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	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
serge0.2	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
serge0.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘))

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.

Step	Hyp	Ref	Expression
1		serge0.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		serge0.2	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
3		serge0.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘))
4		breq2 4790	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑘) → (0 ≤ 𝑥 ↔ 0 ≤ (𝐹‘𝑘)))
5	4	elrab 3515	. . . 4 ⊢ ((𝐹‘𝑘) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((𝐹‘𝑘) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑘)))
6	2, 3, 5	sylanbrc 572	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
7		breq2 4790	. . . . . 6 ⊢ (𝑥 = 𝑘 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑘))
8	7	elrab 3515	. . . . 5 ⊢ (𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ (𝑘 ∈ ℝ ∧ 0 ≤ 𝑘))
9		breq2 4790	. . . . . 6 ⊢ (𝑥 = 𝑦 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑦))
10	9	elrab 3515	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦))
11		readdcl 10221	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑘 + 𝑦) ∈ ℝ)
12	11	ad2ant2r 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))
14	13	an4s 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)))
16	15	elrab 3515	. . . . . 6 ⊢ ((𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((𝑘 + 𝑦) ∈ ℝ ∧ 0 ≤ (𝑘 + 𝑦)))
17	12, 14, 16	sylanbrc 572	. . . . 5 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
18	8, 10, 17	syl2anb 585	. . . 4 ⊢ ((𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∧ 𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
19	18	adantl 467	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∧ 𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
20	1, 6, 19	seqcl 13028	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
21		breq2 4790	. . . 4 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) → (0 ≤ 𝑥 ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)))
22	21	elrab 3515	. . 3 ⊢ ((seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((seq𝑀( + , 𝐹)‘𝑁) ∈ ℝ ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)))
23	22	simprbi 484	. 2 ⊢ ((seq𝑀( + , 𝐹)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))
24	20, 23	syl 17	1 ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 382   ∈ wcel 2145  {crab 3065   class class class wbr 4786  ‘cfv 6031  (class class class)co 6793  ℝcr 10137  0cc0 10138   + caddc 10141   ≤ cle 10277  ℤ_≥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