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Theorem ser3ge0 10503
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 10018 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 5511 . . . . 5 (𝑤 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑀))
54breq2d 4012 . . . 4 (𝑤 = 𝑀 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑀)))
65imbi2d 230 . . 3 (𝑤 = 𝑀 → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀))))
7 fveq2 5511 . . . . 5 (𝑤 = 𝑗 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑗))
87breq2d 4012 . . . 4 (𝑤 = 𝑗 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)))
98imbi2d 230 . . 3 (𝑤 = 𝑗 → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑗))))
10 fveq2 5511 . . . . 5 (𝑤 = (𝑗 + 1) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘(𝑗 + 1)))
1110breq2d 4012 . . . 4 (𝑤 = (𝑗 + 1) → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))))
1211imbi2d 230 . . 3 (𝑤 = (𝑗 + 1) → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))))
13 fveq2 5511 . . . . 5 (𝑤 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑁))
1413breq2d 4012 . . . 4 (𝑤 = 𝑁 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)))
1514imbi2d 230 . . 3 (𝑤 = 𝑁 → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))))
16 fveq2 5511 . . . . . . 7 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
1716breq2d 4012 . . . . . 6 (𝑘 = 𝑀 → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹𝑀)))
18 ser3ge0.3 . . . . . . 7 ((𝜑𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹𝑘))
1918ralrimiva 2550 . . . . . 6 (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)0 ≤ (𝐹𝑘))
20 eluzfz1 10017 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
211, 20syl 14 . . . . . 6 (𝜑𝑀 ∈ (𝑀...𝑁))
2217, 19, 21rspcdva 2846 . . . . 5 (𝜑 → 0 ≤ (𝐹𝑀))
23 eluzel2 9522 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
241, 23syl 14 . . . . . 6 (𝜑𝑀 ∈ ℤ)
25 ser3ge0.2 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
26 readdcl 7928 . . . . . . 7 ((𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑘 + 𝑣) ∈ ℝ)
2726adantl 277 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑘 + 𝑣) ∈ ℝ)
2824, 25, 27seq3-1 10446 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
2922, 28breqtrrd 4028 . . . 4 (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀))
3029a1i 9 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀)))
31 eqid 2177 . . . . . . . . . . 11 (ℤ𝑀) = (ℤ𝑀)
3231, 24, 25, 27seqf 10447 . . . . . . . . . 10 (𝜑 → seq𝑀( + , 𝐹):(ℤ𝑀)⟶ℝ)
3332ad2antrr 488 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → seq𝑀( + , 𝐹):(ℤ𝑀)⟶ℝ)
34 elfzouz 10137 . . . . . . . . . 10 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (ℤ𝑀))
3534ad2antlr 489 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 𝑗 ∈ (ℤ𝑀))
3633, 35ffvelcdmd 5648 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ)
37 fveq2 5511 . . . . . . . . . . 11 (𝑘 = (𝑗 + 1) → (𝐹𝑘) = (𝐹‘(𝑗 + 1)))
3837eleq1d 2246 . . . . . . . . . 10 (𝑘 = (𝑗 + 1) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(𝑗 + 1)) ∈ ℝ))
3925ralrimiva 2550 . . . . . . . . . . 11 (𝜑 → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ ℝ)
4039adantr 276 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ𝑀)(𝐹𝑘) ∈ ℝ)
41 peano2uz 9572 . . . . . . . . . . . 12 (𝑗 ∈ (ℤ𝑀) → (𝑗 + 1) ∈ (ℤ𝑀))
4234, 41syl 14 . . . . . . . . . . 11 (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (ℤ𝑀))
4342adantl 277 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ𝑀))
4438, 40, 43rspcdva 2846 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
4544adantr 276 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
46 simpr 110 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (seq𝑀( + , 𝐹)‘𝑗))
4737breq2d 4012 . . . . . . . . 9 (𝑘 = (𝑗 + 1) → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹‘(𝑗 + 1))))
4819ad2antrr 488 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → ∀𝑘 ∈ (𝑀...𝑁)0 ≤ (𝐹𝑘))
49 fzofzp1 10213 . . . . . . . . . 10 (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁))
5049ad2antlr 489 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (𝑗 + 1) ∈ (𝑀...𝑁))
5147, 48, 50rspcdva 2846 . . . . . . . 8 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (𝐹‘(𝑗 + 1)))
5236, 45, 46, 51addge0d 8469 . . . . . . 7 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
5325adantlr 477 . . . . . . . . 9 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
5453adantlr 477 . . . . . . . 8 ((((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)
5526adantl 277 . . . . . . . 8 ((((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) ∧ (𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑘 + 𝑣) ∈ ℝ)
5635, 54, 55seq3p1 10448 . . . . . . 7 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (seq𝑀( + , 𝐹)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
5752, 56breqtrrd 4028 . . . . . 6 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))
5857ex 115 . . . . 5 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (0 ≤ (seq𝑀( + , 𝐹)‘𝑗) → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))))
5958expcom 116 . . . 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 10225 . 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 104   = wceq 1353  wcel 2148  wral 2455   class class class wbr 4000  wf 5208  cfv 5212  (class class class)co 5869  cr 7801  0cc0 7802  1c1 7803   + caddc 7805  cle 7983  cz 9242  cuz 9517  ...cfz 9995  ..^cfzo 10128  seqcseq 10431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-fzo 10129  df-seqfrec 10432
This theorem is referenced by:  ser3le  10504
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