xrminltinf - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xrminltinf		GIF version

Theorem xrminltinf 11044

Description: Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)

**Assertion**
Ref	Expression
xrminltinf	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (inf({𝐵, 𝐶}, ℝ^, < ) < 𝐴 ↔ (𝐵 < 𝐴 ∨ 𝐶 < 𝐴)))

**Proof of Theorem xrminltinf**
Step	Hyp	Ref	Expression
1		xnegcl 9618	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
2	1	3ad2ant2 1003	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
3		xnegcl 9618	. . . 4 ⊢ (𝐶 ∈ ℝ^* → -_𝑒𝐶 ∈ ℝ^*)
4	3	3ad2ant3 1004	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
5		xnegcl 9618	. . . 4 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
6	5	3ad2ant1 1002	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
7		xrltmaxsup 11029	. . 3 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^* ∧ -_𝑒𝐴 ∈ ℝ^) → (-_𝑒𝐴 < sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ (-_𝑒𝐴 < -_𝑒𝐵 ∨ -_𝑒𝐴 < -_𝑒𝐶)))
8	2, 4, 6, 7	syl3anc 1216	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (-_𝑒𝐴 < sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ (-_𝑒𝐴 < -_𝑒𝐵 ∨ -_𝑒𝐴 < -_𝑒𝐶)))
9		xrminmax 11037	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
10	9	3adant1 999	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
11		xnegneg 9619	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
12	11	eqcomd 2145	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → 𝐴 = -_𝑒-_𝑒𝐴)
13	12	3ad2ant1 1002	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 = -_𝑒-_𝑒𝐴)
14	10, 13	breq12d 3942	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (inf({𝐵, 𝐶}, ℝ^, < ) < 𝐴 ↔ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ) < -_𝑒-_𝑒𝐴))
15		xrmaxcl 11024	. . . . 5 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
16	2, 4, 15	syl2anc 408	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
17		xltneg 9622	. . . 4 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^) → (-_𝑒𝐴 < sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) < -_𝑒-_𝑒𝐴))
18	6, 16, 17	syl2anc 408	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (-_𝑒𝐴 < sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ) < -_𝑒-_𝑒𝐴))
19	14, 18	bitr4d 190	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (inf({𝐵, 𝐶}, ℝ^, < ) < 𝐴 ↔ -_𝑒𝐴 < sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
20		simp2 982	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
21		simp1 981	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
22		xltneg 9622	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ -_𝑒𝐴 < -_𝑒𝐵))
23	20, 21, 22	syl2anc 408	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ -_𝑒𝐴 < -_𝑒𝐵))
24		simp3 983	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
25		xltneg 9622	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ -_𝑒𝐴 < -_𝑒𝐶))
26	24, 21, 25	syl2anc 408	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ -_𝑒𝐴 < -_𝑒𝐶))
27	23, 26	orbi12d 782	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐵 < 𝐴 ∨ 𝐶 < 𝐴) ↔ (-_𝑒𝐴 < -_𝑒𝐵 ∨ -_𝑒𝐴 < -_𝑒𝐶)))
28	8, 19, 27	3bitr4d 219	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (inf({𝐵, 𝐶}, ℝ^, < ) < 𝐴 ↔ (𝐵 < 𝐴 ∨ 𝐶 < 𝐴)))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 104 ∨ wo 697 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 {cpr 3528 class class class wbr 3929 supcsup 6869 infcinf 6870 ℝ^*cxr 7802 < clt 7803 -_𝑒cxne 9559

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-sup 6871 df-inf 6872 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-n0 8981 df-z 9058 df-uz 9330 df-rp 9445 df-xneg 9562 df-seqfrec 10222 df-exp 10296 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774

This theorem is referenced by: bdbl 12675

W3C validator