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Theorem monoord2 10508
Description: Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
Hypotheses
Ref Expression
monoord2.1 (𝜑𝑁 ∈ (ℤ𝑀))
monoord2.2 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)
monoord2.3 ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
Assertion
Ref Expression
monoord2 (𝜑 → (𝐹𝑁) ≤ (𝐹𝑀))
Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑁   𝜑,𝑘

Proof of Theorem monoord2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 monoord2.1 . . . 4 (𝜑𝑁 ∈ (ℤ𝑀))
2 monoord2.2 . . . . . . 7 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)
32renegcld 8367 . . . . . 6 ((𝜑𝑘 ∈ (𝑀...𝑁)) → -(𝐹𝑘) ∈ ℝ)
4 eqid 2189 . . . . . 6 (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘)) = (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))
53, 4fmptd 5691 . . . . 5 (𝜑 → (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘)):(𝑀...𝑁)⟶ℝ)
65ffvelcdmda 5672 . . . 4 ((𝜑𝑛 ∈ (𝑀...𝑁)) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑛) ∈ ℝ)
7 monoord2.3 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
87ralrimiva 2563 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
9 oveq1 5903 . . . . . . . . . . 11 (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
109fveq2d 5538 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
11 fveq2 5534 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
1210, 11breq12d 4031 . . . . . . . . 9 (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛)))
1312cbvralv 2718 . . . . . . . 8 (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ↔ ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
148, 13sylib 122 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
1514r19.21bi 2578 . . . . . 6 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
16 fveq2 5534 . . . . . . . . 9 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
1716eleq1d 2258 . . . . . . . 8 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
182ralrimiva 2563 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ ℝ)
1918adantr 276 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ ℝ)
20 fzp1elp1 10105 . . . . . . . . . 10 (𝑛 ∈ (𝑀...(𝑁 − 1)) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
2120adantl 277 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
22 eluzelz 9567 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
231, 22syl 14 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℤ)
2423zcnd 9406 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
25 ax-1cn 7934 . . . . . . . . . . . 12 1 ∈ ℂ
26 npcan 8196 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
2724, 25, 26sylancl 413 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
2827oveq2d 5912 . . . . . . . . . 10 (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
2928adantr 276 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
3021, 29eleqtrd 2268 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...𝑁))
3117, 19, 30rspcdva 2861 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
3211eleq1d 2258 . . . . . . . 8 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹𝑛) ∈ ℝ))
33 fzssp1 10097 . . . . . . . . . 10 (𝑀...(𝑁 − 1)) ⊆ (𝑀...((𝑁 − 1) + 1))
3433, 28sseqtrid 3220 . . . . . . . . 9 (𝜑 → (𝑀...(𝑁 − 1)) ⊆ (𝑀...𝑁))
3534sselda 3170 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝑛 ∈ (𝑀...𝑁))
3632, 19, 35rspcdva 2861 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑛) ∈ ℝ)
3731, 36lenegd 8511 . . . . . 6 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛) ↔ -(𝐹𝑛) ≤ -(𝐹‘(𝑛 + 1))))
3815, 37mpbid 147 . . . . 5 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹𝑛) ≤ -(𝐹‘(𝑛 + 1)))
3936renegcld 8367 . . . . . 6 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹𝑛) ∈ ℝ)
4011negeqd 8182 . . . . . . 7 (𝑘 = 𝑛 → -(𝐹𝑘) = -(𝐹𝑛))
4140, 4fvmptg 5613 . . . . . 6 ((𝑛 ∈ (𝑀...𝑁) ∧ -(𝐹𝑛) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑛) = -(𝐹𝑛))
4235, 39, 41syl2anc 411 . . . . 5 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑛) = -(𝐹𝑛))
4331renegcld 8367 . . . . . 6 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹‘(𝑛 + 1)) ∈ ℝ)
4416negeqd 8182 . . . . . . 7 (𝑘 = (𝑛 + 1) → -(𝐹𝑘) = -(𝐹‘(𝑛 + 1)))
4544, 4fvmptg 5613 . . . . . 6 (((𝑛 + 1) ∈ (𝑀...𝑁) ∧ -(𝐹‘(𝑛 + 1)) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘(𝑛 + 1)) = -(𝐹‘(𝑛 + 1)))
4630, 43, 45syl2anc 411 . . . . 5 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘(𝑛 + 1)) = -(𝐹‘(𝑛 + 1)))
4738, 42, 463brtr4d 4050 . . . 4 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑛) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘(𝑛 + 1)))
481, 6, 47monoord 10507 . . 3 (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑀) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑁))
49 eluzfz1 10061 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
501, 49syl 14 . . . 4 (𝜑𝑀 ∈ (𝑀...𝑁))
51 fveq2 5534 . . . . . . 7 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
5251eleq1d 2258 . . . . . 6 (𝑘 = 𝑀 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹𝑀) ∈ ℝ))
5352, 18, 50rspcdva 2861 . . . . 5 (𝜑 → (𝐹𝑀) ∈ ℝ)
5453renegcld 8367 . . . 4 (𝜑 → -(𝐹𝑀) ∈ ℝ)
5551negeqd 8182 . . . . 5 (𝑘 = 𝑀 → -(𝐹𝑘) = -(𝐹𝑀))
5655, 4fvmptg 5613 . . . 4 ((𝑀 ∈ (𝑀...𝑁) ∧ -(𝐹𝑀) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑀) = -(𝐹𝑀))
5750, 54, 56syl2anc 411 . . 3 (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑀) = -(𝐹𝑀))
58 eluzfz2 10062 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
591, 58syl 14 . . . 4 (𝜑𝑁 ∈ (𝑀...𝑁))
60 fveq2 5534 . . . . . . 7 (𝑘 = 𝑁 → (𝐹𝑘) = (𝐹𝑁))
6160eleq1d 2258 . . . . . 6 (𝑘 = 𝑁 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹𝑁) ∈ ℝ))
6261, 18, 59rspcdva 2861 . . . . 5 (𝜑 → (𝐹𝑁) ∈ ℝ)
6362renegcld 8367 . . . 4 (𝜑 → -(𝐹𝑁) ∈ ℝ)
6460negeqd 8182 . . . . 5 (𝑘 = 𝑁 → -(𝐹𝑘) = -(𝐹𝑁))
6564, 4fvmptg 5613 . . . 4 ((𝑁 ∈ (𝑀...𝑁) ∧ -(𝐹𝑁) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑁) = -(𝐹𝑁))
6659, 63, 65syl2anc 411 . . 3 (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑁) = -(𝐹𝑁))
6748, 57, 663brtr3d 4049 . 2 (𝜑 → -(𝐹𝑀) ≤ -(𝐹𝑁))
6862, 53lenegd 8511 . 2 (𝜑 → ((𝐹𝑁) ≤ (𝐹𝑀) ↔ -(𝐹𝑀) ≤ -(𝐹𝑁)))
6967, 68mpbird 167 1 (𝜑 → (𝐹𝑁) ≤ (𝐹𝑀))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wral 2468   class class class wbr 4018  cmpt 4079  cfv 5235  (class class class)co 5896  cc 7839  cr 7840  1c1 7842   + caddc 7844  cle 8023  cmin 8158  -cneg 8159  cz 9283  cuz 9558  ...cfz 10038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-ltadd 7957
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-inn 8950  df-n0 9207  df-z 9284  df-uz 9559  df-fz 10039
This theorem is referenced by: (None)
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