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Mirrors > Home > MPE Home > Th. List > iserabs | Structured version Visualization version GIF version |
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 Mario Carneiro, 27-May-2014.) |
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))) |
Ref | Expression |
---|---|
iserabs | ⊢ (𝜑 → (abs‘𝐴) ≤ 𝐵) |
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 | 1 | fvexi 6920 | . . . . 5 ⊢ 𝑍 ∈ V |
5 | 4 | mptex 7242 | . . . 4 ⊢ (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) ∈ V |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) ∈ V) |
7 | iserabs.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
8 | 1, 2, 7 | serf 14067 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
9 | 8 | ffvelcdmda 7103 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ) |
10 | 2fveq3 6911 | . . . . 5 ⊢ (𝑚 = 𝑛 → (abs‘(seq𝑀( + , 𝐹)‘𝑚)) = (abs‘(seq𝑀( + , 𝐹)‘𝑛))) | |
11 | eqid 2734 | . . . . 5 ⊢ (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) | |
12 | fvex 6919 | . . . . 5 ⊢ (abs‘(seq𝑀( + , 𝐹)‘𝑛)) ∈ V | |
13 | 10, 11, 12 | fvmpt 7015 | . . . 4 ⊢ (𝑛 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))‘𝑛) = (abs‘(seq𝑀( + , 𝐹)‘𝑛))) |
14 | 13 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))‘𝑛) = (abs‘(seq𝑀( + , 𝐹)‘𝑛))) |
15 | 1, 3, 6, 2, 9, 14 | climabs 15636 | . 2 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚))) ⇝ (abs‘𝐴)) |
16 | iserabs.3 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵) | |
17 | 9 | abscld 15471 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑀( + , 𝐹)‘𝑛)) ∈ ℝ) |
18 | 14, 17 | eqeltrd 2838 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))‘𝑛) ∈ ℝ) |
19 | iserabs.7 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘))) | |
20 | 7 | abscld 15471 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ∈ ℝ) |
21 | 19, 20 | eqeltrd 2838 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) |
22 | 1, 2, 21 | serfre 14068 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ) |
23 | 22 | ffvelcdmda 7103 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ) |
24 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) | |
25 | 24, 1 | eleqtrdi 2848 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑀)) |
26 | elfzuz 13556 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
27 | 26, 1 | eleqtrrdi 2849 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍) |
28 | 27, 7 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ) |
29 | 28 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ) |
30 | 27, 19 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘))) |
31 | 30 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘))) |
32 | 25, 29, 31 | seqabs 15846 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑀( + , 𝐹)‘𝑛)) ≤ (seq𝑀( + , 𝐺)‘𝑛)) |
33 | 14, 32 | eqbrtrd 5169 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(seq𝑀( + , 𝐹)‘𝑚)))‘𝑛) ≤ (seq𝑀( + , 𝐺)‘𝑛)) |
34 | 1, 2, 15, 16, 18, 23, 33 | climle 15672 | 1 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 + caddc 11155 ≤ cle 11293 ℤcz 12610 ℤ≥cuz 12875 ...cfz 13543 seqcseq 14038 abscabs 15269 ⇝ cli 15516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 df-sum 15719 |
This theorem is referenced by: eftlub 16141 abelthlem7 26496 |
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