Theorem monoord 14035

Description: Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)

Hypotheses

Ref	Expression
monoord.1	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
monoord.2	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
monoord.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))

Assertion

Ref	Expression
monoord	⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁))

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

Proof of Theorem monoord

Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		monoord.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 13547	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2816	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6900	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀))
6	5	breq2d 5162	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑀)))
7	4, 6	imbi12d 343	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀))))
8	7	imbi2d 339	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀)))))
9		eleq1 2816	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
10		fveq2 6900	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛))
11	10	breq2d 5162	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑛)))
12	9, 11	imbi12d 343	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛))))
13	12	imbi2d 339	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛)))))
14		eleq1 2816	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
15		fveq2 6900	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
16	15	breq2d 5162	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))
17	14, 16	imbi12d 343	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1)))))
18	17	imbi2d 339	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))))
19		eleq1 2816	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
20		fveq2 6900	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐹‘𝑥) = (𝐹‘𝑁))
21	20	breq2d 5162	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑁)))
22	19, 21	imbi12d 343	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁))))
23	22	imbi2d 339	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁)))))
24		fveq2 6900	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
25	24	eleq1d 2813	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑀) ∈ ℝ))
26		monoord.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
27	26	ralrimiva 3142	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
28		eluzfz1 13546	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
29	1, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
30	25, 27, 29	rspcdva 3610	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ)
31	30	leidd 11816	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑀))
32	31	a1d 25	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀)))
33		peano2fzr 13552	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
34	33	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
35	34	expr 455	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
36	35	imim1d 82	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛))))
37		fveq2 6900	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
38		fvoveq1 7447	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
39	37, 38	breq12d 5163	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))))
40		monoord.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
41	40	ralrimiva 3142	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
42	41	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
43		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
44		eluzelz 12868	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
45	43, 44	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
46		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
47		elfzuz3 13536	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
48	46, 47	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
49		eluzp1m1 12884	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑛))
50	45, 48, 49	syl2anc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑁 − 1) ∈ (ℤ≥‘𝑛))
51		elfzuzb 13533	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑁 − 1) ∈ (ℤ≥‘𝑛)))
52	43, 50, 51	sylanbrc 581	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...(𝑁 − 1)))
53	39, 42, 52	rspcdva 3610	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
54	30	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑀) ∈ ℝ)
55	37	eleq1d 2813	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑛) ∈ ℝ))
56	27	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
57	55, 56, 34	rspcdva 3610	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑛) ∈ ℝ)
58		fveq2 6900	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
59	58	eleq1d 2813	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
60	59, 56, 46	rspcdva 3610	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
61		letr 11344	. . . . . . 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))))
62	54, 57, 60, 61	syl3anc 1368	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (((𝐹‘𝑀) ≤ (𝐹‘𝑛) ∧ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))
63	53, 62	mpan2d 692	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝐹‘𝑀) ≤ (𝐹‘𝑛) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))
64	36, 63	animpimp2impd 844	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))))
65	8, 13, 18, 23, 32, 64	uzind4i 12930	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁))))
66	1, 65	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁)))
67	3, 66	mpd 15	1 ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3057   class class class wbr 5150  ‘cfv 6551  (class class class)co 7424  ℝcr 11143  1c1 11145   + caddc 11147   ≤ cle 11285   − cmin 11480  ℤcz 12594  ℤ_≥cuz 12858  ...cfz 13522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-er 8729  df-en 8969  df-dom 8970  df-sdom 8971  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-n0 12509  df-z 12595  df-uz 12859  df-fz 13523

This theorem is referenced by:  monoord2  14036  sermono  14037  climub  15646  isercolllem1  15649  climsup  15654  dvfsumlem3  25981  emcllem7  26952  lmdvg  33559  monoords  44681  iblspltprt  45363  itgspltprt  45369  fourierdlem11  45508  fourierdlem12  45509  fourierdlem15  45512  fourierdlem50  45546  fourierdlem79  45575