Theorem serige0 9473

Description: A finite sum of nonnegative terms is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)

Hypotheses

Ref	Expression
serige0.1	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
serige0.2	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
serige0.3	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 0 ≤ (𝐹‘𝑘))

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.

Step	Hyp	Ref	Expression
1		serige0.1	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzel2 8624	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4		cnex 7097	. . . . . 6 ⊢ ℂ ∈ V
5	4	a1i 9	. . . . 5 ⊢ (𝜑 → ℂ ∈ V)
6		ssrab2 3079	. . . . . . 7 ⊢ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ⊆ ℝ
7		ax-resscn 7068	. . . . . . 7 ⊢ ℝ ⊆ ℂ
8	6, 7	sstri 3008	. . . . . 6 ⊢ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ⊆ ℂ
9	8	a1i 9	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ⊆ ℂ)
10		serige0.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
11		serige0.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 0 ≤ (𝐹‘𝑘))
12		breq2 3789	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → (0 ≤ 𝑥 ↔ 0 ≤ (𝐹‘𝑘)))
13	12	elrab 2749	. . . . . 6 ⊢ ((𝐹‘𝑘) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((𝐹‘𝑘) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑘)))
14	10, 11, 13	sylanbrc 408	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
15		breq2 3789	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑘))
16	15	elrab 2749	. . . . . . 7 ⊢ (𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ (𝑘 ∈ ℝ ∧ 0 ≤ 𝑘))
17		breq2 3789	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑦))
18	17	elrab 2749	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦))
19		readdcl 7099	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑘 + 𝑦) ∈ ℝ)
20	19	ad2ant2r 492	. . . . . . . 8 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑘 + 𝑦) ∈ ℝ)
21		addge0 7555	. . . . . . . . 9 ⊢ (((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (0 ≤ 𝑘 ∧ 0 ≤ 𝑦)) → 0 ≤ (𝑘 + 𝑦))
22	21	an4s 552	. . . . . . . 8 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → 0 ≤ (𝑘 + 𝑦))
23		breq2 3789	. . . . . . . . 9 ⊢ (𝑥 = (𝑘 + 𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (𝑘 + 𝑦)))
24	23	elrab 2749	. . . . . . . 8 ⊢ ((𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((𝑘 + 𝑦) ∈ ℝ ∧ 0 ≤ (𝑘 + 𝑦)))
25	20, 22, 24	sylanbrc 408	. . . . . . 7 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
26	16, 18, 25	syl2anb 285	. . . . . 6 ⊢ ((𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∧ 𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
27	26	adantl 271	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∧ 𝑦 ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})) → (𝑘 + 𝑦) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
28		addcl 7098	. . . . . 6 ⊢ ((𝑘 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑘 + 𝑦) ∈ ℂ)
29	28	adantl 271	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑘 + 𝑦) ∈ ℂ)
30	3, 5, 9, 14, 27, 29	iseqss 9446	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹, {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥}) = seq𝑀( + , 𝐹, ℂ))
31	30	fveq1d 5200	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹, {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})‘𝑁) = (seq𝑀( + , 𝐹, ℂ)‘𝑁))
32		reex 7107	. . . . . 6 ⊢ ℝ ∈ V
33	32	rabex 3922	. . . . 5 ⊢ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∈ V
34	33	a1i 9	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ∈ V)
35	1, 34, 14, 27	iseqcl 9443	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹, {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
36	31, 35	eqeltrrd 2156	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥})
37		breq2 3789	. . . 4 ⊢ (𝑥 = (seq𝑀( + , 𝐹, ℂ)‘𝑁) → (0 ≤ 𝑥 ↔ 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁)))
38	37	elrab 2749	. . 3 ⊢ ((seq𝑀( + , 𝐹, ℂ)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} ↔ ((seq𝑀( + , 𝐹, ℂ)‘𝑁) ∈ ℝ ∧ 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁)))
39	38	simprbi 269	. 2 ⊢ ((seq𝑀( + , 𝐹, ℂ)‘𝑁) ∈ {𝑥 ∈ ℝ ∣ 0 ≤ 𝑥} → 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁))
40	36, 39	syl 14	1 ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1433  {crab 2352  Vcvv 2601   ⊆ wss 2973   class class class wbr 3785  ‘cfv 4922  (class class class)co 5532  ℂcc 6979  ℝcr 6980  0cc0 6981   + caddc 6984   ≤ cle 7154  ℤcz 8351  ℤ_≥cuz 8619  seqcseq 9431

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  serile  9474