Theorem ser3ge0 10995

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 10454	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5698	. . . . 5 ⊢ (𝑤 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑀))
5	4	breq2d 4143	. . . 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 5698	. . . . 5 ⊢ (𝑤 = 𝑗 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑗))
8	7	breq2d 4143	. . . 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 5698	. . . . 5 ⊢ (𝑤 = (𝑗 + 1) → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘(𝑗 + 1)))
11	10	breq2d 4143	. . . 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 5698	. . . . 5 ⊢ (𝑤 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑤) = (seq𝑀( + , 𝐹)‘𝑁))
14	13	breq2d 4143	. . . 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 5698	. . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
17	16	breq2d 4143	. . . . . 6 ⊢ (𝑘 = 𝑀 → (0 ≤ (𝐹‘𝑘) ↔ 0 ≤ (𝐹‘𝑀)))
18		ser3ge0.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘))
19	18	ralrimiva 2623	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)0 ≤ (𝐹‘𝑘))
20		eluzfz1 10453	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
21	1, 20	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
22	17, 19, 21	rspcdva 2934	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝐹‘𝑀))
23		eluzel2 9943	. . . . . . 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 8312	. . . . . . 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 10921	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
29	22, 28	breqtrrd 4159	. . . 4 ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀))
30	29	a1i 9	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑀)))
31		eqid 2238	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
32	31, 24, 25, 27	seqf 10923	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶ℝ)
33	32	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶ℝ)
34		elfzouz 10576	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (ℤ≥‘𝑀))
35	34	ad2antlr 493	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 𝑗 ∈ (ℤ≥‘𝑀))
36	33, 35	ffvelcdmd 5847	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ)
37		fveq2 5698	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑗 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑗 + 1)))
38	37	eleq1d 2307	. . . . . . . . . 10 ⊢ (𝑘 = (𝑗 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑗 + 1)) ∈ ℝ))
39	25	ralrimiva 2623	. . . . . . . . . . 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 10000	. . . . . . . . . . . 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 2934	. . . . . . . . 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 4143	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (0 ≤ (𝐹‘𝑘) ↔ 0 ≤ (𝐹‘(𝑗 + 1))))
48	19	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → ∀𝑘 ∈ (𝑀...𝑁)0 ≤ (𝐹‘𝑘))
49		fzofzp1 10663	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁))
50	49	ad2antlr 493	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (𝑗 + 1) ∈ (𝑀...𝑁))
51	47, 48, 50	rspcdva 2934	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ (𝐹‘(𝑗 + 1)))
52	36, 45, 46, 51	addge0d 8858	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → 0 ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
53	25	adantlr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
54	53	adantlr 481	. . . . . . . 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 10924	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 0 ≤ (seq𝑀( + , 𝐹)‘𝑗)) → (seq𝑀( + , 𝐹)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
57	52, 56	breqtrrd 4159	. . . . . 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 10676	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)))
62	3, 61	mpcom 36	1 ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528   class class class wbr 4131  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088  ℝcr 8185  0cc0 8186  1c1 8187   + caddc 8189   ≤ cle 8368  ℤcz 9656  ℤ_≥cuz 9938  ...cfz 10429  ..^cfzo 10567  seqcseq 10906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-ltadd 8302

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-inn 9315  df-n0 9576  df-z 9657  df-uz 9939  df-fz 10430  df-fzo 10568  df-seqfrec 10907

This theorem is used by:  ser3le  10996