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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 15260 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) | |
4 | 1, 2, 3 | syl2anc 585 | 1 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 − cmin 11431 abscabs 15168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-2 12262 df-cj 15033 df-re 15034 df-im 15035 df-abs 15170 |
This theorem is referenced by: rlimuni 15481 climuni 15483 2clim 15503 rlimrecl 15511 subcn2 15526 reccn2 15528 climcau 15604 caucvgrlem 15606 serf0 15614 mertenslem2 15818 xrsxmet 24294 elcncf2 24375 cnllycmp 24441 dvlip 25479 c1lip1 25483 dvfsumrlim2 25518 dvfsum2 25520 ftc1a 25523 aalioulem3 25816 ulmcaulem 25875 ulmcau 25876 ulmbdd 25879 ulmcn 25880 ulmdvlem1 25881 logcnlem4 26122 ssscongptld 26294 chordthmlem3 26306 chordthmlem4 26307 lgamucov 26509 ftalem2 26545 logfacrlim 26694 dchrisumlem3 26961 dchrisum0lem1b 26985 mulog2sumlem2 27005 pntrlog2bndlem3 27049 smcnlem 29915 qqhucn 32903 dnibndlem2 35260 dnibndlem6 35264 dnibndlem8 35266 dnibnd 35272 unbdqndv2lem1 35290 knoppndvlem10 35302 knoppndvlem15 35307 ftc1anclem8 36473 irrapxlem3 41433 irrapxlem5 41435 pell14qrgt0 41468 acongeq 41593 absimlere 44063 limcrecl 44218 islpcn 44228 lptre2pt 44229 0ellimcdiv 44238 limclner 44240 dvbdfbdioolem2 44518 ioodvbdlimc1lem1 44520 ioodvbdlimc1lem2 44521 ioodvbdlimc2lem 44523 fourierdlem42 44738 ioorrnopnlem 44893 smflimlem4 45363 |
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