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Theorem minabs 11228

Description: The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)

**Assertion**
Ref	Expression
minabs	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵))) / 2))

**Proof of Theorem minabs**
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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