Theorem iserabs 12189

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 9606	. . . . . . 7 ⊢ ℤ ∈ V
5		uzssz 9895	. . . . . . 7 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
6	4, 5	ssexi 4253	. . . . . 6 ⊢ (ℤ≥‘𝑀) ∈ V
7	1, 6	eqeltri 2307	. . . . 5 ⊢ 𝑍 ∈ V
8	7	mptex 5917	. . . 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 10872	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
12	11	ffvelcdmda 5817	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
13		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
14	12	abscld 11894	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑀( + , 𝐹)‘𝑛)) ∈ ℝ)
15		2fveq3 5680	. . . . 5 ⊢ (𝑚 = 𝑛 → (abs‘(seq𝑀( + , 𝐹)‘𝑚)) = (abs‘(seq𝑀( + , 𝐹)‘𝑛)))
16		eqid 2234	. . . . 5 ⊢ (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))
17	15, 16	fvmptg 5758	. . . 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 12033	. 2 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) ⇝ (abs‘𝐴))
20		iserabs.3	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)
21	18, 14	eqeltrd 2311	. 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 11894	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
24	22, 23	eqeltrd 2311	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
25	1, 2, 24	serfre 10873	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
26	25	ffvelcdmda 5817	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
27	2	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ ℤ)
28		eluzelz 9884	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
29	28, 1	eleq2s 2329	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
30	29	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
31	27, 30	fzfigd 10820	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀...𝑛) ∈ Fin)
32		elfzuz 10377	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
33	32, 1	eleqtrrdi 2328	. . . . . . 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 12179	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘Σ𝑘 ∈ (𝑀...𝑛)(𝐹‘𝑘)) ≤ Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐹‘𝑘)))
37		eqidd 2235	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐹‘𝑘))
38	1	eleq2i 2301	. . . . . . . 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 2301	. . . . . . . 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 12111	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)(𝐹‘𝑘) = (seq𝑀( + , 𝐹)‘𝑛))
45	44	fveq2d 5679	. . . 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 8318	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℂ)
51	47, 40, 50	fsum3ser 12111	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐹‘𝑘)) = (seq𝑀( + , 𝐺)‘𝑛))
52	36, 45, 51	3brtr3d 4145	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑀( + , 𝐹)‘𝑛)) ≤ (seq𝑀( + , 𝐺)‘𝑛))
53	18, 52	eqbrtrd 4136	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))‘𝑛) ≤ (seq𝑀( + , 𝐺)‘𝑛))
54	1, 2, 19, 20, 21, 26, 53	climle 12047	1 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815   class class class wbr 4114   ↦ cmpt 4176  ‘cfv 5357  (class class class)co 6058  ℂcc 8141  ℝcr 8142   + caddc 8146   ≤ cle 8325  ℤcz 9597  ℤ_≥cuz 9874  ...cfz 10364  seqcseq 10836  abscabs 11710   ⇝ cli 11991  Σcsu 12066

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-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  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-rmo 2530  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-if 3625  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-po 4422  df-iso 4423  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-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-n0 9517  df-z 9598  df-uz 9875  df-q 9973  df-rp 10008  df-fz 10365  df-fzo 10502  df-seqfrec 10837  df-exp 10928  df-ihash 11167  df-cj 11555  df-re 11556  df-im 11557  df-rsqrt 11711  df-abs 11712  df-clim 11992  df-sumdc 12067

This theorem is referenced by:  eftlub  12404