monoordxrv - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > monoordxrv		Structured version Visualization version GIF version

Theorem monoordxrv 46764

Description: Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.)

**Hypotheses**
Ref	Expression
monoordxrv.1	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
monoordxrv.2	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ^*)
monoordxrv.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))

**Assertion**
Ref	Expression
monoordxrv	⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁))

Distinct variable groups: 𝑘,𝐹 𝑘,𝑀 𝑘,𝑁 𝜑,𝑘

Proof of Theorem monoordxrv Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		monoordxrv.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		eluzfz2 13705	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2852	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6916	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀))
6	5	breq2d 5120	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑀)))
7	4, 6	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀))))
8	7	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀)))))
9		eleq1 2852	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
10		fveq2 6916	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛))
11	10	breq2d 5120	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑛)))
12	9, 11	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛))))
13	12	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛)))))
14		eleq1 2852	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
15		fveq2 6916	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
16	15	breq2d 5120	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))
17	14, 16	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1)))))
18	17	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))))
19		eleq1 2852	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
20		fveq2 6916	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐹‘𝑥) = (𝐹‘𝑁))
21	20	breq2d 5120	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑁)))
22	19, 21	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁))))
23	22	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁)))))
24		eluzfz1 13704	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
25	1, 24	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
26		monoordxrv.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ^*)
27	26	ralrimiva 3158	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^*)
28		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
29	28	eleq1d 2849	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ^* ↔ (𝐹‘𝑀) ∈ ℝ^*))
30	29	rspcv 3576	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^* → (𝐹‘𝑀) ∈ ℝ^*))
31	25, 27, 30	sylc 66	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ^*)
32	31	xrleidd 13321	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑀))
33	32	a1d 26	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀)))
34	33	a1i 11	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀))))
35		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ_≥‘𝑀))
36		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
37		peano2fzr 13710	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
38	35, 36, 37	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
39	38	expr 462	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
40	39	imim1d 83	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛))))
41		eluzelz 13015	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → 𝑛 ∈ ℤ)
42	35, 41	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
43		elfzuz3 13693	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘(𝑛 + 1)))
44	36, 43	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑁 ∈ (ℤ_≥‘(𝑛 + 1)))
45		eluzp1m1 13031	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝑁 − 1) ∈ (ℤ_≥‘𝑛))
46	42, 44, 45	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑁 − 1) ∈ (ℤ_≥‘𝑛))
47		elfzuzb 13690	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) ↔ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑁 − 1) ∈ (ℤ_≥‘𝑛)))
48	35, 46, 47	sylanbrc 595	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...(𝑁 − 1)))
49		monoordxrv.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
50	49	ralrimiva 3158	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
51	50	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
52		fveq2 6916	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
53		fvoveq1 7474	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
54	52, 53	breq12d 5121	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))))
55	54	rspcv 3576	. . . . . . 7 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) → (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))))
56	48, 51, 55	sylc 66	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
57	31	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑀) ∈ ℝ^*)
58	27	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^*)
59	52	eleq1d 2849	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ^* ↔ (𝐹‘𝑛) ∈ ℝ^*))
60	59	rspcv 3576	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^* → (𝐹‘𝑛) ∈ ℝ^*))
61	38, 58, 60	sylc 66	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑛) ∈ ℝ^*)
62		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
63	62	eleq1d 2849	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ^* ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ^*))
64	63	rspcv 3576	. . . . . . . 8 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ^* → (𝐹‘(𝑛 + 1)) ∈ ℝ^*))
65	36, 58, 64	sylc 66	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ ℝ^*)
66		xrletr 13327	. . . . . . 7 ⊢ (((𝐹‘𝑀) ∈ ℝ^* ∧ (𝐹‘𝑛) ∈ ℝ^* ∧ (𝐹‘(𝑛 + 1)) ∈ ℝ^*) → (((𝐹‘𝑀) ≤ (𝐹‘𝑛) ∧ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))
67	57, 61, 65, 66	syl3anc 1399	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (((𝐹‘𝑀) ≤ (𝐹‘𝑛) ∧ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))
68	56, 67	mpan2d 707	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝐹‘𝑀) ≤ (𝐹‘𝑛) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))
69	40, 68	animpimp2impd 860	. . . 4 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))))
70	8, 13, 18, 23, 34, 69	uzind4 13073	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁))))
71	1, 70	mpcom 39	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁)))
72	3, 71	mpd 16	1 ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 1c1 11240 + caddc 11242 ℝ^*cxr 11381 ≤ cle 11383 − cmin 11580 ℤcz 12733 ℤ_≥cuz 13005 ...cfz 13679

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-n0 12647 df-z 12734 df-uz 13006 df-fz 13680

This theorem is used by: monoordxr 46765 monoord2xrv 46766

W3C validator