Theorem iserabs 11952

Description: Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.)

Hypotheses

Ref	Expression
iserabs.1	⊢ 𝑍 = (ℤ≥‘𝑀)
iserabs.2	⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)
iserabs.3	⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)
iserabs.5	⊢ (𝜑 → 𝑀 ∈ ℤ)
iserabs.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
iserabs.7	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘)))

Assertion

Ref	Expression
iserabs	⊢ (𝜑 → (abs‘𝐴) ≤ 𝐵)

Distinct variable groups:   𝑘,𝐹   𝑘,𝐺   𝑘,𝑀   𝜑,𝑘   𝑘,𝑍

Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)

Proof of Theorem iserabs

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iserabs.1	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		iserabs.5	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		iserabs.2	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)
4		zex 9423	. . . . . . 7 ⊢ ℤ ∈ V
5		uzssz 9710	. . . . . . 7 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
6	4, 5	ssexi 4201	. . . . . 6 ⊢ (ℤ≥‘𝑀) ∈ V
7	1, 6	eqeltri 2282	. . . . 5 ⊢ 𝑍 ∈ V
8	7	mptex 5838	. . . 4 ⊢ (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) ∈ V
9	8	a1i 9	. . 3 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) ∈ V)
10		iserabs.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
11	1, 2, 10	serf 10672	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
12	11	ffvelcdmda 5743	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
13		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
14	12	abscld 11658	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑀( + , 𝐹)‘𝑛)) ∈ ℝ)
15		2fveq3 5608	. . . . 5 ⊢ (𝑚 = 𝑛 → (abs‘(seq𝑀( + , 𝐹)‘𝑚)) = (abs‘(seq𝑀( + , 𝐹)‘𝑛)))
16		eqid 2209	. . . . 5 ⊢ (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))
17	15, 16	fvmptg 5683	. . . 4 ⊢ ((𝑛 ∈ 𝑍 ∧ (abs‘(seq𝑀( + , 𝐹)‘𝑛)) ∈ ℝ) → ((𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))‘𝑛) = (abs‘(seq𝑀( + , 𝐹)‘𝑛)))
18	13, 14, 17	syl2anc 411	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))‘𝑛) = (abs‘(seq𝑀( + , 𝐹)‘𝑛)))
19	1, 3, 9, 2, 12, 18	climabs 11797	. 2 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) ⇝ (abs‘𝐴))
20		iserabs.3	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)
21	18, 14	eqeltrd 2286	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))‘𝑛) ∈ ℝ)
22		iserabs.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘)))
23	10	abscld 11658	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
24	22, 23	eqeltrd 2286	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
25	1, 2, 24	serfre 10673	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
26	25	ffvelcdmda 5743	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
27	2	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ ℤ)
28		eluzelz 9699	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
29	28, 1	eleq2s 2304	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
30	29	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
31	27, 30	fzfigd 10620	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀...𝑛) ∈ Fin)
32		elfzuz 10185	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
33	32, 1	eleqtrrdi 2303	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
34	33, 10	sylan2 286	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ)
35	34	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ)
36	31, 35	fsumabs 11942	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘Σ𝑘 ∈ (𝑀...𝑛)(𝐹‘𝑘)) ≤ Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐹‘𝑘)))
37		eqidd 2210	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐹‘𝑘))
38	1	eleq2i 2276	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
39	38	biimpi 120	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
40	39	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑀))
41	1	eleq2i 2276	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀))
42	41, 10	sylan2br 288	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
43	42	adantlr 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
44	37, 40, 43	fsum3ser 11874	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)(𝐹‘𝑘) = (seq𝑀( + , 𝐹)‘𝑛))
45	44	fveq2d 5607	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘Σ𝑘 ∈ (𝑀...𝑛)(𝐹‘𝑘)) = (abs‘(seq𝑀( + , 𝐹)‘𝑛)))
46	22	adantlr 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘)))
47	41, 46	sylan2br 288	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘)))
48	23	adantlr 477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
49	41, 48	sylan2br 288	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
50	49	recnd 8143	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℂ)
51	47, 40, 50	fsum3ser 11874	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐹‘𝑘)) = (seq𝑀( + , 𝐺)‘𝑛))
52	36, 45, 51	3brtr3d 4093	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑀( + , 𝐹)‘𝑛)) ≤ (seq𝑀( + , 𝐺)‘𝑛))
53	18, 52	eqbrtrd 4084	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))‘𝑛) ≤ (seq𝑀( + , 𝐺)‘𝑛))
54	1, 2, 19, 20, 21, 26, 53	climle 11811	1 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180  Vcvv 2779   class class class wbr 4062   ↦ cmpt 4124  ‘cfv 5294  (class class class)co 5974  ℂcc 7965  ℝcr 7966   + caddc 7970   ≤ cle 8150  ℤcz 9414  ℤ_≥cuz 9690  ...cfz 10172  seqcseq 10636  abscabs 11474   ⇝ cli 11755  Σcsu 11830

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086  ax-caucvg 8087

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-isom 5303  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-frec 6507  df-1o 6532  df-oadd 6536  df-er 6650  df-en 6858  df-dom 6859  df-fin 6860  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-n0 9338  df-z 9415  df-uz 9691  df-q 9783  df-rp 9818  df-fz 10173  df-fzo 10307  df-seqfrec 10637  df-exp 10728  df-ihash 10965  df-cj 11319  df-re 11320  df-im 11321  df-rsqrt 11475  df-abs 11476  df-clim 11756  df-sumdc 11831

This theorem is referenced by:  eftlub  12167