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Theorem monoord2 10477
Description: Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
Hypotheses
Ref Expression
monoord2.1 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
monoord2.2 ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ ℝ)
monoord2.3 ((πœ‘ ∧ π‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜(π‘˜ + 1)) ≀ (πΉβ€˜π‘˜))
Assertion
Ref Expression
monoord2 (πœ‘ β†’ (πΉβ€˜π‘) ≀ (πΉβ€˜π‘€))
Distinct variable groups:   π‘˜,𝐹   π‘˜,𝑀   π‘˜,𝑁   πœ‘,π‘˜

Proof of Theorem monoord2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 monoord2.1 . . . 4 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
2 monoord2.2 . . . . . . 7 ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ ℝ)
32renegcld 8337 . . . . . 6 ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ -(πΉβ€˜π‘˜) ∈ ℝ)
4 eqid 2177 . . . . . 6 (π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜)) = (π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))
53, 4fmptd 5671 . . . . 5 (πœ‘ β†’ (π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜)):(𝑀...𝑁)βŸΆβ„)
65ffvelcdmda 5652 . . . 4 ((πœ‘ ∧ 𝑛 ∈ (𝑀...𝑁)) β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜π‘›) ∈ ℝ)
7 monoord2.3 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜(π‘˜ + 1)) ≀ (πΉβ€˜π‘˜))
87ralrimiva 2550 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))(πΉβ€˜(π‘˜ + 1)) ≀ (πΉβ€˜π‘˜))
9 oveq1 5882 . . . . . . . . . . 11 (π‘˜ = 𝑛 β†’ (π‘˜ + 1) = (𝑛 + 1))
109fveq2d 5520 . . . . . . . . . 10 (π‘˜ = 𝑛 β†’ (πΉβ€˜(π‘˜ + 1)) = (πΉβ€˜(𝑛 + 1)))
11 fveq2 5516 . . . . . . . . . 10 (π‘˜ = 𝑛 β†’ (πΉβ€˜π‘˜) = (πΉβ€˜π‘›))
1210, 11breq12d 4017 . . . . . . . . 9 (π‘˜ = 𝑛 β†’ ((πΉβ€˜(π‘˜ + 1)) ≀ (πΉβ€˜π‘˜) ↔ (πΉβ€˜(𝑛 + 1)) ≀ (πΉβ€˜π‘›)))
1312cbvralv 2704 . . . . . . . 8 (βˆ€π‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))(πΉβ€˜(π‘˜ + 1)) ≀ (πΉβ€˜π‘˜) ↔ βˆ€π‘› ∈ (𝑀...(𝑁 βˆ’ 1))(πΉβ€˜(𝑛 + 1)) ≀ (πΉβ€˜π‘›))
148, 13sylib 122 . . . . . . 7 (πœ‘ β†’ βˆ€π‘› ∈ (𝑀...(𝑁 βˆ’ 1))(πΉβ€˜(𝑛 + 1)) ≀ (πΉβ€˜π‘›))
1514r19.21bi 2565 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜(𝑛 + 1)) ≀ (πΉβ€˜π‘›))
16 fveq2 5516 . . . . . . . . 9 (π‘˜ = (𝑛 + 1) β†’ (πΉβ€˜π‘˜) = (πΉβ€˜(𝑛 + 1)))
1716eleq1d 2246 . . . . . . . 8 (π‘˜ = (𝑛 + 1) β†’ ((πΉβ€˜π‘˜) ∈ ℝ ↔ (πΉβ€˜(𝑛 + 1)) ∈ ℝ))
182ralrimiva 2550 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘˜ ∈ (𝑀...𝑁)(πΉβ€˜π‘˜) ∈ ℝ)
1918adantr 276 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ βˆ€π‘˜ ∈ (𝑀...𝑁)(πΉβ€˜π‘˜) ∈ ℝ)
20 fzp1elp1 10075 . . . . . . . . . 10 (𝑛 ∈ (𝑀...(𝑁 βˆ’ 1)) β†’ (𝑛 + 1) ∈ (𝑀...((𝑁 βˆ’ 1) + 1)))
2120adantl 277 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (𝑛 + 1) ∈ (𝑀...((𝑁 βˆ’ 1) + 1)))
22 eluzelz 9537 . . . . . . . . . . . . . 14 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑁 ∈ β„€)
231, 22syl 14 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑁 ∈ β„€)
2423zcnd 9376 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑁 ∈ β„‚)
25 ax-1cn 7904 . . . . . . . . . . . 12 1 ∈ β„‚
26 npcan 8166 . . . . . . . . . . . 12 ((𝑁 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
2724, 25, 26sylancl 413 . . . . . . . . . . 11 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
2827oveq2d 5891 . . . . . . . . . 10 (πœ‘ β†’ (𝑀...((𝑁 βˆ’ 1) + 1)) = (𝑀...𝑁))
2928adantr 276 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (𝑀...((𝑁 βˆ’ 1) + 1)) = (𝑀...𝑁))
3021, 29eleqtrd 2256 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (𝑛 + 1) ∈ (𝑀...𝑁))
3117, 19, 30rspcdva 2847 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜(𝑛 + 1)) ∈ ℝ)
3211eleq1d 2246 . . . . . . . 8 (π‘˜ = 𝑛 β†’ ((πΉβ€˜π‘˜) ∈ ℝ ↔ (πΉβ€˜π‘›) ∈ ℝ))
33 fzssp1 10067 . . . . . . . . . 10 (𝑀...(𝑁 βˆ’ 1)) βŠ† (𝑀...((𝑁 βˆ’ 1) + 1))
3433, 28sseqtrid 3206 . . . . . . . . 9 (πœ‘ β†’ (𝑀...(𝑁 βˆ’ 1)) βŠ† (𝑀...𝑁))
3534sselda 3156 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ 𝑛 ∈ (𝑀...𝑁))
3632, 19, 35rspcdva 2847 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜π‘›) ∈ ℝ)
3731, 36lenegd 8481 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ ((πΉβ€˜(𝑛 + 1)) ≀ (πΉβ€˜π‘›) ↔ -(πΉβ€˜π‘›) ≀ -(πΉβ€˜(𝑛 + 1))))
3815, 37mpbid 147 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ -(πΉβ€˜π‘›) ≀ -(πΉβ€˜(𝑛 + 1)))
3936renegcld 8337 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ -(πΉβ€˜π‘›) ∈ ℝ)
4011negeqd 8152 . . . . . . 7 (π‘˜ = 𝑛 β†’ -(πΉβ€˜π‘˜) = -(πΉβ€˜π‘›))
4140, 4fvmptg 5593 . . . . . 6 ((𝑛 ∈ (𝑀...𝑁) ∧ -(πΉβ€˜π‘›) ∈ ℝ) β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜π‘›) = -(πΉβ€˜π‘›))
4235, 39, 41syl2anc 411 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜π‘›) = -(πΉβ€˜π‘›))
4331renegcld 8337 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ -(πΉβ€˜(𝑛 + 1)) ∈ ℝ)
4416negeqd 8152 . . . . . . 7 (π‘˜ = (𝑛 + 1) β†’ -(πΉβ€˜π‘˜) = -(πΉβ€˜(𝑛 + 1)))
4544, 4fvmptg 5593 . . . . . 6 (((𝑛 + 1) ∈ (𝑀...𝑁) ∧ -(πΉβ€˜(𝑛 + 1)) ∈ ℝ) β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜(𝑛 + 1)) = -(πΉβ€˜(𝑛 + 1)))
4630, 43, 45syl2anc 411 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜(𝑛 + 1)) = -(πΉβ€˜(𝑛 + 1)))
4738, 42, 463brtr4d 4036 . . . 4 ((πœ‘ ∧ 𝑛 ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜π‘›) ≀ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜(𝑛 + 1)))
481, 6, 47monoord 10476 . . 3 (πœ‘ β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜π‘€) ≀ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜π‘))
49 eluzfz1 10031 . . . . 5 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑀 ∈ (𝑀...𝑁))
501, 49syl 14 . . . 4 (πœ‘ β†’ 𝑀 ∈ (𝑀...𝑁))
51 fveq2 5516 . . . . . . 7 (π‘˜ = 𝑀 β†’ (πΉβ€˜π‘˜) = (πΉβ€˜π‘€))
5251eleq1d 2246 . . . . . 6 (π‘˜ = 𝑀 β†’ ((πΉβ€˜π‘˜) ∈ ℝ ↔ (πΉβ€˜π‘€) ∈ ℝ))
5352, 18, 50rspcdva 2847 . . . . 5 (πœ‘ β†’ (πΉβ€˜π‘€) ∈ ℝ)
5453renegcld 8337 . . . 4 (πœ‘ β†’ -(πΉβ€˜π‘€) ∈ ℝ)
5551negeqd 8152 . . . . 5 (π‘˜ = 𝑀 β†’ -(πΉβ€˜π‘˜) = -(πΉβ€˜π‘€))
5655, 4fvmptg 5593 . . . 4 ((𝑀 ∈ (𝑀...𝑁) ∧ -(πΉβ€˜π‘€) ∈ ℝ) β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜π‘€) = -(πΉβ€˜π‘€))
5750, 54, 56syl2anc 411 . . 3 (πœ‘ β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜π‘€) = -(πΉβ€˜π‘€))
58 eluzfz2 10032 . . . . 5 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑁 ∈ (𝑀...𝑁))
591, 58syl 14 . . . 4 (πœ‘ β†’ 𝑁 ∈ (𝑀...𝑁))
60 fveq2 5516 . . . . . . 7 (π‘˜ = 𝑁 β†’ (πΉβ€˜π‘˜) = (πΉβ€˜π‘))
6160eleq1d 2246 . . . . . 6 (π‘˜ = 𝑁 β†’ ((πΉβ€˜π‘˜) ∈ ℝ ↔ (πΉβ€˜π‘) ∈ ℝ))
6261, 18, 59rspcdva 2847 . . . . 5 (πœ‘ β†’ (πΉβ€˜π‘) ∈ ℝ)
6362renegcld 8337 . . . 4 (πœ‘ β†’ -(πΉβ€˜π‘) ∈ ℝ)
6460negeqd 8152 . . . . 5 (π‘˜ = 𝑁 β†’ -(πΉβ€˜π‘˜) = -(πΉβ€˜π‘))
6564, 4fvmptg 5593 . . . 4 ((𝑁 ∈ (𝑀...𝑁) ∧ -(πΉβ€˜π‘) ∈ ℝ) β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜π‘) = -(πΉβ€˜π‘))
6659, 63, 65syl2anc 411 . . 3 (πœ‘ β†’ ((π‘˜ ∈ (𝑀...𝑁) ↦ -(πΉβ€˜π‘˜))β€˜π‘) = -(πΉβ€˜π‘))
6748, 57, 663brtr3d 4035 . 2 (πœ‘ β†’ -(πΉβ€˜π‘€) ≀ -(πΉβ€˜π‘))
6862, 53lenegd 8481 . 2 (πœ‘ β†’ ((πΉβ€˜π‘) ≀ (πΉβ€˜π‘€) ↔ -(πΉβ€˜π‘€) ≀ -(πΉβ€˜π‘)))
6967, 68mpbird 167 1 (πœ‘ β†’ (πΉβ€˜π‘) ≀ (πΉβ€˜π‘€))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455   class class class wbr 4004   ↦ cmpt 4065  β€˜cfv 5217  (class class class)co 5875  β„‚cc 7809  β„cr 7810  1c1 7812   + caddc 7814   ≀ cle 7993   βˆ’ cmin 8128  -cneg 8129  β„€cz 9253  β„€β‰₯cuz 9528  ...cfz 10008
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-fz 10009
This theorem is referenced by: (None)
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