Theorem monoord 13938

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 13449	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2825	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀))
6	5	breq2d 5117	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑀)))
7	4, 6	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀))))
8	7	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀)))))
9		eleq1 2825	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
10		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛))
11	10	breq2d 5117	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑛)))
12	9, 11	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛))))
13	12	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛)))))
14		eleq1 2825	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
15		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
16	15	breq2d 5117	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))
17	14, 16	imbi12d 344	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1)))))
18	17	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))))
19		eleq1 2825	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
20		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐹‘𝑥) = (𝐹‘𝑁))
21	20	breq2d 5117	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑁)))
22	19, 21	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁))))
23	22	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁)))))
24		fveq2 6842	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
25	24	eleq1d 2822	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑀) ∈ ℝ))
26		monoord.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
27	26	ralrimiva 3143	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
28		eluzfz1 13448	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
29	1, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
30	25, 27, 29	rspcdva 3582	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ)
31	30	leidd 11721	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑀))
32	31	a1d 25	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀)))
33		peano2fzr 13454	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
34	33	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
35	34	expr 457	. . . . . 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 6842	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
38		fvoveq1 7380	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
39	37, 38	breq12d 5118	. . . . . . 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 3143	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
42	41	adantr 481	. . . . . . 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 12773	. . . . . . . . . 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 13438	. . . . . . . . . 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 12789	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑛))
50	45, 48, 49	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑁 − 1) ∈ (ℤ≥‘𝑛))
51		elfzuzb 13435	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑁 − 1) ∈ (ℤ≥‘𝑛)))
52	43, 50, 51	sylanbrc 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...(𝑁 − 1)))
53	39, 42, 52	rspcdva 3582	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
54	30	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑀) ∈ ℝ)
55	37	eleq1d 2822	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑛) ∈ ℝ))
56	27	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
57	55, 56, 34	rspcdva 3582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑛) ∈ ℝ)
58		fveq2 6842	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
59	58	eleq1d 2822	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
60	59, 56, 46	rspcdva 3582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
61		letr 11249	. . . . . . 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 1371	. . . . . 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 12835	. . 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  ℝcr 11050  1c1 11052   + caddc 11054   ≤ cle 11190   − cmin 11385  ℤcz 12499  ℤ_≥cuz 12763  ...cfz 13424

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425

This theorem is referenced by:  monoord2  13939  sermono  13940  climub  15546  isercolllem1  15549  climsup  15554  dvfsumlem3  25392  emcllem7  26351  lmdvg  32534  monoords  43521  iblspltprt  44204  itgspltprt  44210  fourierdlem11  44349  fourierdlem12  44350  fourierdlem15  44353  fourierdlem50  44387  fourierdlem79  44416