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Mirrors > Home > MPE Home > Th. List > Mathboxes > reabssgn | Structured version Visualization version GIF version |
Description: Alternate expression for the absolute value of a real number. (Contributed by RP, 22-May-2024.) |
Ref | Expression |
---|---|
reabssgn | ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = ((sgn‘𝐴) · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 10844 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | sgnval 14616 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) |
4 | 3 | oveq1d 7206 | . 2 ⊢ (𝐴 ∈ ℝ → ((sgn‘𝐴) · 𝐴) = (if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) · 𝐴)) |
5 | ovif 7286 | . . . 4 ⊢ (if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) · 𝐴) = if(𝐴 = 0, (0 · 𝐴), (if(𝐴 < 0, -1, 1) · 𝐴)) | |
6 | ovif 7286 | . . . . 5 ⊢ (if(𝐴 < 0, -1, 1) · 𝐴) = if(𝐴 < 0, (-1 · 𝐴), (1 · 𝐴)) | |
7 | ifeq2 4430 | . . . . 5 ⊢ ((if(𝐴 < 0, -1, 1) · 𝐴) = if(𝐴 < 0, (-1 · 𝐴), (1 · 𝐴)) → if(𝐴 = 0, (0 · 𝐴), (if(𝐴 < 0, -1, 1) · 𝐴)) = if(𝐴 = 0, (0 · 𝐴), if(𝐴 < 0, (-1 · 𝐴), (1 · 𝐴)))) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ if(𝐴 = 0, (0 · 𝐴), (if(𝐴 < 0, -1, 1) · 𝐴)) = if(𝐴 = 0, (0 · 𝐴), if(𝐴 < 0, (-1 · 𝐴), (1 · 𝐴))) |
9 | 5, 8 | eqtri 2759 | . . 3 ⊢ (if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) · 𝐴) = if(𝐴 = 0, (0 · 𝐴), if(𝐴 < 0, (-1 · 𝐴), (1 · 𝐴))) |
10 | mul02lem2 10974 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0) | |
11 | 10 | adantr 484 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (0 · 𝐴) = 0) |
12 | simpr 488 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → 𝐴 = 0) | |
13 | 12 | abs00bd 14820 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (abs‘𝐴) = 0) |
14 | 11, 13 | eqtr4d 2774 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 0) → (0 · 𝐴) = (abs‘𝐴)) |
15 | recn 10784 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
16 | 15 | mulm1d 11249 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-1 · 𝐴) = -𝐴) |
17 | 15 | mulid2d 10816 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴) |
18 | 16, 17 | ifeq12d 4446 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → if(𝐴 < 0, (-1 · 𝐴), (1 · 𝐴)) = if(𝐴 < 0, -𝐴, 𝐴)) |
19 | reabsifneg 40857 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 < 0, -𝐴, 𝐴)) | |
20 | 18, 19 | eqtr4d 2774 | . . . . 5 ⊢ (𝐴 ∈ ℝ → if(𝐴 < 0, (-1 · 𝐴), (1 · 𝐴)) = (abs‘𝐴)) |
21 | 20 | adantr 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 = 0) → if(𝐴 < 0, (-1 · 𝐴), (1 · 𝐴)) = (abs‘𝐴)) |
22 | 14, 21 | ifeqda 4461 | . . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = 0, (0 · 𝐴), if(𝐴 < 0, (-1 · 𝐴), (1 · 𝐴))) = (abs‘𝐴)) |
23 | 9, 22 | syl5eq 2783 | . 2 ⊢ (𝐴 ∈ ℝ → (if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) · 𝐴) = (abs‘𝐴)) |
24 | 4, 23 | eqtr2d 2772 | 1 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = ((sgn‘𝐴) · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ifcif 4425 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 0cc0 10694 1c1 10695 · cmul 10699 ℝ*cxr 10831 < clt 10832 -cneg 11028 sgncsgn 14614 abscabs 14762 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-seq 13540 df-exp 13601 df-sgn 14615 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 |
This theorem is referenced by: (None) |
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