Theorem monoord 14005

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 13516	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2820	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀))
6	5	breq2d 5160	. . . . . 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 2820	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
10		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛))
11	10	breq2d 5160	. . . . . 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 2820	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
15		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
16	15	breq2d 5160	. . . . . 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 2820	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
20		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐹‘𝑥) = (𝐹‘𝑁))
21	20	breq2d 5160	. . . . . 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 6891	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
25	24	eleq1d 2817	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑀) ∈ ℝ))
26		monoord.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
27	26	ralrimiva 3145	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
28		eluzfz1 13515	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
29	1, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
30	25, 27, 29	rspcdva 3613	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ)
31	30	leidd 11787	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑀))
32	31	a1d 25	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀)))
33		peano2fzr 13521	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
34	33	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
35	34	expr 456	. . . . . 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 6891	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
38		fvoveq1 7435	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
39	37, 38	breq12d 5161	. . . . . . 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 3145	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
42	41	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
43		simprl 768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
44		eluzelz 12839	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
45	43, 44	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
46		simprr 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
47		elfzuz3 13505	. . . . . . . . . 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 12855	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑁 − 1) ∈ (ℤ≥‘𝑛))
50	45, 48, 49	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑁 − 1) ∈ (ℤ≥‘𝑛))
51		elfzuzb 13502	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑁 − 1) ∈ (ℤ≥‘𝑛)))
52	43, 50, 51	sylanbrc 582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...(𝑁 − 1)))
53	39, 42, 52	rspcdva 3613	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
54	30	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑀) ∈ ℝ)
55	37	eleq1d 2817	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑛) ∈ ℝ))
56	27	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ)
57	55, 56, 34	rspcdva 3613	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑛) ∈ ℝ)
58		fveq2 6891	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
59	58	eleq1d 2817	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
60	59, 56, 46	rspcdva 3613	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
61		letr 11315	. . . . . . 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 1370	. . . . . 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 691	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝐹‘𝑀) ≤ (𝐹‘𝑛) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))
64	36, 63	animpimp2impd 843	. . . 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 12901	. . 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 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060   class class class wbr 5148  ‘cfv 6543  (class class class)co 7412  ℝcr 11115  1c1 11117   + caddc 11119   ≤ cle 11256   − cmin 11451  ℤcz 12565  ℤ_≥cuz 12829  ...cfz 13491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-n0 12480  df-z 12566  df-uz 12830  df-fz 13492

This theorem is referenced by:  monoord2  14006  sermono  14007  climub  15615  isercolllem1  15618  climsup  15623  dvfsumlem3  25883  emcllem7  26847  lmdvg  33397  monoords  44466  iblspltprt  45148  itgspltprt  45154  fourierdlem11  45293  fourierdlem12  45294  fourierdlem15  45297  fourierdlem50  45331  fourierdlem79  45360