦋(𝑑 + 1) / 𝑥⦌𝐶 ∈ ℝ)
102		simp3 1135	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶)
103		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦((𝜑 ∧ 𝑑 ∈ 𝐻) → ⦋𝑑 / 𝑥⦌𝐶 < ⦋(𝑑 + 1) / 𝑥⦌𝐶)
104		eleq1 2877	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → (𝑦 ∈ 𝐻 ↔ 𝑑 ∈ 𝐻))
105	104	anbi2d 631	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → ((𝜑 ∧ 𝑦 ∈ 𝐻) ↔ (𝜑 ∧ 𝑑 ∈ 𝐻)))
106		csbeq1 3831	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑑 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝑑 / 𝑥⦌𝐶)
107	44, 106	syl5eqr 2847	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → 𝐹 = ⦋𝑑 / 𝑥⦌𝐶)
108		oveq1 7142	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑑 → (𝑦 + 1) = (𝑑 + 1))
109	108	csbeq1d 3832	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑑 → ⦋(𝑦 + 1) / 𝑥⦌𝐶 = ⦋(𝑑 + 1) / 𝑥⦌𝐶)
110	49, 109	syl5eqr 2847	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑑 → 𝐺 = ⦋(𝑑 + 1) / 𝑥⦌𝐶)
111	107, 110	breq12d 5043	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑑 → (𝐹 < 𝐺 ↔ ⦋𝑑 / 𝑥⦌𝐶 < ⦋(𝑑 + 1) / 𝑥⦌𝐶))
112	105, 111	imbi12d 348	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑑 → (((𝜑 ∧ 𝑦 ∈ 𝐻) → 𝐹 < 𝐺) ↔ ((𝜑 ∧ 𝑑 ∈ 𝐻) → ⦋𝑑 / 𝑥⦌𝐶 < ⦋(𝑑 + 1) / 𝑥⦌𝐶)))
113	103, 112, 55	chvarfv 2240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐻) → ⦋𝑑 / 𝑥⦌𝐶 < ⦋(𝑑 + 1) / 𝑥⦌𝐶)
114	58, 75, 113	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → ⦋𝑑 / 𝑥⦌𝐶 < ⦋(𝑑 + 1) / 𝑥⦌𝐶)
115	57, 86, 101, 102, 114	lttrd 10790	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) ∧ (𝜑 ∧ 𝑎 ∈ 𝐻) ∧ ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋(𝑑 + 1) / 𝑥⦌𝐶)
116	115	3exp 1116	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) → ((𝜑 ∧ 𝑎 ∈ 𝐻) → (⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶 → ⦋𝑎 / 𝑥⦌𝐶 < ⦋(𝑑 + 1) / 𝑥⦌𝐶)))
117	116	a2d 29	. . . . . . 7 ⊢ ((𝑎 ∈ ℤ ∧ 𝑑 ∈ ℤ ∧ 𝑎 < 𝑑) → (((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑑 / 𝑥⦌𝐶) → ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋(𝑑 + 1) / 𝑥⦌𝐶)))
118	30, 33, 36, 39, 56, 117	uzind2 12063	. . . . . 6 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑎 < 𝑏) → ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑏 / 𝑥⦌𝐶))
119	24, 26, 27, 118	syl3anc 1368	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) ∧ 𝑎 < 𝑏) → ((𝜑 ∧ 𝑎 ∈ 𝐻) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑏 / 𝑥⦌𝐶))
120	21, 119	mpd 15	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) ∧ 𝑎 < 𝑏) → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑏 / 𝑥⦌𝐶)
121	120	ex 416	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑎 < 𝑏 → ⦋𝑎 / 𝑥⦌𝐶 < ⦋𝑏 / 𝑥⦌𝐶))
122	1, 2, 3, 8, 19, 121	ltord1 11155	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻)) → (𝐴 < 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝐶 < ⦋𝐵 / 𝑥⦌𝐶))
123		nfcvd 2956	. . . . 5 ⊢ (𝐴 ∈ 𝐻 → Ⅎ𝑥𝐷)
124		monotuz.6	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
125	123, 124	csbiegf 3861	. . . 4 ⊢ (𝐴 ∈ 𝐻 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷)
126		nfcvd 2956	. . . . 5 ⊢ (𝐵 ∈ 𝐻 → Ⅎ𝑥𝐸)
127		monotuz.7	. . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)
128	126, 127	csbiegf 3861	. . . 4 ⊢ (𝐵 ∈ 𝐻 → ⦋𝐵 / 𝑥⦌𝐶 = 𝐸)
129	125, 128	breqan12d 5046	. . 3 ⊢ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (⦋𝐴 / 𝑥⦌𝐶 < ⦋𝐵 / 𝑥⦌𝐶 ↔ 𝐷 < 𝐸))
130	129	adantl 485	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻)) → (⦋𝐴 / 𝑥⦌𝐶 < ⦋𝐵 / 𝑥⦌𝐶 ↔ 𝐷 < 𝐸))
131	122, 130	bitrd 282	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻)) → (𝐴 < 𝐵 ↔ 𝐷 < 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⦋csb 3828 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 ℤcz 11969 ℤ_≥cuz 12231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232

This theorem is referenced by: ltrmynn0 39889 ltrmxnn0 39890

W3C validator