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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 14890 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 − cmin 11062 abscabs 14797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-2 11893 df-cj 14662 df-re 14663 df-im 14664 df-abs 14799 |
This theorem is referenced by: rlimuni 15111 climuni 15113 2clim 15133 rlimrecl 15141 subcn2 15156 reccn2 15158 climcau 15234 caucvgrlem 15236 serf0 15244 mertenslem2 15449 xrsxmet 23706 elcncf2 23787 cnllycmp 23853 dvlip 24890 c1lip1 24894 dvfsumrlim2 24929 dvfsum2 24931 ftc1a 24934 aalioulem3 25227 ulmcaulem 25286 ulmcau 25287 ulmbdd 25290 ulmcn 25291 ulmdvlem1 25292 logcnlem4 25533 ssscongptld 25705 chordthmlem3 25717 chordthmlem4 25718 lgamucov 25920 ftalem2 25956 logfacrlim 26105 dchrisumlem3 26372 dchrisum0lem1b 26396 mulog2sumlem2 26416 pntrlog2bndlem3 26460 smcnlem 28778 qqhucn 31654 dnibndlem2 34396 dnibndlem6 34400 dnibndlem8 34402 dnibnd 34408 unbdqndv2lem1 34426 knoppndvlem10 34438 knoppndvlem15 34443 ftc1anclem8 35594 irrapxlem3 40349 irrapxlem5 40351 pell14qrgt0 40384 acongeq 40508 absimlere 42695 limcrecl 42845 islpcn 42855 lptre2pt 42856 0ellimcdiv 42865 limclner 42867 dvbdfbdioolem2 43145 ioodvbdlimc1lem1 43147 ioodvbdlimc1lem2 43148 ioodvbdlimc2lem 43150 fourierdlem42 43365 ioorrnopnlem 43520 smflimlem4 43981 |
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