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Mirrors > Home > MPE Home > Th. List > absneg | Structured version Visualization version GIF version |
Description: Absolute value of negative. (Contributed by NM, 27-Feb-2005.) |
Ref | Expression |
---|---|
absneg | ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjneg 14007 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴)) | |
2 | 1 | oveq2d 6781 | . . . 4 ⊢ (𝐴 ∈ ℂ → (-𝐴 · (∗‘-𝐴)) = (-𝐴 · -(∗‘𝐴))) |
3 | cjcl 13965 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
4 | mul2neg 10582 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (-𝐴 · -(∗‘𝐴)) = (𝐴 · (∗‘𝐴))) | |
5 | 3, 4 | mpdan 705 | . . . 4 ⊢ (𝐴 ∈ ℂ → (-𝐴 · -(∗‘𝐴)) = (𝐴 · (∗‘𝐴))) |
6 | 2, 5 | eqtrd 2758 | . . 3 ⊢ (𝐴 ∈ ℂ → (-𝐴 · (∗‘-𝐴)) = (𝐴 · (∗‘𝐴))) |
7 | 6 | fveq2d 6308 | . 2 ⊢ (𝐴 ∈ ℂ → (√‘(-𝐴 · (∗‘-𝐴))) = (√‘(𝐴 · (∗‘𝐴)))) |
8 | negcl 10394 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
9 | absval 14098 | . . 3 ⊢ (-𝐴 ∈ ℂ → (abs‘-𝐴) = (√‘(-𝐴 · (∗‘-𝐴)))) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (√‘(-𝐴 · (∗‘-𝐴)))) |
11 | absval 14098 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
12 | 7, 10, 11 | 3eqtr4d 2768 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1596 ∈ wcel 2103 ‘cfv 6001 (class class class)co 6765 ℂcc 10047 · cmul 10054 -cneg 10380 ∗ccj 13956 √csqrt 14093 abscabs 14094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-op 4292 df-uni 4545 df-br 4761 df-opab 4821 df-mpt 4838 df-id 5128 df-po 5139 df-so 5140 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-er 7862 df-en 8073 df-dom 8074 df-sdom 8075 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-div 10798 df-2 11192 df-cj 13959 df-re 13960 df-im 13961 df-abs 14096 |
This theorem is referenced by: absnid 14158 absimle 14169 abslt 14174 absle 14175 abssub 14186 abs2dif2 14193 sqreulem 14219 absnegi 14259 absnegd 14308 cnheibor 22876 ftalem3 24921 qqhcn 30265 jm2.26lem3 37987 |
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