Theorem iserabs 11659

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 9354	. . . . . . 7 ⊢ ℤ ∈ V
5		uzssz 9640	. . . . . . 7 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
6	4, 5	ssexi 4172	. . . . . 6 ⊢ (ℤ≥‘𝑀) ∈ V
7	1, 6	eqeltri 2269	. . . . 5 ⊢ 𝑍 ∈ V
8	7	mptex 5791	. . . 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 10594	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
12	11	ffvelcdmda 5700	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
13		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
14	12	abscld 11365	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑀( + , 𝐹)‘𝑛)) ∈ ℝ)
15		2fveq3 5566	. . . . 5 ⊢ (𝑚 = 𝑛 → (abs‘(seq𝑀( + , 𝐹)‘𝑚)) = (abs‘(seq𝑀( + , 𝐹)‘𝑛)))
16		eqid 2196	. . . . 5 ⊢ (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))
17	15, 16	fvmptg 5640	. . . 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 11504	. 2 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) ⇝ (abs‘𝐴))
20		iserabs.3	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)
21	18, 14	eqeltrd 2273	. 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 11365	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
24	22, 23	eqeltrd 2273	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
25	1, 2, 24	serfre 10595	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
26	25	ffvelcdmda 5700	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
27	2	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ ℤ)
28		eluzelz 9629	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
29	28, 1	eleq2s 2291	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
30	29	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
31	27, 30	fzfigd 10542	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀...𝑛) ∈ Fin)
32		elfzuz 10115	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
33	32, 1	eleqtrrdi 2290	. . . . . . 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 11649	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘Σ𝑘 ∈ (𝑀...𝑛)(𝐹‘𝑘)) ≤ Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐹‘𝑘)))
37		eqidd 2197	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐹‘𝑘))
38	1	eleq2i 2263	. . . . . . . 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 2263	. . . . . . . 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 11581	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)(𝐹‘𝑘) = (seq𝑀( + , 𝐹)‘𝑛))
45	44	fveq2d 5565	. . . 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 8074	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℂ)
51	47, 40, 50	fsum3ser 11581	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐹‘𝑘)) = (seq𝑀( + , 𝐺)‘𝑛))
52	36, 45, 51	3brtr3d 4065	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑀( + , 𝐹)‘𝑛)) ≤ (seq𝑀( + , 𝐺)‘𝑛))
53	18, 52	eqbrtrd 4056	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))‘𝑛) ≤ (seq𝑀( + , 𝐺)‘𝑛))
54	1, 2, 19, 20, 21, 26, 53	climle 11518	1 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   class class class wbr 4034   ↦ cmpt 4095  ‘cfv 5259  (class class class)co 5925  ℂcc 7896  ℝcr 7897   + caddc 7901   ≤ cle 8081  ℤcz 9345  ℤ_≥cuz 9620  ...cfz 10102  seqcseq 10558  abscabs 11181   ⇝ cli 11462  Σcsu 11537

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016  ax-arch 8017  ax-caucvg 8018

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-n0 9269  df-z 9346  df-uz 9621  df-q 9713  df-rp 9748  df-fz 10103  df-fzo 10237  df-seqfrec 10559  df-exp 10650  df-ihash 10887  df-cj 11026  df-re 11027  df-im 11028  df-rsqrt 11182  df-abs 11183  df-clim 11463  df-sumdc 11538

This theorem is referenced by:  eftlub  11874