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Theorem serige0 9789
 Description: A finite sum of nonnegative terms is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
Hypotheses
Ref Expression
serige0.1 (𝜑𝑁 ∈ (ℤ𝑀))
serige0.2 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
serige0.3 ((𝜑𝑘 ∈ (ℤ𝑀)) → 0 ≤ (𝐹𝑘))
Assertion
Ref Expression
serige0 (𝜑 → 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁))
Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑁   𝜑,𝑘

Proof of Theorem serige0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 serige0.1 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzel2 8919 . . . . . 6 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
31, 2syl 14 . . . . 5 (𝜑𝑀 ∈ ℤ)
4 cnex 7369 . . . . . 6 ℂ ∈ V
54a1i 9 . . . . 5 (𝜑 → ℂ ∈ V)
6 ssrab2 3090 . . . . . . 7 {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ⊆ ℝ
7 ax-resscn 7340 . . . . . . 7 ℝ ⊆ ℂ
86, 7sstri 3019 . . . . . 6 {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ⊆ ℂ
98a1i 9 . . . . 5 (𝜑 → {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ⊆ ℂ)
10 serige0.2 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
11 serige0.3 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑀)) → 0 ≤ (𝐹𝑘))
12 breq2 3815 . . . . . . 7 (𝑥 = (𝐹𝑘) → (0 ≤ 𝑥 ↔ 0 ≤ (𝐹𝑘)))
1312elrab 2759 . . . . . 6 ((𝐹𝑘) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((𝐹𝑘) ∈ ℝ ∧ 0 ≤ (𝐹𝑘)))
1410, 11, 13sylanbrc 408 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
15 breq2 3815 . . . . . . . 8 (𝑥 = 𝑘 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑘))
1615elrab 2759 . . . . . . 7 (𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ (𝑘 ∈ ℝ ∧ 0 ≤ 𝑘))
17 breq2 3815 . . . . . . . 8 (𝑥 = 𝑦 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑦))
1817elrab 2759 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦))
19 readdcl 7371 . . . . . . . . 9 ((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑘 + 𝑦) ∈ ℝ)
2019ad2ant2r 493 . . . . . . . 8 (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑘 + 𝑦) ∈ ℝ)
21 addge0 7832 . . . . . . . . 9 (((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (0 ≤ 𝑘 ∧ 0 ≤ 𝑦)) → 0 ≤ (𝑘 + 𝑦))
2221an4s 553 . . . . . . . 8 (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → 0 ≤ (𝑘 + 𝑦))
23 breq2 3815 . . . . . . . . 9 (𝑥 = (𝑘 + 𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (𝑘 + 𝑦)))
2423elrab 2759 . . . . . . . 8 ((𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((𝑘 + 𝑦) ∈ ℝ ∧ 0 ≤ (𝑘 + 𝑦)))
2520, 22, 24sylanbrc 408 . . . . . . 7 (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
2616, 18, 25syl2anb 285 . . . . . 6 ((𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∧ 𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
2726adantl 271 . . . . 5 ((𝜑 ∧ (𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∧ 𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
28 addcl 7370 . . . . . 6 ((𝑘 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑘 + 𝑦) ∈ ℂ)
2928adantl 271 . . . . 5 ((𝜑 ∧ (𝑘 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑘 + 𝑦) ∈ ℂ)
303, 5, 9, 14, 27, 29iseqss 9760 . . . 4 (𝜑 → seq𝑀( + , 𝐹, {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) = seq𝑀( + , 𝐹, ℂ))
3130fveq1d 5255 . . 3 (𝜑 → (seq𝑀( + , 𝐹, {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})‘𝑁) = (seq𝑀( + , 𝐹, ℂ)‘𝑁))
321, 14, 27iseqcl 9756 . . 3 (𝜑 → (seq𝑀( + , 𝐹, {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
3331, 32eqeltrrd 2160 . 2 (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
34 breq2 3815 . . . 4 (𝑥 = (seq𝑀( + , 𝐹, ℂ)‘𝑁) → (0 ≤ 𝑥 ↔ 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁)))
3534elrab 2759 . . 3 ((seq𝑀( + , 𝐹, ℂ)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((seq𝑀( + , 𝐹, ℂ)‘𝑁) ∈ ℝ ∧ 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁)))
3635simprbi 269 . 2 ((seq𝑀( + , 𝐹, ℂ)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} → 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁))
3733, 36syl 14 1 (𝜑 → 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1434  {crab 2357  Vcvv 2612   ⊆ wss 2984   class class class wbr 3811  ‘cfv 4969  (class class class)co 5591  ℂcc 7251  ℝcr 7252  0cc0 7253   + caddc 7256   ≤ cle 7426  ℤcz 8646  ℤ≥cuz 8914  seqcseq 9740 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-addcom 7348  ax-addass 7350  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-0id 7356  ax-rnegex 7357  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-ltadd 7364 This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-ilim 4160  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-frec 6088  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-inn 8317  df-n0 8566  df-z 8647  df-uz 8915  df-iseq 9741 This theorem is referenced by:  serile  9790
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