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Theorem monoord2 10143
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 8061 . . . . . 6 ((𝜑𝑘 ∈ (𝑀...𝑁)) → -(𝐹𝑘) ∈ ℝ)
4 eqid 2115 . . . . . 6 (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘)) = (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))
53, 4fmptd 5528 . . . . 5 (𝜑 → (𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘)):(𝑀...𝑁)⟶ℝ)
65ffvelrnda 5509 . . . 4 ((𝜑𝑛 ∈ (𝑀...𝑁)) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑛) ∈ ℝ)
7 monoord2.3 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
87ralrimiva 2479 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
9 oveq1 5735 . . . . . . . . . . 11 (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
109fveq2d 5379 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
11 fveq2 5375 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
1210, 11breq12d 3908 . . . . . . . . 9 (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛)))
1312cbvralv 2628 . . . . . . . 8 (∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ↔ ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
148, 13sylib 121 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (𝑀...(𝑁 − 1))(𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
1514r19.21bi 2494 . . . . . 6 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
16 fveq2 5375 . . . . . . . . 9 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
1716eleq1d 2183 . . . . . . . 8 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ))
182ralrimiva 2479 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ ℝ)
1918adantr 272 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) ∈ ℝ)
20 fzp1elp1 9748 . . . . . . . . . 10 (𝑛 ∈ (𝑀...(𝑁 − 1)) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
2120adantl 273 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...((𝑁 − 1) + 1)))
22 eluzelz 9237 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
231, 22syl 14 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℤ)
2423zcnd 9078 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
25 ax-1cn 7638 . . . . . . . . . . . 12 1 ∈ ℂ
26 npcan 7894 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
2724, 25, 26sylancl 407 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
2827oveq2d 5744 . . . . . . . . . 10 (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
2928adantr 272 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
3021, 29eleqtrd 2193 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝑛 + 1) ∈ (𝑀...𝑁))
3117, 19, 30rspcdva 2765 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
3211eleq1d 2183 . . . . . . . 8 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹𝑛) ∈ ℝ))
33 fzssp1 9740 . . . . . . . . . 10 (𝑀...(𝑁 − 1)) ⊆ (𝑀...((𝑁 − 1) + 1))
3433, 28sseqtrid 3113 . . . . . . . . 9 (𝜑 → (𝑀...(𝑁 − 1)) ⊆ (𝑀...𝑁))
3534sselda 3063 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝑛 ∈ (𝑀...𝑁))
3632, 19, 35rspcdva 2765 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑛) ∈ ℝ)
3731, 36lenegd 8204 . . . . . 6 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛) ↔ -(𝐹𝑛) ≤ -(𝐹‘(𝑛 + 1))))
3815, 37mpbid 146 . . . . 5 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹𝑛) ≤ -(𝐹‘(𝑛 + 1)))
3936renegcld 8061 . . . . . 6 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹𝑛) ∈ ℝ)
4011negeqd 7880 . . . . . . 7 (𝑘 = 𝑛 → -(𝐹𝑘) = -(𝐹𝑛))
4140, 4fvmptg 5451 . . . . . 6 ((𝑛 ∈ (𝑀...𝑁) ∧ -(𝐹𝑛) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑛) = -(𝐹𝑛))
4235, 39, 41syl2anc 406 . . . . 5 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑛) = -(𝐹𝑛))
4331renegcld 8061 . . . . . 6 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → -(𝐹‘(𝑛 + 1)) ∈ ℝ)
4416negeqd 7880 . . . . . . 7 (𝑘 = (𝑛 + 1) → -(𝐹𝑘) = -(𝐹‘(𝑛 + 1)))
4544, 4fvmptg 5451 . . . . . 6 (((𝑛 + 1) ∈ (𝑀...𝑁) ∧ -(𝐹‘(𝑛 + 1)) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘(𝑛 + 1)) = -(𝐹‘(𝑛 + 1)))
4630, 43, 45syl2anc 406 . . . . 5 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘(𝑛 + 1)) = -(𝐹‘(𝑛 + 1)))
4738, 42, 463brtr4d 3925 . . . 4 ((𝜑𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑛) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘(𝑛 + 1)))
481, 6, 47monoord 10142 . . 3 (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑀) ≤ ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑁))
49 eluzfz1 9704 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
501, 49syl 14 . . . 4 (𝜑𝑀 ∈ (𝑀...𝑁))
51 fveq2 5375 . . . . . . 7 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
5251eleq1d 2183 . . . . . 6 (𝑘 = 𝑀 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹𝑀) ∈ ℝ))
5352, 18, 50rspcdva 2765 . . . . 5 (𝜑 → (𝐹𝑀) ∈ ℝ)
5453renegcld 8061 . . . 4 (𝜑 → -(𝐹𝑀) ∈ ℝ)
5551negeqd 7880 . . . . 5 (𝑘 = 𝑀 → -(𝐹𝑘) = -(𝐹𝑀))
5655, 4fvmptg 5451 . . . 4 ((𝑀 ∈ (𝑀...𝑁) ∧ -(𝐹𝑀) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑀) = -(𝐹𝑀))
5750, 54, 56syl2anc 406 . . 3 (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑀) = -(𝐹𝑀))
58 eluzfz2 9705 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
591, 58syl 14 . . . 4 (𝜑𝑁 ∈ (𝑀...𝑁))
60 fveq2 5375 . . . . . . 7 (𝑘 = 𝑁 → (𝐹𝑘) = (𝐹𝑁))
6160eleq1d 2183 . . . . . 6 (𝑘 = 𝑁 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹𝑁) ∈ ℝ))
6261, 18, 59rspcdva 2765 . . . . 5 (𝜑 → (𝐹𝑁) ∈ ℝ)
6362renegcld 8061 . . . 4 (𝜑 → -(𝐹𝑁) ∈ ℝ)
6460negeqd 7880 . . . . 5 (𝑘 = 𝑁 → -(𝐹𝑘) = -(𝐹𝑁))
6564, 4fvmptg 5451 . . . 4 ((𝑁 ∈ (𝑀...𝑁) ∧ -(𝐹𝑁) ∈ ℝ) → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑁) = -(𝐹𝑁))
6659, 63, 65syl2anc 406 . . 3 (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ↦ -(𝐹𝑘))‘𝑁) = -(𝐹𝑁))
6748, 57, 663brtr3d 3924 . 2 (𝜑 → -(𝐹𝑀) ≤ -(𝐹𝑁))
6862, 53lenegd 8204 . 2 (𝜑 → ((𝐹𝑁) ≤ (𝐹𝑀) ↔ -(𝐹𝑀) ≤ -(𝐹𝑁)))
6967, 68mpbird 166 1 (𝜑 → (𝐹𝑁) ≤ (𝐹𝑀))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  wral 2390   class class class wbr 3895  cmpt 3949  cfv 5081  (class class class)co 5728  cc 7545  cr 7546  1c1 7548   + caddc 7550  cle 7725  cmin 7856  -cneg 7857  cz 8958  cuz 9228  ...cfz 9683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959  df-uz 9229  df-fz 9684
This theorem is referenced by: (None)
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