Theorem ser3ge0 10922

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	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
ser3ge0.2	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
ser3ge0.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘))

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.

Step	Hyp	Ref	Expression
1		ser3ge0.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 10386	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5675	. . . . 5 ⊢ (𝑤 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑀))
5	4	breq2d 4126	. . . 4 ⊢ (𝑤 = 𝑀 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑀)))
6	5	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑀 → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀))))
7		fveq2 5675	. . . . 5 ⊢ (𝑤 = 𝑗 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑗))
8	7	breq2d 4126	. . . 4 ⊢ (𝑤 = 𝑗 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)))
9	8	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑗 → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑗))))
10		fveq2 5675	. . . . 5 ⊢ (𝑤 = (𝑗 + 1) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘(𝑗 + 1)))
11	10	breq2d 4126	. . . 4 ⊢ (𝑤 = (𝑗 + 1) → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))))
12	11	imbi2d 230	. . 3 ⊢ (𝑤 = (𝑗 + 1) → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))))
13		fveq2 5675	. . . . 5 ⊢ (𝑤 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑁))
14	13	breq2d 4126	. . . 4 ⊢ (𝑤 = 𝑁 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑤) ↔ 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)))
15	14	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑤)) ↔ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))))
16		fveq2 5675	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
17	16	breq2d 4126	. . . . . 6 ⊢ (𝑘 = 𝑀 → (0 ≤ (𝐹‘𝑘) ↔ 0 ≤ (𝐹‘𝑀)))
18		ser3ge0.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘))
19	18	ralrimiva 2617	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)0 ≤ (𝐹‘𝑘))
20		eluzfz1 10385	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
21	1, 20	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
22	17, 19, 21	rspcdva 2928	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘𝑀))
23		eluzel2 9876	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
24	1, 23	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
25		ser3ge0.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
26		readdcl 8269	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑘 + 𝑣) ∈ ℝ)
27	26	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑘 + 𝑣) ∈ ℝ)
28	24, 25, 27	seq3-1 10848	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
29	22, 28	breqtrrd 4142	. . . 4 ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀))
30	29	a1i 9	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀)))
31		eqid 2234	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
32	31, 24, 25, 27	seqf 10850	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶ℝ)
33	32	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶ℝ)
34		elfzouz 10507	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (ℤ≥‘𝑀))
35	34	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 𝑗 ∈ (ℤ≥‘𝑀))
36	33, 35	ffvelcdmd 5818	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ)
37		fveq2 5675	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑗 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑗 + 1)))
38	37	eleq1d 2303	. . . . . . . . . 10 ⊢ (𝑘 = (𝑗 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑗 + 1)) ∈ ℝ))
39	25	ralrimiva 2617	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ ℝ)
40	39	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ∀𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ ℝ)
41		peano2uz 9933	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ (ℤ≥‘𝑀))
42	34, 41	syl 14	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (ℤ≥‘𝑀))
43	42	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ≥‘𝑀))
44	38, 40, 43	rspcdva 2928	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
45	44	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
46		simpr 110	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (seq𝑀( + , 𝐹)‘𝑗))
47	37	breq2d 4126	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (0 ≤ (𝐹‘𝑘) ↔ 0 ≤ (𝐹‘(𝑗 + 1))))
48	19	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → ∀𝑘 ∈ (𝑀...𝑁)0 ≤ (𝐹‘𝑘))
49		fzofzp1 10594	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁))
50	49	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (𝑗 + 1) ∈ (𝑀...𝑁))
51	47, 48, 50	rspcdva 2928	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (𝐹‘(𝑗 + 1)))
52	36, 45, 46, 51	addge0d 8813	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
53	25	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
54	53	adantlr 477	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
55	26	adantl 277	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) ∧ (𝑘 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑘 + 𝑣) ∈ ℝ)
56	35, 54, 55	seq3p1 10851	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (seq𝑀( + , 𝐹)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
57	52, 56	breqtrrd 4142	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))
58	57	ex 115	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (0 ≤ (seq𝑀( + , 𝐹)‘𝑗) → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1))))
59	58	expcom 116	. . . 4 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝜑 → (0 ≤ (seq𝑀( + , 𝐹)‘𝑗) → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))))
60	59	a2d 26	. . 3 ⊢ (𝑗 ∈ (𝑀..^𝑁) → ((𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))))
61	6, 9, 12, 15, 30, 60	fzind2 10607	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)))
62	3, 61	mpcom 36	1 ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ∀wral 2522   class class class wbr 4114  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  ℝcr 8142  0cc0 8143  1c1 8144   + caddc 8146   ≤ cle 8325  ℤcz 9594  ℤ_≥cuz 9871  ...cfz 10361  ..^cfzo 10498  seqcseq 10833

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834

This theorem is referenced by:  ser3le  10923