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Theorem xrltmininf 11980

Description: Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)

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

**Proof of Theorem xrltmininf**
Step	Hyp	Ref	Expression
1		xrminmax 11975	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
2	1	3adant1 1042	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
3	2	breq2d 4126	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 < inf({𝐵, 𝐶}, ℝ^, < ) ↔ 𝐴 < -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
4		simp2 1025	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
5	4	xnegcld 10207	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
6		simp3 1026	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
7	6	xnegcld 10207	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
8		simp1 1024	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
9	8	xnegcld 10207	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
10		xrmaxltsup 11968	. . . 4 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^* ∧ -_𝑒𝐴 ∈ ℝ^) → (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) < -_𝑒𝐴 ↔ (-_𝑒𝐵 < -_𝑒𝐴 ∧ -_𝑒𝐶 < -_𝑒𝐴)))
11	5, 7, 9, 10	syl3anc 1274	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) < -_𝑒𝐴 ↔ (-_𝑒𝐵 < -_𝑒𝐴 ∧ -_𝑒𝐶 < -_𝑒𝐴)))
12		xrmaxcl 11962	. . . . . . 7 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
13	5, 7, 12	syl2anc 411	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
14	13	xnegcld 10207	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
15		xltneg 10188	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^) → (𝐴 < -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ -_𝑒-_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) < -_𝑒𝐴))
16	8, 14, 15	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 < -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ -_𝑒-_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ) < -_𝑒𝐴))
17		xnegneg 10185	. . . . . 6 ⊢ (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^ → -_𝑒-_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) = sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ))
18	13, 17	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒-_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) = sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
19	18	breq1d 4124	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (-_𝑒-_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) < -_𝑒𝐴 ↔ sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ) < -_𝑒𝐴))
20	16, 19	bitrd 188	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 < -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ) < -_𝑒𝐴))
21		xltneg 10188	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ -_𝑒𝐵 < -_𝑒𝐴))
22	21	3adant3 1044	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ -_𝑒𝐵 < -_𝑒𝐴))
23		xltneg 10188	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐶 ↔ -_𝑒𝐶 < -_𝑒𝐴))
24	23	3adant2 1043	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐶 ↔ -_𝑒𝐶 < -_𝑒𝐴))
25	22, 24	anbi12d 473	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐴 < 𝐶) ↔ (-_𝑒𝐵 < -_𝑒𝐴 ∧ -_𝑒𝐶 < -_𝑒𝐴)))
26	11, 20, 25	3bitr4d 220	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 < -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))
27	3, 26	bitrd 188	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 < inf({𝐵, 𝐶}, ℝ^, < ) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 {cpr 3695 class class class wbr 4114 supcsup 7286 infcinf 7287 ℝ^*cxr 8323 < clt 8324 -_𝑒cxne 10121

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-rp 10005 df-xneg 10124 df-seqfrec 10834 df-exp 10925 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709

This theorem is referenced by: xrminrpcl 11984 iooinsup 11987 blininf 15401 bdxmet 15478 bdmopn 15481

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