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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 20800 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld)) |
4 | cnfldadd 20801 | . . . 4 ⊢ + = (+g‘ℂfld) | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → + = (+g‘ℂfld)) |
6 | cnfldmul 20802 | . . . 4 ⊢ · = (.r‘ℂfld) | |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → · = (.r‘ℂfld)) |
8 | cnfld0 20821 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
9 | 8 | a1i 11 | . . 3 ⊢ (⊤ → 0 = (0g‘ℂfld)) |
10 | cnring 20819 | . . . 4 ⊢ ℂfld ∈ Ring | |
11 | 10 | a1i 11 | . . 3 ⊢ (⊤ → ℂfld ∈ Ring) |
12 | absf 15222 | . . . 4 ⊢ abs:ℂ⟶ℝ | |
13 | 12 | a1i 11 | . . 3 ⊢ (⊤ → abs:ℂ⟶ℝ) |
14 | abs0 15170 | . . . 4 ⊢ (abs‘0) = 0 | |
15 | 14 | a1i 11 | . . 3 ⊢ (⊤ → (abs‘0) = 0) |
16 | absgt0 15209 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 ≠ 0 ↔ 0 < (abs‘𝑥))) | |
17 | 16 | biimpa 477 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → 0 < (abs‘𝑥)) |
18 | 17 | 3adant1 1130 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → 0 < (abs‘𝑥)) |
19 | absmul 15179 | . . . . 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 1130 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) |
22 | abstri 15215 | . . . . 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 1130 | . . 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 20279 | . 2 ⊢ (⊤ → abs ∈ (AbsVal‘ℂfld)) |
26 | 25 | mptru 1548 | 1 ⊢ abs ∈ (AbsVal‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 + caddc 11054 · cmul 11056 < clt 11189 ≤ cle 11190 abscabs 15119 Basecbs 17083 +gcplusg 17133 .rcmulr 17134 0gc0g 17321 Ringcrg 19964 AbsValcabv 20275 ℂfldccnfld 20796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-ico 13270 df-fz 13425 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-plusg 17146 df-mulr 17147 df-starv 17148 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-cmn 19564 df-mgp 19897 df-ring 19966 df-cring 19967 df-abv 20276 df-cnfld 20797 |
This theorem is referenced by: cnnrg 24144 cnindmet 24526 qabsabv 26977 |
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