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Mirrors > Home > MPE Home > Th. List > absabv | Structured version Visualization version GIF version |
Description: The regular absolute value function on the complex numbers is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
absabv | ⊢ abs ∈ (AbsVal‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2822 | . . 3 ⊢ (⊤ → (AbsVal‘ℂfld) = (AbsVal‘ℂfld)) | |
2 | cnfldbas 20543 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld)) |
4 | cnfldadd 20544 | . . . 4 ⊢ + = (+g‘ℂfld) | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → + = (+g‘ℂfld)) |
6 | cnfldmul 20545 | . . . 4 ⊢ · = (.r‘ℂfld) | |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → · = (.r‘ℂfld)) |
8 | cnfld0 20563 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
9 | 8 | a1i 11 | . . 3 ⊢ (⊤ → 0 = (0g‘ℂfld)) |
10 | cnring 20561 | . . . 4 ⊢ ℂfld ∈ Ring | |
11 | 10 | a1i 11 | . . 3 ⊢ (⊤ → ℂfld ∈ Ring) |
12 | absf 14691 | . . . 4 ⊢ abs:ℂ⟶ℝ | |
13 | 12 | a1i 11 | . . 3 ⊢ (⊤ → abs:ℂ⟶ℝ) |
14 | abs0 14639 | . . . 4 ⊢ (abs‘0) = 0 | |
15 | 14 | a1i 11 | . . 3 ⊢ (⊤ → (abs‘0) = 0) |
16 | absgt0 14678 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 ≠ 0 ↔ 0 < (abs‘𝑥))) | |
17 | 16 | biimpa 479 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → 0 < (abs‘𝑥)) |
18 | 17 | 3adant1 1126 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → 0 < (abs‘𝑥)) |
19 | absmul 14648 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) | |
20 | 19 | ad2ant2r 745 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) |
21 | 20 | 3adant1 1126 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) |
22 | abstri 14684 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 + 𝑦)) ≤ ((abs‘𝑥) + (abs‘𝑦))) | |
23 | 22 | ad2ant2r 745 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 + 𝑦)) ≤ ((abs‘𝑥) + (abs‘𝑦))) |
24 | 23 | 3adant1 1126 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 + 𝑦)) ≤ ((abs‘𝑥) + (abs‘𝑦))) |
25 | 1, 3, 5, 7, 9, 11, 13, 15, 18, 21, 24 | isabvd 19585 | . 2 ⊢ (⊤ → abs ∈ (AbsVal‘ℂfld)) |
26 | 25 | mptru 1540 | 1 ⊢ abs ∈ (AbsVal‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 abscabs 14587 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 0gc0g 16707 Ringcrg 19291 AbsValcabv 19581 ℂfldccnfld 20539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-ico 12738 df-fz 12887 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-cmn 18902 df-mgp 19234 df-ring 19293 df-cring 19294 df-abv 19582 df-cnfld 20540 |
This theorem is referenced by: cnnrg 23383 cnindmet 23760 qabsabv 26199 |
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