Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ser3ge0 GIF version

Theorem ser3ge0 10083
 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 9595 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 5340 . . . . 5 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑀))
54breq2d 3879 . . . 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 5340 . . . . 5 (𝑤 = 𝑗 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑗))
87breq2d 3879 . . . 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 5340 . . . . 5 (𝑤 = (𝑗 + 1) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘(𝑗 + 1)))
1110breq2d 3879 . . . 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 5340 . . . . 5 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑁))
1413breq2d 3879 . . . 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 5340 . . . . . . 7 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
1716breq2d 3879 . . . . . 6 (𝑘 = 𝑀 → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹𝑀)))
18 ser3ge0.3 . . . . . . 7 ((𝜑𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹𝑘))
1918ralrimiva 2458 . . . . . 6 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)0 ≤ (𝐹𝑘))
20 eluzfz1 9594 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
211, 20syl 14 . . . . . 6 (𝜑𝑀 ∈ (𝑀...𝑁))
2217, 19, 21rspcdva 2741 . . . . 5 (𝜑 → 0 ≤ (𝐹𝑀))
23 eluzel2 9123 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
241, 23syl 14 . . . . . 6 (𝜑𝑀 ∈ ℤ)
25 ser3ge0.2 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
26 readdcl 7565 . . . . . . 7 ((𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑘 + 𝑣) ∈ ℝ)
2726adantl 272 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑘 + 𝑣) ∈ ℝ)
2824, 25, 27seq3-1 10023 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
2922, 28breqtrrd 3893 . . . 4 (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀))
3029a1i 9 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀)))
31 eqid 2095 . . . . . . . . . . 11 (ℤ𝑀) = (ℤ𝑀)
3231, 24, 25, 27seqf 10026 . . . . . . . . . 10 (𝜑 → seq𝑀( + , 𝐹):(ℤ𝑀)⟶ℝ)
3332ad2antrr 473 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → seq𝑀( + , 𝐹):(ℤ𝑀)⟶ℝ)
34 elfzouz 9711 . . . . . . . . . 10 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (ℤ𝑀))
3534ad2antlr 474 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 𝑗 ∈ (ℤ𝑀))
3633, 35ffvelrnd 5474 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ)
37 fveq2 5340 . . . . . . . . . . 11 (𝑘 = (𝑗 + 1) → (𝐹𝑘) = (𝐹‘(𝑗 + 1)))
3837eleq1d 2163 . . . . . . . . . 10 (𝑘 = (𝑗 + 1) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(𝑗 + 1)) ∈ ℝ))
3925ralrimiva 2458 . . . . . . . . . . 11 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ ℝ)
4039adantr 271 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ ℝ)
41 peano2uz 9170 . . . . . . . . . . . 12 (𝑗 ∈ (ℤ𝑀) → (𝑗 + 1) ∈ (ℤ𝑀))
4234, 41syl 14 . . . . . . . . . . 11 (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (ℤ𝑀))
4342adantl 272 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ𝑀))
4438, 40, 43rspcdva 2741 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
4544adantr 271 . . . . . . . 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 3879 . . . . . . . . 9 (𝑘 = (𝑗 + 1) → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹‘(𝑗 + 1))))
4819ad2antrr 473 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → ∀𝑘 ∈ (𝑀...𝑁)0 ≤ (𝐹𝑘))
49 fzofzp1 9787 . . . . . . . . . 10 (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁))
5049ad2antlr 474 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (𝑗 + 1) ∈ (𝑀...𝑁))
5147, 48, 50rspcdva 2741 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (𝐹‘(𝑗 + 1)))
5236, 45, 46, 51addge0d 8096 . . . . . . 7 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
5325adantlr 462 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
5453adantlr 462 . . . . . . . 8 ((((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
5526adantl 272 . . . . . . . 8 ((((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) ∧ (𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑘 + 𝑣) ∈ ℝ)
5635, 54, 55seq3p1 10030 . . . . . . 7 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (seq𝑀( + , 𝐹)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
5752, 56breqtrrd 3893 . . . . . 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 9799 . 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 1296   ∈ wcel 1445  ∀wral 2370   class class class wbr 3867  ⟶wf 5045  ‘cfv 5049  (class class class)co 5690  ℝcr 7446  0cc0 7447  1c1 7448   + caddc 7450   ≤ cle 7620  ℤcz 8848  ℤ≥cuz 9118  ...cfz 9573  ..^cfzo 9702  seqcseq 10001 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-addass 7544  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-ltadd 7558 This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-n0 8772  df-z 8849  df-uz 9119  df-fz 9574  df-fzo 9703  df-iseq 10002  df-seq3 10003 This theorem is referenced by:  ser3le  10084
 Copyright terms: Public domain W3C validator