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Theorem ser3ge0 10297
 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
ser3ge0.1 (𝜑𝑁 ∈ (ℤ𝑀))
ser3ge0.2 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
ser3ge0.3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹𝑘))
Assertion
Ref Expression
ser3ge0 (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))
Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑁   𝜑,𝑘

Proof of Theorem ser3ge0
Dummy variables 𝑗 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ser3ge0.1 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 9819 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 5421 . . . . 5 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑀))
54breq2d 3941 . . . 4 (𝑤 = 𝑀 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑀)))
65imbi2d 229 . . 3 (𝑤 = 𝑀 → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀))))
7 fveq2 5421 . . . . 5 (𝑤 = 𝑗 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑗))
87breq2d 3941 . . . 4 (𝑤 = 𝑗 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)))
98imbi2d 229 . . 3 (𝑤 = 𝑗 → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑗))))
10 fveq2 5421 . . . . 5 (𝑤 = (𝑗 + 1) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘(𝑗 + 1)))
1110breq2d 3941 . . . 4 (𝑤 = (𝑗 + 1) → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))))
1211imbi2d 229 . . 3 (𝑤 = (𝑗 + 1) → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))))
13 fveq2 5421 . . . . 5 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑁))
1413breq2d 3941 . . . 4 (𝑤 = 𝑁 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)))
1514imbi2d 229 . . 3 (𝑤 = 𝑁 → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))))
16 fveq2 5421 . . . . . . 7 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
1716breq2d 3941 . . . . . 6 (𝑘 = 𝑀 → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹𝑀)))
18 ser3ge0.3 . . . . . . 7 ((𝜑𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹𝑘))
1918ralrimiva 2505 . . . . . 6 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)0 ≤ (𝐹𝑘))
20 eluzfz1 9818 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
211, 20syl 14 . . . . . 6 (𝜑𝑀 ∈ (𝑀...𝑁))
2217, 19, 21rspcdva 2794 . . . . 5 (𝜑 → 0 ≤ (𝐹𝑀))
23 eluzel2 9338 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
241, 23syl 14 . . . . . 6 (𝜑𝑀 ∈ ℤ)
25 ser3ge0.2 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
26 readdcl 7753 . . . . . . 7 ((𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑘 + 𝑣) ∈ ℝ)
2726adantl 275 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑘 + 𝑣) ∈ ℝ)
2824, 25, 27seq3-1 10240 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
2922, 28breqtrrd 3956 . . . 4 (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀))
3029a1i 9 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀)))
31 eqid 2139 . . . . . . . . . . 11 (ℤ𝑀) = (ℤ𝑀)
3231, 24, 25, 27seqf 10241 . . . . . . . . . 10 (𝜑 → seq𝑀( + , 𝐹):(ℤ𝑀)⟶ℝ)
3332ad2antrr 479 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → seq𝑀( + , 𝐹):(ℤ𝑀)⟶ℝ)
34 elfzouz 9935 . . . . . . . . . 10 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (ℤ𝑀))
3534ad2antlr 480 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 𝑗 ∈ (ℤ𝑀))
3633, 35ffvelrnd 5556 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ)
37 fveq2 5421 . . . . . . . . . . 11 (𝑘 = (𝑗 + 1) → (𝐹𝑘) = (𝐹‘(𝑗 + 1)))
3837eleq1d 2208 . . . . . . . . . 10 (𝑘 = (𝑗 + 1) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(𝑗 + 1)) ∈ ℝ))
3925ralrimiva 2505 . . . . . . . . . . 11 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ ℝ)
4039adantr 274 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ ℝ)
41 peano2uz 9385 . . . . . . . . . . . 12 (𝑗 ∈ (ℤ𝑀) → (𝑗 + 1) ∈ (ℤ𝑀))
4234, 41syl 14 . . . . . . . . . . 11 (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (ℤ𝑀))
4342adantl 275 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ𝑀))
4438, 40, 43rspcdva 2794 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
4544adantr 274 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
46 simpr 109 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (seq𝑀( + , 𝐹)‘𝑗))
4737breq2d 3941 . . . . . . . . 9 (𝑘 = (𝑗 + 1) → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹‘(𝑗 + 1))))
4819ad2antrr 479 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → ∀𝑘 ∈ (𝑀...𝑁)0 ≤ (𝐹𝑘))
49 fzofzp1 10011 . . . . . . . . . 10 (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁))
5049ad2antlr 480 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (𝑗 + 1) ∈ (𝑀...𝑁))
5147, 48, 50rspcdva 2794 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (𝐹‘(𝑗 + 1)))
5236, 45, 46, 51addge0d 8291 . . . . . . 7 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
5325adantlr 468 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
5453adantlr 468 . . . . . . . 8 ((((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
5526adantl 275 . . . . . . . 8 ((((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) ∧ (𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑘 + 𝑣) ∈ ℝ)
5635, 54, 55seq3p1 10242 . . . . . . 7 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (seq𝑀( + , 𝐹)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
5752, 56breqtrrd 3956 . . . . . 6 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))
5857ex 114 . . . . 5 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (0 ≤ (seq𝑀( + , 𝐹)‘𝑗) → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))))
5958expcom 115 . . . 4 (𝑗 ∈ (𝑀..^𝑁) → (𝜑 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑗) → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))))
6059a2d 26 . . 3 (𝑗 ∈ (𝑀..^𝑁) → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))))
616, 9, 12, 15, 30, 60fzind2 10023 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)))
623, 61mpcom 36 1 (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2416   class class class wbr 3929  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  ℝcr 7626  0cc0 7627  1c1 7628   + caddc 7630   ≤ cle 7808  ℤcz 9061  ℤ≥cuz 9333  ...cfz 9797  ..^cfzo 9926  seqcseq 10225 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743 This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798  df-fzo 9927  df-seqfrec 10226 This theorem is referenced by:  ser3le  10298
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