Theorem climserile 10183

Description: The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by Jim Kingdon, 22-Aug-2021.)

Hypotheses

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

Assertion

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

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

Proof of Theorem climserile

Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clim2ser.1	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		climserile.2	. 2 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
3		climserile.3	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)
4	2, 1	syl6eleq 2171	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
5		eluzel2 8624	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
6	4, 5	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		climserile.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
8	1, 6, 7	iserfre 9454	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℝ):𝑍⟶ℝ)
9		cnex 7097	. . . . . . 7 ⊢ ℂ ∈ V
10	9	a1i 9	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
11		ax-resscn 7068	. . . . . . 7 ⊢ ℝ ⊆ ℂ
12	11	a1i 9	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
13	1	eleq2i 2145	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀))
14	13, 7	sylan2br 282	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
15		readdcl 7099	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 + 𝑥) ∈ ℝ)
16	15	adantl 271	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ)) → (𝑘 + 𝑥) ∈ ℝ)
17		addcl 7098	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 + 𝑥) ∈ ℂ)
18	17	adantl 271	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ)
19	6, 10, 12, 14, 16, 18	iseqss 9446	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℝ) = seq𝑀( + , 𝐹, ℂ))
20	19	feq1d 5054	. . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹, ℝ):𝑍⟶ℝ ↔ seq𝑀( + , 𝐹, ℂ):𝑍⟶ℝ))
21	8, 20	mpbid 145	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℝ)
22	21	ffvelrnda 5323	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℂ)‘𝑗) ∈ ℝ)
23	1	peano2uzs 8672	. . . . 5 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
24		fveq2 5198	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑗 + 1)))
25	24	breq2d 3797	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → (0 ≤ (𝐹‘𝑘) ↔ 0 ≤ (𝐹‘(𝑗 + 1))))
26	25	imbi2d 228	. . . . . . 7 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → 0 ≤ (𝐹‘𝑘)) ↔ (𝜑 → 0 ≤ (𝐹‘(𝑗 + 1)))))
27		climserile.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))
28	27	expcom 114	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → 0 ≤ (𝐹‘𝑘)))
29	26, 28	vtoclga 2664	. . . . . 6 ⊢ ((𝑗 + 1) ∈ 𝑍 → (𝜑 → 0 ≤ (𝐹‘(𝑗 + 1))))
30	29	impcom 123	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ 𝑍) → 0 ≤ (𝐹‘(𝑗 + 1)))
31	23, 30	sylan2 280	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 0 ≤ (𝐹‘(𝑗 + 1)))
32	24	eleq1d 2147	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑗 + 1)) ∈ ℝ))
33	32	imbi2d 228	. . . . . . . 8 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → (𝐹‘𝑘) ∈ ℝ) ↔ (𝜑 → (𝐹‘(𝑗 + 1)) ∈ ℝ)))
34	7	expcom 114	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → (𝐹‘𝑘) ∈ ℝ))
35	33, 34	vtoclga 2664	. . . . . . 7 ⊢ ((𝑗 + 1) ∈ 𝑍 → (𝜑 → (𝐹‘(𝑗 + 1)) ∈ ℝ))
36	35	impcom 123	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
37	23, 36	sylan2 280	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘(𝑗 + 1)) ∈ ℝ)
38	22, 37	addge01d 7633	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (0 ≤ (𝐹‘(𝑗 + 1)) ↔ (seq𝑀( + , 𝐹, ℂ)‘𝑗) ≤ ((seq𝑀( + , 𝐹, ℂ)‘𝑗) + (𝐹‘(𝑗 + 1)))))
39	31, 38	mpbid 145	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℂ)‘𝑗) ≤ ((seq𝑀( + , 𝐹, ℂ)‘𝑗) + (𝐹‘(𝑗 + 1))))
40		simpr 108	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
41	40, 1	syl6eleq 2171	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
42	9	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ℂ ∈ V)
43	14	adantlr 460	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
44	43	recnd 7147	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
45	17	adantl 271	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 + 𝑥) ∈ ℂ)
46	41, 42, 44, 45	iseqp1 9445	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℂ)‘(𝑗 + 1)) = ((seq𝑀( + , 𝐹, ℂ)‘𝑗) + (𝐹‘(𝑗 + 1))))
47	39, 46	breqtrrd 3811	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹, ℂ)‘𝑗) ≤ (seq𝑀( + , 𝐹, ℂ)‘(𝑗 + 1)))
48	1, 2, 3, 22, 47	climub 10182	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ≤ 𝐴)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ⊆ wss 2973   class class class wbr 3785  ⟶wf 4918  ‘cfv 4922  (class class class)co 5532  ℂcc 6979  ℝcr 6980  0cc0 6981  1c1 6982   + caddc 6984   ≤ cle 7154  ℤcz 8351  ℤ_≥cuz 8619  seqcseq 9431   ⇝ cli 10117

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-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  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-rmo 2356  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-if 3352  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-po 4051  df-iso 4052  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-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-rp 8735  df-fz 9030  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-clim 10118

This theorem is referenced by: (None)