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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 11033 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = -sup({-𝐴, -𝐵}, ℝ, < )) | |
2 | renegcl 8047 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
3 | renegcl 8047 | . . . . 5 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
4 | maxabs 11013 | . . . . 5 ⊢ ((-𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → sup({-𝐴, -𝐵}, ℝ, < ) = (((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2)) | |
5 | 2, 3, 4 | syl2an 287 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({-𝐴, -𝐵}, ℝ, < ) = (((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2)) |
6 | 5 | negeqd 7981 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -sup({-𝐴, -𝐵}, ℝ, < ) = -(((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2)) |
7 | 1, 6 | eqtrd 2173 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = -(((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2)) |
8 | simpl 108 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
9 | 8 | recnd 7818 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
10 | 9 | negcld 8084 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -𝐴 ∈ ℂ) |
11 | simpr 109 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) | |
12 | 11 | recnd 7818 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℂ) |
13 | 12 | negcld 8084 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -𝐵 ∈ ℂ) |
14 | 10, 13 | addcld 7809 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 + -𝐵) ∈ ℂ) |
15 | 10, 13 | subcld 8097 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 − -𝐵) ∈ ℂ) |
16 | 15 | abscld 10985 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(-𝐴 − -𝐵)) ∈ ℝ) |
17 | 16 | recnd 7818 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(-𝐴 − -𝐵)) ∈ ℂ) |
18 | 14, 17 | addcld 7809 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) ∈ ℂ) |
19 | 2cnd 8817 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 2 ∈ ℂ) | |
20 | 2ap0 8837 | . . . 4 ⊢ 2 # 0 | |
21 | 20 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 2 # 0) |
22 | 18, 19, 21 | divnegapd 8587 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2) = (-((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2)) |
23 | 14, 17 | negdi2d 8111 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) = (-(-𝐴 + -𝐵) − (abs‘(-𝐴 − -𝐵)))) |
24 | 10, 13 | negdid 8110 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(-𝐴 + -𝐵) = (--𝐴 + --𝐵)) |
25 | 9 | negnegd 8088 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → --𝐴 = 𝐴) |
26 | 12 | negnegd 8088 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → --𝐵 = 𝐵) |
27 | 25, 26 | oveq12d 5800 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (--𝐴 + --𝐵) = (𝐴 + 𝐵)) |
28 | 24, 27 | eqtrd 2173 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(-𝐴 + -𝐵) = (𝐴 + 𝐵)) |
29 | 9, 12 | neg2subd 8114 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 − -𝐵) = (𝐵 − 𝐴)) |
30 | 29 | fveq2d 5433 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(-𝐴 − -𝐵)) = (abs‘(𝐵 − 𝐴))) |
31 | 9, 12 | abssubd 10997 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
32 | 30, 31 | eqtr4d 2176 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(-𝐴 − -𝐵)) = (abs‘(𝐴 − 𝐵))) |
33 | 28, 32 | oveq12d 5800 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-(-𝐴 + -𝐵) − (abs‘(-𝐴 − -𝐵))) = ((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵)))) |
34 | 23, 33 | eqtrd 2173 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) = ((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵)))) |
35 | 34 | oveq1d 5797 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-((-𝐴 + -𝐵) + (abs‘(-𝐴 − -𝐵))) / 2) = (((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵))) / 2)) |
36 | 7, 22, 35 | 3eqtrd 2177 | 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 103 = wceq 1332 ∈ wcel 1481 {cpr 3533 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 supcsup 6877 infcinf 6878 ℝcr 7643 0cc0 7644 + caddc 7647 < clt 7824 − cmin 7957 -cneg 7958 # cap 8367 / cdiv 8456 2c2 8795 abscabs 10801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-rp 9471 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 |
This theorem is referenced by: bdtri 11043 |
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