MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlimrege0 Structured version   Visualization version   GIF version

Theorem rlimrege0 14915
Description: The limit of a sequence of complex numbers with nonnegative real part has nonnegative real part. (Contributed by Mario Carneiro, 10-May-2016.)
Hypotheses
Ref Expression
rlimcld2.1 (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
rlimcld2.2 (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐶)
rlimrege0.4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
rlimrege0.5 ((𝜑𝑥𝐴) → 0 ≤ (ℜ‘𝐵))
Assertion
Ref Expression
rlimrege0 (𝜑 → 0 ≤ (ℜ‘𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rlimrege0
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimcld2.1 . . 3 (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
2 rlimcld2.2 . . 3 (𝜑 → (𝑥𝐴𝐵) ⇝𝑟 𝐶)
3 ssrab2 4032 . . . 4 {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ⊆ ℂ
43a1i 11 . . 3 (𝜑 → {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ⊆ ℂ)
5 eldifi 4079 . . . . . 6 (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑦 ∈ ℂ)
65adantl 485 . . . . 5 ((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → 𝑦 ∈ ℂ)
76recld 14532 . . . 4 ((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → (ℜ‘𝑦) ∈ ℝ)
8 fveq2 6643 . . . . . . . . . 10 (𝑤 = 𝑦 → (ℜ‘𝑤) = (ℜ‘𝑦))
98breq2d 5051 . . . . . . . . 9 (𝑤 = 𝑦 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝑦)))
109notbid 321 . . . . . . . 8 (𝑤 = 𝑦 → (¬ 0 ≤ (ℜ‘𝑤) ↔ ¬ 0 ≤ (ℜ‘𝑦)))
11 notrab 4255 . . . . . . . 8 (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) = {𝑤 ∈ ℂ ∣ ¬ 0 ≤ (ℜ‘𝑤)}
1210, 11elrab2 3660 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) ↔ (𝑦 ∈ ℂ ∧ ¬ 0 ≤ (ℜ‘𝑦)))
1312simprbi 500 . . . . . 6 (𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → ¬ 0 ≤ (ℜ‘𝑦))
1413adantl 485 . . . . 5 ((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → ¬ 0 ≤ (ℜ‘𝑦))
15 0re 10620 . . . . . 6 0 ∈ ℝ
16 ltnle 10697 . . . . . 6 (((ℜ‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘𝑦) < 0 ↔ ¬ 0 ≤ (ℜ‘𝑦)))
177, 15, 16sylancl 589 . . . . 5 ((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → ((ℜ‘𝑦) < 0 ↔ ¬ 0 ≤ (ℜ‘𝑦)))
1814, 17mpbird 260 . . . 4 ((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → (ℜ‘𝑦) < 0)
197, 18negelrpd 12401 . . 3 ((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → -(ℜ‘𝑦) ∈ ℝ+)
207renegcld 11044 . . . . 5 ((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) → -(ℜ‘𝑦) ∈ ℝ)
2120adantr 484 . . . 4 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ∈ ℝ)
22 elrabi 3652 . . . . . . 7 (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 𝑧 ∈ ℂ)
2322adantl 485 . . . . . 6 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑧 ∈ ℂ)
246adantr 484 . . . . . 6 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 𝑦 ∈ ℂ)
2523, 24subcld 10974 . . . . 5 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (𝑧𝑦) ∈ ℂ)
2625recld 14532 . . . 4 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘(𝑧𝑦)) ∈ ℝ)
2725abscld 14775 . . . 4 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (abs‘(𝑧𝑦)) ∈ ℝ)
28 0red 10621 . . . . . 6 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 0 ∈ ℝ)
2923recld 14532 . . . . . 6 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘𝑧) ∈ ℝ)
3024recld 14532 . . . . . 6 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘𝑦) ∈ ℝ)
31 fveq2 6643 . . . . . . . . . 10 (𝑤 = 𝑧 → (ℜ‘𝑤) = (ℜ‘𝑧))
3231breq2d 5051 . . . . . . . . 9 (𝑤 = 𝑧 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝑧)))
3332elrab 3657 . . . . . . . 8 (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ↔ (𝑧 ∈ ℂ ∧ 0 ≤ (ℜ‘𝑧)))
3433simprbi 500 . . . . . . 7 (𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 0 ≤ (ℜ‘𝑧))
3534adantl 485 . . . . . 6 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → 0 ≤ (ℜ‘𝑧))
3628, 29, 30, 35lesub1dd 11233 . . . . 5 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (0 − (ℜ‘𝑦)) ≤ ((ℜ‘𝑧) − (ℜ‘𝑦)))
37 df-neg 10850 . . . . . 6 -(ℜ‘𝑦) = (0 − (ℜ‘𝑦))
3837a1i 11 . . . . 5 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) = (0 − (ℜ‘𝑦)))
3923, 24resubd 14554 . . . . 5 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘(𝑧𝑦)) = ((ℜ‘𝑧) − (ℜ‘𝑦)))
4036, 38, 393brtr4d 5071 . . . 4 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ≤ (ℜ‘(𝑧𝑦)))
4125releabsd 14790 . . . 4 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → (ℜ‘(𝑧𝑦)) ≤ (abs‘(𝑧𝑦)))
4221, 26, 27, 40, 41letrd 10774 . . 3 (((𝜑𝑦 ∈ (ℂ ∖ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})) ∧ 𝑧 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)}) → -(ℜ‘𝑦) ≤ (abs‘(𝑧𝑦)))
43 fveq2 6643 . . . . 5 (𝑤 = 𝐵 → (ℜ‘𝑤) = (ℜ‘𝐵))
4443breq2d 5051 . . . 4 (𝑤 = 𝐵 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝐵)))
45 rlimrege0.4 . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
46 rlimrege0.5 . . . 4 ((𝜑𝑥𝐴) → 0 ≤ (ℜ‘𝐵))
4744, 45, 46elrabd 3659 . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})
481, 2, 4, 19, 42, 47rlimcld2 14914 . 2 (𝜑𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)})
49 fveq2 6643 . . . . 5 (𝑤 = 𝐶 → (ℜ‘𝑤) = (ℜ‘𝐶))
5049breq2d 5051 . . . 4 (𝑤 = 𝐶 → (0 ≤ (ℜ‘𝑤) ↔ 0 ≤ (ℜ‘𝐶)))
5150elrab 3657 . . 3 (𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} ↔ (𝐶 ∈ ℂ ∧ 0 ≤ (ℜ‘𝐶)))
5251simprbi 500 . 2 (𝐶 ∈ {𝑤 ∈ ℂ ∣ 0 ≤ (ℜ‘𝑤)} → 0 ≤ (ℜ‘𝐶))
5348, 52syl 17 1 (𝜑 → 0 ≤ (ℜ‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {crab 3130  cdif 3907  wss 3910   class class class wbr 5039  cmpt 5119  cfv 6328  (class class class)co 7130  supcsup 8880  cc 10512  cr 10513  0cc0 10514  +∞cpnf 10649  *cxr 10651   < clt 10652  cle 10653  cmin 10847  -cneg 10848  cre 14435  abscabs 14572  𝑟 crli 14821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-pm 8384  df-en 8485  df-dom 8486  df-sdom 8487  df-sup 8882  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-2 11678  df-3 11679  df-n0 11876  df-z 11960  df-uz 12222  df-rp 12368  df-seq 13353  df-exp 13414  df-cj 14437  df-re 14438  df-im 14439  df-sqrt 14573  df-abs 14574  df-rlim 14825
This theorem is referenced by:  rlimge0  14917
  Copyright terms: Public domain W3C validator