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Theorem serge0 13062
Description: A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
serge0.1 (𝜑𝑁 ∈ (ℤ𝑀))
serge0.2 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)
serge0.3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹𝑘))
Assertion
Ref Expression
serge0 (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))
Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑁   𝜑,𝑘

Proof of Theorem serge0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 serge0.1 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 serge0.2 . . . 4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)
3 serge0.3 . . . 4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹𝑘))
4 breq2 4790 . . . . 5 (𝑥 = (𝐹𝑘) → (0 ≤ 𝑥 ↔ 0 ≤ (𝐹𝑘)))
54elrab 3515 . . . 4 ((𝐹𝑘) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((𝐹𝑘) ∈ ℝ ∧ 0 ≤ (𝐹𝑘)))
62, 3, 5sylanbrc 572 . . 3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
7 breq2 4790 . . . . . 6 (𝑥 = 𝑘 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑘))
87elrab 3515 . . . . 5 (𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ (𝑘 ∈ ℝ ∧ 0 ≤ 𝑘))
9 breq2 4790 . . . . . 6 (𝑥 = 𝑦 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑦))
109elrab 3515 . . . . 5 (𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦))
11 readdcl 10221 . . . . . . 7 ((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑘 + 𝑦) ∈ ℝ)
1211ad2ant2r 741 . . . . . 6 (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑘 + 𝑦) ∈ ℝ)
13 addge0 10719 . . . . . . 7 (((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (0 ≤ 𝑘 ∧ 0 ≤ 𝑦)) → 0 ≤ (𝑘 + 𝑦))
1413an4s 639 . . . . . 6 (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → 0 ≤ (𝑘 + 𝑦))
15 breq2 4790 . . . . . . 7 (𝑥 = (𝑘 + 𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (𝑘 + 𝑦)))
1615elrab 3515 . . . . . 6 ((𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((𝑘 + 𝑦) ∈ ℝ ∧ 0 ≤ (𝑘 + 𝑦)))
1712, 14, 16sylanbrc 572 . . . . 5 (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
188, 10, 17syl2anb 585 . . . 4 ((𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∧ 𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
1918adantl 467 . . 3 ((𝜑 ∧ (𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∧ 𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
201, 6, 19seqcl 13028 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
21 breq2 4790 . . . 4 (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) → (0 ≤ 𝑥 ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)))
2221elrab 3515 . . 3 ((seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((seq𝑀( + , 𝐹)‘𝑁) ∈ ℝ ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)))
2322simprbi 484 . 2 ((seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))
2420, 23syl 17 1 (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  {crab 3065   class class class wbr 4786  cfv 6031  (class class class)co 6793  cr 10137  0cc0 10138   + caddc 10141  cle 10277  cuz 11888  ...cfz 12533  seqcseq 13008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-seq 13009
This theorem is referenced by:  serle  13063
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