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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 2735 | . . 3 ⊢ (⊤ → (AbsVal‘ℂfld) = (AbsVal‘ℂfld)) | |
2 | cnfldbas 21385 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld)) |
4 | cnfldadd 21387 | . . . 4 ⊢ + = (+g‘ℂfld) | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → + = (+g‘ℂfld)) |
6 | cnfldmul 21389 | . . . 4 ⊢ · = (.r‘ℂfld) | |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → · = (.r‘ℂfld)) |
8 | cnfld0 21422 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
9 | 8 | a1i 11 | . . 3 ⊢ (⊤ → 0 = (0g‘ℂfld)) |
10 | cnring 21420 | . . . 4 ⊢ ℂfld ∈ Ring | |
11 | 10 | a1i 11 | . . 3 ⊢ (⊤ → ℂfld ∈ Ring) |
12 | absf 15372 | . . . 4 ⊢ abs:ℂ⟶ℝ | |
13 | 12 | a1i 11 | . . 3 ⊢ (⊤ → abs:ℂ⟶ℝ) |
14 | abs0 15320 | . . . 4 ⊢ (abs‘0) = 0 | |
15 | 14 | a1i 11 | . . 3 ⊢ (⊤ → (abs‘0) = 0) |
16 | absgt0 15359 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 ≠ 0 ↔ 0 < (abs‘𝑥))) | |
17 | 16 | biimpa 476 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → 0 < (abs‘𝑥)) |
18 | 17 | 3adant1 1129 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → 0 < (abs‘𝑥)) |
19 | absmul 15329 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) | |
20 | 19 | ad2ant2r 747 | . . . 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 15365 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 + 𝑦)) ≤ ((abs‘𝑥) + (abs‘𝑦))) | |
23 | 22 | ad2ant2r 747 | . . . 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 20829 | . 2 ⊢ (⊤ → abs ∈ (AbsVal‘ℂfld)) |
26 | 25 | mptru 1543 | 1 ⊢ abs ∈ (AbsVal‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 ⊤wtru 1537 ∈ wcel 2105 ≠ wne 2937 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 + caddc 11155 · cmul 11157 < clt 11292 ≤ cle 11293 abscabs 15269 Basecbs 17244 +gcplusg 17297 .rcmulr 17298 0gc0g 17485 Ringcrg 20250 AbsValcabv 20825 ℂfldccnfld 21381 |
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-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 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 ax-addf 11231 ax-mulf 11232 |
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-tp 4635 df-op 4637 df-uni 4912 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-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-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-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 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-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-ico 13389 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-abv 20826 df-cnfld 21382 |
This theorem is referenced by: cnnrg 24816 cnindmet 25209 qabsabv 27687 |
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