Theorem climserle 14241

Description: The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)

Hypotheses

Ref	Expression
clim2ser.1	⊢ 𝑍 = (ℤ≥‘𝑀)
climserle.2	⊢ (𝜑 → 𝑁 ∈ 𝑍)
climserle.3	⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)
climserle.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
climserle.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))

Assertion

Ref	Expression
climserle	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴)

Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑀   𝑘,𝑁   𝜑,𝑘   𝑘,𝑍

Proof of Theorem climserle

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clim2ser.1	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		climserle.2	. 2 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
3		climserle.3	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)
4	2, 1	syl6eleq 2698	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
5		eluzel2 11568	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
6	4, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		climserle.4	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
8	1, 6, 7	serfre 12692	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
9	8	ffvelrnda 6267	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ)
10	1	peano2uzs 11618	. . . . 5 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
11		fveq2 6103	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑗 + 1)))
12	11	breq2d 4595	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (0 ≤ (𝐹‘𝑘) ↔ 0 ≤ (𝐹‘(𝑗 + 1))))
13	12	imbi2d 329	. . . . . . 7 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → 0 ≤ (𝐹‘𝑘)) ↔ (𝜑 → 0 ≤ (𝐹‘(𝑗 + 1)))))
14		climserle.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))
15	14	expcom 450	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → 0 ≤ (𝐹‘𝑘)))
16	13, 15	vtoclga 3245	. . . . . 6 ⊢ ((𝑗 + 1) ∈ 𝑍 → (𝜑 → 0 ≤ (𝐹‘(𝑗 + 1))))
17	16	impcom 445	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ 𝑍) → 0 ≤ (𝐹‘(𝑗 + 1)))
18	10, 17	sylan2 490	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 0 ≤ (𝐹‘(𝑗 + 1)))
19	11	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑗 + 1)) ∈ ℝ))
20	19	imbi2d 329	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → (𝐹‘𝑘) ∈ ℝ) ↔ (𝜑 → (𝐹‘(𝑗 + 1)) ∈ ℝ)))
21	7	expcom 450	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → (𝐹‘𝑘) ∈ ℝ))
22	20, 21	vtoclga 3245	. . . . . . 7 ⊢ ((𝑗 + 1) ∈ 𝑍 → (𝜑 → (𝐹‘(𝑗 + 1)) ∈ ℝ))
23	22	impcom 445	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
24	10, 23	sylan2 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
25	9, 24	addge01d 10494	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (0 ≤ (𝐹‘(𝑗 + 1)) ↔ (seq𝑀( + , 𝐹)‘𝑗) ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1)))))
26	18, 25	mpbid 221	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ≤ ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
27		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
28	27, 1	syl6eleq 2698	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
29		seqp1 12678	. . . 4 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
30	28, 29	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹)‘𝑗) + (𝐹‘(𝑗 + 1))))
31	26, 30	breqtrrd 4611	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ≤ (seq𝑀( + , 𝐹)‘(𝑗 + 1)))
32	1, 2, 3, 9, 31	climub 14240	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954  ℤcz 11254  ℤ_≥cuz 11563  seqcseq 12663   ⇝ cli 14063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fl 12455  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068

This theorem is referenced by:  isumrpcl  14414  ege2le3  14659  prmreclem6  15463  ioombl1lem4  23136  rge0scvg  29323