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Theorem monoord 12693
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 (𝜑𝑁 ∈ (ℤ𝑀))
monoord.2 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)
monoord.3 ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
Assertion
Ref Expression
monoord (𝜑 → (𝐹𝑀) ≤ (𝐹𝑁))
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.
StepHypRef Expression
1 monoord.1 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 12220 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 eleq1 2676 . . . . . 6 (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5 fveq2 6103 . . . . . . 7 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
65breq2d 4595 . . . . . 6 (𝑥 = 𝑀 → ((𝐹𝑀) ≤ (𝐹𝑥) ↔ (𝐹𝑀) ≤ (𝐹𝑀)))
74, 6imbi12d 333 . . . . 5 (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑥)) ↔ (𝑀 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑀))))
87imbi2d 329 . . . 4 (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑥))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑀)))))
9 eleq1 2676 . . . . . 6 (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
10 fveq2 6103 . . . . . . 7 (𝑥 = 𝑛 → (𝐹𝑥) = (𝐹𝑛))
1110breq2d 4595 . . . . . 6 (𝑥 = 𝑛 → ((𝐹𝑀) ≤ (𝐹𝑥) ↔ (𝐹𝑀) ≤ (𝐹𝑛)))
129, 11imbi12d 333 . . . . 5 (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑥)) ↔ (𝑛 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑛))))
1312imbi2d 329 . . . 4 (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑛)))))
14 eleq1 2676 . . . . . 6 (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
15 fveq2 6103 . . . . . . 7 (𝑥 = (𝑛 + 1) → (𝐹𝑥) = (𝐹‘(𝑛 + 1)))
1615breq2d 4595 . . . . . 6 (𝑥 = (𝑛 + 1) → ((𝐹𝑀) ≤ (𝐹𝑥) ↔ (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1))))
1714, 16imbi12d 333 . . . . 5 (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑥)) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1)))))
1817imbi2d 329 . . . 4 (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1))))))
19 eleq1 2676 . . . . . 6 (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
20 fveq2 6103 . . . . . . 7 (𝑥 = 𝑁 → (𝐹𝑥) = (𝐹𝑁))
2120breq2d 4595 . . . . . 6 (𝑥 = 𝑁 → ((𝐹𝑀) ≤ (𝐹𝑥) ↔ (𝐹𝑀) ≤ (𝐹𝑁)))
2219, 21imbi12d 333 . . . . 5 (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑥)) ↔ (𝑁 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑁))))
2322imbi2d 329 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑁)))))
24 eluzfz1 12219 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
251, 24syl 17 . . . . . . . 8 (𝜑𝑀 ∈ (𝑀...𝑁))
26 monoord.2 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)
2726ralrimiva 2949 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ ℝ)
28 fveq2 6103 . . . . . . . . . 10 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
2928eleq1d 2672 . . . . . . . . 9 (𝑘 = 𝑀 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹𝑀) ∈ ℝ))
3029rspcv 3278 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ ℝ → (𝐹𝑀) ∈ ℝ))
3125, 27, 30sylc 63 . . . . . . 7 (𝜑 → (𝐹𝑀) ∈ ℝ)
3231leidd 10473 . . . . . 6 (𝜑 → (𝐹𝑀) ≤ (𝐹𝑀))
3332a1d 25 . . . . 5 (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑀)))
3433a1i 11 . . . 4 (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑀))))
35 simprl 790 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ𝑀))
36 simprr 792 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁))
37 peano2fzr 12225 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
3835, 36, 37syl2anc 691 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
3938expr 641 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
4039imim1d 80 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑛))))
41 eluzelz 11573 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
4235, 41syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ)
43 elfzuz3 12210 . . . . . . . . . . . . . 14 ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
4436, 43syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑁 ∈ (ℤ‘(𝑛 + 1)))
45 eluzp1m1 11587 . . . . . . . . . . . . 13 ((𝑛 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑛 + 1))) → (𝑁 − 1) ∈ (ℤ𝑛))
4642, 44, 45syl2anc 691 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑁 − 1) ∈ (ℤ𝑛))
47 elfzuzb 12207 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀...(𝑁 − 1)) ↔ (𝑛 ∈ (ℤ𝑀) ∧ (𝑁 − 1) ∈ (ℤ𝑛)))
4835, 46, 47sylanbrc 695 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...(𝑁 − 1)))
49 monoord.3 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
5049ralrimiva 2949 . . . . . . . . . . . 12 (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
5150adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
52 fveq2 6103 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
53 oveq1 6556 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
5453fveq2d 6107 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
5552, 54breq12d 4596 . . . . . . . . . . . 12 (𝑘 = 𝑛 → ((𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1))))
5655rspcv 3278 . . . . . . . . . . 11 (𝑛 ∈ (𝑀...(𝑁 − 1)) → (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1))))
5748, 51, 56sylc 63 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
5831adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹𝑀) ∈ ℝ)
5927adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ ℝ)
6052eleq1d 2672 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹𝑛) ∈ ℝ))
6160rspcv 3278 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ ℝ → (𝐹𝑛) ∈ ℝ))
6238, 59, 61sylc 63 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹𝑛) ∈ ℝ)
63 fveq2 6103 . . . . . . . . . . . . . 14 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
6463eleq1d 2672 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
6564rspcv 3278 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ ℝ → (𝐹‘(𝑛 + 1)) ∈ ℝ))
6636, 59, 65sylc 63 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
67 letr 10010 . . . . . . . . . . 11 (((𝐹𝑀) ∈ ℝ ∧ (𝐹𝑛) ∈ ℝ ∧ (𝐹‘(𝑛 + 1)) ∈ ℝ) → (((𝐹𝑀) ≤ (𝐹𝑛) ∧ (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1))) → (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1))))
6858, 62, 66, 67syl3anc 1318 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (((𝐹𝑀) ≤ (𝐹𝑛) ∧ (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1))) → (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1))))
6957, 68mpan2d 706 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝐹𝑀) ≤ (𝐹𝑛) → (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1))))
7069expr 641 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹𝑀) ≤ (𝐹𝑛) → (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1)))))
7170a2d 29 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1)))))
7240, 71syld 46 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1)))))
7372expcom 450 . . . . 5 (𝑛 ∈ (ℤ𝑀) → (𝜑 → ((𝑛 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1))))))
7473a2d 29 . . . 4 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹‘(𝑛 + 1))))))
758, 13, 18, 23, 34, 74uzind4 11622 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑁))))
761, 75mpcom 37 . 2 (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹𝑀) ≤ (𝐹𝑁)))
773, 76mpd 15 1 (𝜑 → (𝐹𝑀) ≤ (𝐹𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896   class class class wbr 4583  cfv 5804  (class class class)co 6549  cr 9814  1c1 9816   + caddc 9818  cle 9954  cmin 10145  cz 11254  cuz 11563  ...cfz 12197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198
This theorem is referenced by:  monoord2  12694  sermono  12695  climub  14240  isercolllem1  14243  climsup  14248  dvfsumlem3  23595  emcllem7  24528  lmdvg  29327  monoords  38452  iblspltprt  38865  itgspltprt  38871  fourierdlem11  39011  fourierdlem12  39012  fourierdlem15  39015  fourierdlem50  39049  fourierdlem79  39078
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