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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 2737 | . . 3 ⊢ (⊤ → (AbsVal‘ℂfld) = (AbsVal‘ℂfld)) | |
2 | cnfldbas 20681 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld)) |
4 | cnfldadd 20682 | . . . 4 ⊢ + = (+g‘ℂfld) | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → + = (+g‘ℂfld)) |
6 | cnfldmul 20683 | . . . 4 ⊢ · = (.r‘ℂfld) | |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → · = (.r‘ℂfld)) |
8 | cnfld0 20702 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
9 | 8 | a1i 11 | . . 3 ⊢ (⊤ → 0 = (0g‘ℂfld)) |
10 | cnring 20700 | . . . 4 ⊢ ℂfld ∈ Ring | |
11 | 10 | a1i 11 | . . 3 ⊢ (⊤ → ℂfld ∈ Ring) |
12 | absf 15125 | . . . 4 ⊢ abs:ℂ⟶ℝ | |
13 | 12 | a1i 11 | . . 3 ⊢ (⊤ → abs:ℂ⟶ℝ) |
14 | abs0 15073 | . . . 4 ⊢ (abs‘0) = 0 | |
15 | 14 | a1i 11 | . . 3 ⊢ (⊤ → (abs‘0) = 0) |
16 | absgt0 15112 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 ≠ 0 ↔ 0 < (abs‘𝑥))) | |
17 | 16 | biimpa 477 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → 0 < (abs‘𝑥)) |
18 | 17 | 3adant1 1129 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → 0 < (abs‘𝑥)) |
19 | absmul 15082 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) | |
20 | 19 | ad2ant2r 744 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) |
21 | 20 | 3adant1 1129 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) |
22 | abstri 15118 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 + 𝑦)) ≤ ((abs‘𝑥) + (abs‘𝑦))) | |
23 | 22 | ad2ant2r 744 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 + 𝑦)) ≤ ((abs‘𝑥) + (abs‘𝑦))) |
24 | 23 | 3adant1 1129 | . . 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 20160 | . 2 ⊢ (⊤ → abs ∈ (AbsVal‘ℂfld)) |
26 | 25 | mptru 1547 | 1 ⊢ abs ∈ (AbsVal‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1540 ⊤wtru 1541 ∈ wcel 2105 ≠ wne 2940 class class class wbr 5086 ⟶wf 6461 ‘cfv 6465 (class class class)co 7316 ℂcc 10948 ℝcr 10949 0cc0 10950 + caddc 10953 · cmul 10955 < clt 11088 ≤ cle 11089 abscabs 15021 Basecbs 16986 +gcplusg 17036 .rcmulr 17037 0gc0g 17224 Ringcrg 19855 AbsValcabv 20156 ℂfldccnfld 20677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027 ax-pre-sup 11028 ax-addf 11029 ax-mulf 11030 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4850 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-om 7759 df-1st 7877 df-2nd 7878 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-1o 8345 df-er 8547 df-map 8666 df-en 8783 df-dom 8784 df-sdom 8785 df-fin 8786 df-sup 9277 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-div 11712 df-nn 12053 df-2 12115 df-3 12116 df-4 12117 df-5 12118 df-6 12119 df-7 12120 df-8 12121 df-9 12122 df-n0 12313 df-z 12399 df-dec 12517 df-uz 12662 df-rp 12810 df-ico 13164 df-fz 13319 df-seq 13801 df-exp 13862 df-cj 14886 df-re 14887 df-im 14888 df-sqrt 15022 df-abs 15023 df-struct 16922 df-sets 16939 df-slot 16957 df-ndx 16969 df-base 16987 df-plusg 17049 df-mulr 17050 df-starv 17051 df-tset 17055 df-ple 17056 df-ds 17058 df-unif 17059 df-0g 17226 df-mgm 18400 df-sgrp 18449 df-mnd 18460 df-grp 18653 df-minusg 18654 df-cmn 19460 df-mgp 19793 df-ring 19857 df-cring 19858 df-abv 20157 df-cnfld 20678 |
This theorem is referenced by: cnnrg 24024 cnindmet 24406 qabsabv 26857 |
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