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Mirrors > Home > ILE Home > Th. List > minabs | GIF version |
Description: The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.) |
Ref | Expression |
---|---|
minabs | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵))) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minmax 11222 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = -sup({-𝐴, -𝐵}, ℝ, < )) | |
2 | renegcl 8208 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
3 | renegcl 8208 | . . . . 5 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
4 | maxabs 11202 | . . . . 5 ⊢ ((-𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → sup({-𝐴, -𝐵}, ℝ, < ) = (((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2)) | |
5 | 2, 3, 4 | syl2an 289 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({-𝐴, -𝐵}, ℝ, < ) = (((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2)) |
6 | 5 | negeqd 8142 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -sup({-𝐴, -𝐵}, ℝ, < ) = -(((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2)) |
7 | 1, 6 | eqtrd 2210 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = -(((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2)) |
8 | simpl 109 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
9 | 8 | recnd 7976 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
10 | 9 | negcld 8245 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -𝐴 ∈ ℂ) |
11 | simpr 110 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) | |
12 | 11 | recnd 7976 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℂ) |
13 | 12 | negcld 8245 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -𝐵 ∈ ℂ) |
14 | 10, 13 | addcld 7967 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 + -𝐵) ∈ ℂ) |
15 | 10, 13 | subcld 8258 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 − -𝐵) ∈ ℂ) |
16 | 15 | abscld 11174 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(-𝐴 − -𝐵)) ∈ ℝ) |
17 | 16 | recnd 7976 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(-𝐴 − -𝐵)) ∈ ℂ) |
18 | 14, 17 | addcld 7967 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) ∈ ℂ) |
19 | 2cnd 8981 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 2 ∈ ℂ) | |
20 | 2ap0 9001 | . . . 4 ⊢ 2 # 0 | |
21 | 20 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 2 # 0) |
22 | 18, 19, 21 | divnegapd 8749 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2) = (-((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2)) |
23 | 14, 17 | negdi2d 8272 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) = (-(-𝐴 + -𝐵) − (abs‘(-𝐴 − -𝐵)))) |
24 | 10, 13 | negdid 8271 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(-𝐴 + -𝐵) = (--𝐴 + --𝐵)) |
25 | 9 | negnegd 8249 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → --𝐴 = 𝐴) |
26 | 12 | negnegd 8249 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → --𝐵 = 𝐵) |
27 | 25, 26 | oveq12d 5887 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (--𝐴 + --𝐵) = (𝐴 + 𝐵)) |
28 | 24, 27 | eqtrd 2210 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(-𝐴 + -𝐵) = (𝐴 + 𝐵)) |
29 | 9, 12 | neg2subd 8275 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 − -𝐵) = (𝐵 − 𝐴)) |
30 | 29 | fveq2d 5515 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(-𝐴 − -𝐵)) = (abs‘(𝐵 − 𝐴))) |
31 | 9, 12 | abssubd 11186 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
32 | 30, 31 | eqtr4d 2213 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(-𝐴 − -𝐵)) = (abs‘(𝐴 − 𝐵))) |
33 | 28, 32 | oveq12d 5887 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-(-𝐴 + -𝐵) − (abs‘(-𝐴 − -𝐵))) = ((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵)))) |
34 | 23, 33 | eqtrd 2210 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) = ((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵)))) |
35 | 34 | oveq1d 5884 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2) = (((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵))) / 2)) |
36 | 7, 22, 35 | 3eqtrd 2214 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵))) / 2)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 {cpr 3592 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 supcsup 6975 infcinf 6976 ℝcr 7801 0cc0 7802 + caddc 7805 < clt 7982 − cmin 8118 -cneg 8119 # cap 8528 / cdiv 8618 2c2 8959 abscabs 10990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-rp 9641 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 |
This theorem is referenced by: bdtri 11232 |
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