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Mirrors > Home > MPE Home > Th. List > absneg | Structured version Visualization version GIF version |
Description: Absolute value of the opposite. (Contributed by NM, 27-Feb-2005.) |
Ref | Expression |
---|---|
absneg | ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjneg 14367 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴)) | |
2 | 1 | oveq2d 6992 | . . . 4 ⊢ (𝐴 ∈ ℂ → (-𝐴 · (∗‘-𝐴)) = (-𝐴 · -(∗‘𝐴))) |
3 | cjcl 14325 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
4 | mul2neg 10880 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (-𝐴 · -(∗‘𝐴)) = (𝐴 · (∗‘𝐴))) | |
5 | 3, 4 | mpdan 674 | . . . 4 ⊢ (𝐴 ∈ ℂ → (-𝐴 · -(∗‘𝐴)) = (𝐴 · (∗‘𝐴))) |
6 | 2, 5 | eqtrd 2814 | . . 3 ⊢ (𝐴 ∈ ℂ → (-𝐴 · (∗‘-𝐴)) = (𝐴 · (∗‘𝐴))) |
7 | 6 | fveq2d 6503 | . 2 ⊢ (𝐴 ∈ ℂ → (√‘(-𝐴 · (∗‘-𝐴))) = (√‘(𝐴 · (∗‘𝐴)))) |
8 | negcl 10686 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
9 | absval 14458 | . . 3 ⊢ (-𝐴 ∈ ℂ → (abs‘-𝐴) = (√‘(-𝐴 · (∗‘-𝐴)))) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (√‘(-𝐴 · (∗‘-𝐴)))) |
11 | absval 14458 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
12 | 7, 10, 11 | 3eqtr4d 2824 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 ℂcc 10333 · cmul 10340 -cneg 10671 ∗ccj 14316 √csqrt 14453 abscabs 14454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-2 11503 df-cj 14319 df-re 14320 df-im 14321 df-abs 14456 |
This theorem is referenced by: absnid 14519 absimle 14530 abslt 14535 absle 14536 abssub 14547 abs2dif2 14554 sqreulem 14580 absnegi 14621 absnegd 14670 cnheibor 23262 ftalem3 25354 qqhcn 30882 jm2.26lem3 39000 |
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