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Theorem xrltmininf 11951

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 11946	. . . 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 4120	. 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 10187	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
6		simp3 1026	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
7	6	xnegcld 10187	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
8		simp1 1024	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
9	8	xnegcld 10187	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
10		xrmaxltsup 11939	. . . 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 11933	. . . . . . 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 10187	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
15		xltneg 10168	. . . . 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 10165	. . . . . 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 4118	. . . 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 10168	. . . . 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 10168	. . . . 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 2203 {cpr 3689 class class class wbr 4108 supcsup 7272 infcinf 7273 ℝ^*cxr 8306 < clt 8307 -_𝑒cxne 10101

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 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8217 ax-resscn 8218 ax-1cn 8219 ax-1re 8220 ax-icn 8221 ax-addcl 8222 ax-addrcl 8223 ax-mulcl 8224 ax-mulrcl 8225 ax-addcom 8226 ax-mulcom 8227 ax-addass 8228 ax-mulass 8229 ax-distr 8230 ax-i2m1 8231 ax-0lt1 8232 ax-1rid 8233 ax-0id 8234 ax-rnegex 8235 ax-precex 8236 ax-cnre 8237 ax-pre-ltirr 8238 ax-pre-ltwlin 8239 ax-pre-lttrn 8240 ax-pre-apti 8241 ax-pre-ltadd 8242 ax-pre-mulgt0 8243 ax-pre-mulext 8244 ax-arch 8245 ax-caucvg 8246

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 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-isom 5360 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-frec 6621 df-sup 7274 df-inf 7275 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312 df-le 8313 df-sub 8445 df-neg 8446 df-reap 8848 df-ap 8855 df-div 8946 df-inn 9237 df-2 9295 df-3 9296 df-4 9297 df-n0 9496 df-z 9577 df-uz 9853 df-rp 9986 df-xneg 10104 df-seqfrec 10809 df-exp 10900 df-cj 11523 df-re 11524 df-im 11525 df-rsqrt 11679 df-abs 11680

This theorem is referenced by: xrminrpcl 11955 iooinsup 11958 blininf 15281 bdxmet 15358 bdmopn 15361

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