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Theorem xrminltinf 11950

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 10161	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
2	1	3ad2ant2 1046	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
3		xnegcl 10161	. . . 4 ⊢ (𝐶 ∈ ℝ^* → -_𝑒𝐶 ∈ ℝ^*)
4	3	3ad2ant3 1047	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
5		xnegcl 10161	. . . 4 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
6	5	3ad2ant1 1045	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
7		xrltmaxsup 11935	. . 3 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^* ∧ -_𝑒𝐴 ∈ ℝ^) → (-_𝑒𝐴 < sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ (-_𝑒𝐴 < -_𝑒𝐵 ∨ -_𝑒𝐴 < -_𝑒𝐶)))
8	2, 4, 6, 7	syl3anc 1274	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (-_𝑒𝐴 < sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ (-_𝑒𝐴 < -_𝑒𝐵 ∨ -_𝑒𝐴 < -_𝑒𝐶)))
9		xrminmax 11943	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
10	9	3adant1 1042	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
11		xnegneg 10162	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
12	11	eqcomd 2238	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → 𝐴 = -_𝑒-_𝑒𝐴)
13	12	3ad2ant1 1045	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 = -_𝑒-_𝑒𝐴)
14	10, 13	breq12d 4121	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (inf({𝐵, 𝐶}, ℝ^, < ) < 𝐴 ↔ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ) < -_𝑒-_𝑒𝐴))
15		xrmaxcl 11930	. . . . 5 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
16	2, 4, 15	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
17		xltneg 10165	. . . 4 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^) → (-_𝑒𝐴 < sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) < -_𝑒-_𝑒𝐴))
18	6, 16, 17	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (-_𝑒𝐴 < sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ) < -_𝑒-_𝑒𝐴))
19	14, 18	bitr4d 191	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (inf({𝐵, 𝐶}, ℝ^, < ) < 𝐴 ↔ -_𝑒𝐴 < sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
20		simp2 1025	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
21		simp1 1024	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
22		xltneg 10165	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ -_𝑒𝐴 < -_𝑒𝐵))
23	20, 21, 22	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ -_𝑒𝐴 < -_𝑒𝐵))
24		simp3 1026	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
25		xltneg 10165	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ -_𝑒𝐴 < -_𝑒𝐶))
26	24, 21, 25	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ -_𝑒𝐴 < -_𝑒𝐶))
27	23, 26	orbi12d 801	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐵 < 𝐴 ∨ 𝐶 < 𝐴) ↔ (-_𝑒𝐴 < -_𝑒𝐵 ∨ -_𝑒𝐴 < -_𝑒𝐶)))
28	8, 19, 27	3bitr4d 220	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 105 ∨ wo 716 ∧ w3a 1005 = wceq 1398 ∈ wcel 2203 {cpr 3689 class class class wbr 4108 supcsup 7272 infcinf 7273 ℝ^*cxr 8303 < clt 8304 -_𝑒cxne 10098

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 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-mulrcl 8222 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-precex 8233 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239 ax-pre-mulgt0 8240 ax-pre-mulext 8241 ax-arch 8242 ax-caucvg 8243

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 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-reap 8845 df-ap 8852 df-div 8943 df-inn 9234 df-2 9292 df-3 9293 df-4 9294 df-n0 9493 df-z 9574 df-uz 9850 df-rp 9983 df-xneg 10101 df-seqfrec 10806 df-exp 10897 df-cj 11520 df-re 11521 df-im 11522 df-rsqrt 11676 df-abs 11677

This theorem is referenced by: bdbl 15355

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