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Mirrors > Home > MPE Home > Th. List > abssuble0d | Structured version Visualization version GIF version |
Description: Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
absltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
absltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
abssubge0d.2 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Ref | Expression |
---|---|
abssuble0d | ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (𝐵 − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absltd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | absltd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | abssubge0d.2 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
4 | abssuble0 14780 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (abs‘(𝐴 − 𝐵)) = (𝐵 − 𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1372 | 1 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (𝐵 − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 class class class wbr 5030 ‘cfv 6339 (class class class)co 7172 ℝcr 10616 ≤ cle 10756 − cmin 10950 abscabs 14685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-sup 8981 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-n0 11979 df-z 12065 df-uz 12327 df-rp 12475 df-seq 13463 df-exp 13524 df-cj 14550 df-re 14551 df-im 14552 df-sqrt 14686 df-abs 14687 |
This theorem is referenced by: climsqz 15090 rlimsqz 15101 climsup 15121 ivthlem3 24207 dvferm2lem 24740 lgamgulmlem3 25770 chtppilim 26213 dchrisumlem2 26228 pntrlog2bndlem2 26316 pntrlog2bndlem4 26318 pntpbnd1 26324 pntibndlem2 26329 unbdqndv2lem2 34335 fourierdlem42 43254 qndenserrnbllem 43399 ovolval2lem 43745 |
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