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Mirrors > Home > MPE Home > Th. List > max0sub | Structured version Visualization version GIF version |
Description: Decompose a real number into positive and negative parts. (Contributed by Mario Carneiro, 6-Aug-2014.) |
Ref | Expression |
---|---|
max0sub | ⊢ (𝐴 ∈ ℝ → (if(0 ≤ 𝐴, 𝐴, 0) − if(0 ≤ -𝐴, -𝐴, 0)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10978 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
2 | id 22 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ) | |
3 | iftrue 4465 | . . . . 5 ⊢ (0 ≤ 𝐴 → if(0 ≤ 𝐴, 𝐴, 0) = 𝐴) | |
4 | 3 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → if(0 ≤ 𝐴, 𝐴, 0) = 𝐴) |
5 | 0xr 11022 | . . . . 5 ⊢ 0 ∈ ℝ* | |
6 | renegcl 11284 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
7 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -𝐴 ∈ ℝ) |
8 | 7 | rexrd 11025 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -𝐴 ∈ ℝ*) |
9 | le0neg2 11484 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0)) | |
10 | 9 | biimpa 477 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -𝐴 ≤ 0) |
11 | xrmaxeq 12913 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ -𝐴 ∈ ℝ* ∧ -𝐴 ≤ 0) → if(0 ≤ -𝐴, -𝐴, 0) = 0) | |
12 | 5, 8, 10, 11 | mp3an2i 1465 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → if(0 ≤ -𝐴, -𝐴, 0) = 0) |
13 | 4, 12 | oveq12d 7293 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (if(0 ≤ 𝐴, 𝐴, 0) − if(0 ≤ -𝐴, -𝐴, 0)) = (𝐴 − 0)) |
14 | recn 10961 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
15 | 14 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℂ) |
16 | 15 | subid1d 11321 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 − 0) = 𝐴) |
17 | 13, 16 | eqtrd 2778 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (if(0 ≤ 𝐴, 𝐴, 0) − if(0 ≤ -𝐴, -𝐴, 0)) = 𝐴) |
18 | rexr 11021 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
19 | 18 | adantr 481 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → 𝐴 ∈ ℝ*) |
20 | simpr 485 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → 𝐴 ≤ 0) | |
21 | xrmaxeq 12913 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐴 ≤ 0) → if(0 ≤ 𝐴, 𝐴, 0) = 0) | |
22 | 5, 19, 20, 21 | mp3an2i 1465 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → if(0 ≤ 𝐴, 𝐴, 0) = 0) |
23 | le0neg1 11483 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴)) | |
24 | 23 | biimpa 477 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → 0 ≤ -𝐴) |
25 | 24 | iftrued 4467 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → if(0 ≤ -𝐴, -𝐴, 0) = -𝐴) |
26 | 22, 25 | oveq12d 7293 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → (if(0 ≤ 𝐴, 𝐴, 0) − if(0 ≤ -𝐴, -𝐴, 0)) = (0 − -𝐴)) |
27 | df-neg 11208 | . . . 4 ⊢ --𝐴 = (0 − -𝐴) | |
28 | 14 | adantr 481 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → 𝐴 ∈ ℂ) |
29 | 28 | negnegd 11323 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → --𝐴 = 𝐴) |
30 | 27, 29 | eqtr3id 2792 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → (0 − -𝐴) = 𝐴) |
31 | 26, 30 | eqtrd 2778 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → (if(0 ≤ 𝐴, 𝐴, 0) − if(0 ≤ -𝐴, -𝐴, 0)) = 𝐴) |
32 | 1, 2, 17, 31 | lecasei 11081 | 1 ⊢ (𝐴 ∈ ℝ → (if(0 ≤ 𝐴, 𝐴, 0) − if(0 ≤ -𝐴, -𝐴, 0)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ifcif 4459 class class class wbr 5074 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 ℝ*cxr 11008 ≤ cle 11010 − cmin 11205 -cneg 11206 |
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 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 |
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-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 |
This theorem is referenced by: mbfi1flimlem 24887 itgitg1 24973 itgconst 24983 itgaddlem2 24988 itgmulc2lem2 24997 itgaddnclem2 35836 itgmulc2nclem2 35844 |
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