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Mirrors > Home > MPE Home > Th. List > abssubi | Structured version Visualization version GIF version |
Description: Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.) |
Ref | Expression |
---|---|
absvalsqi.1 | ⊢ 𝐴 ∈ ℂ |
abssub.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
abssubi | ⊢ (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absvalsqi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | abssub.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | abssub 15027 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) | |
4 | 1, 2, 3 | mp2an 689 | 1 ⊢ (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 ‘cfv 6428 (class class class)co 7269 ℂcc 10858 − cmin 11194 abscabs 14934 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5486 df-po 5500 df-so 5501 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-er 8487 df-en 8723 df-dom 8724 df-sdom 8725 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-div 11622 df-2 12025 df-cj 14799 df-re 14800 df-im 14801 df-abs 14936 |
This theorem is referenced by: (None) |
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