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Mirrors > Home > MPE Home > Th. List > abssubd | Structured version Visualization version GIF version |
Description: Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abscld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
abssubd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
abssubd | ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | abssubd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | abssub 14066 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) | |
4 | 1, 2, 3 | syl2anc 693 | 1 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 − cmin 10266 abscabs 13974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-cj 13839 df-re 13840 df-im 13841 df-abs 13976 |
This theorem is referenced by: rlimuni 14281 climuni 14283 2clim 14303 rlimrecl 14311 subcn2 14325 reccn2 14327 climcau 14401 caucvgrlem 14403 serf0 14411 mertenslem2 14617 xrsxmet 22612 elcncf2 22693 cnllycmp 22755 dvlip 23756 c1lip1 23760 dvfsumrlim2 23795 dvfsum2 23797 ftc1a 23800 aalioulem3 24089 ulmcaulem 24148 ulmcau 24149 ulmbdd 24152 ulmcn 24153 ulmdvlem1 24154 logcnlem4 24391 ssscongptld 24552 chordthmlem3 24561 chordthmlem4 24562 lgamucov 24764 ftalem2 24800 logfacrlim 24949 dchrisumlem3 25180 dchrisum0lem1b 25204 mulog2sumlem2 25224 pntrlog2bndlem3 25268 smcnlem 27552 qqhucn 30036 dnibndlem2 32469 dnibndlem6 32473 dnibndlem8 32475 dnibnd 32481 unbdqndv2lem1 32500 knoppndvlem10 32512 knoppndvlem15 32517 ftc1anclem8 33492 irrapxlem3 37388 irrapxlem5 37390 pell14qrgt0 37423 acongeq 37550 absimlere 39710 limcrecl 39861 islpcn 39871 lptre2pt 39872 0ellimcdiv 39881 limclner 39883 dvbdfbdioolem2 40144 ioodvbdlimc1lem1 40146 ioodvbdlimc1lem2 40147 ioodvbdlimc2lem 40149 fourierdlem42 40366 ioorrnopnlem 40524 smflimlem4 40982 |
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