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Theorem xrminltinf 12064

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 10252	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_e𝐵 ∈ ℝ^*)
2	1	3ad2ant2 1050	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_e𝐵 ∈ ℝ^)
3		xnegcl 10252	. . . 4 ⊢ (𝐶 ∈ ℝ^* → -_e𝐶 ∈ ℝ^*)
4	3	3ad2ant3 1051	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_e𝐶 ∈ ℝ^)
5		xnegcl 10252	. . . 4 ⊢ (𝐴 ∈ ℝ^* → -_e𝐴 ∈ ℝ^*)
6	5	3ad2ant1 1049	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_e𝐴 ∈ ℝ^)
7		xrltmaxsup 12049	. . 3 ⊢ ((-_e𝐵 ∈ ℝ^* ∧ -_e𝐶 ∈ ℝ^* ∧ -_e𝐴 ∈ ℝ^) → (-_e𝐴 < sup({-_e𝐵, -_e𝐶}, ℝ^, < ) ↔ (-_e𝐴 < -_e𝐵 ∨ -_e𝐴 < -_e𝐶)))
8	2, 4, 6, 7	syl3anc 1278	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (-_e𝐴 < sup({-_e𝐵, -_e𝐶}, ℝ^, < ) ↔ (-_e𝐴 < -_e𝐵 ∨ -_e𝐴 < -_e𝐶)))
9		xrminmax 12057	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_esup({-_e𝐵, -_e𝐶}, ℝ^*, < ))
10	9	3adant1 1046	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_esup({-_e𝐵, -_e𝐶}, ℝ^*, < ))
11		xnegneg 10253	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → -_e-_e𝐴 = 𝐴)
12	11	eqcomd 2244	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → 𝐴 = -_e-_e𝐴)
13	12	3ad2ant1 1049	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 = -_e-_e𝐴)
14	10, 13	breq12d 4144	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (inf({𝐵, 𝐶}, ℝ^, < ) < 𝐴 ↔ -_esup({-_e𝐵, -_e𝐶}, ℝ^*, < ) < -_e-_e𝐴))
15		xrmaxcl 12044	. . . . 5 ⊢ ((-_e𝐵 ∈ ℝ^* ∧ -_e𝐶 ∈ ℝ^) → sup({-_e𝐵, -_e𝐶}, ℝ^, < ) ∈ ℝ^*)
16	2, 4, 15	syl2anc 415	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({-_e𝐵, -_e𝐶}, ℝ^, < ) ∈ ℝ^*)
17		xltneg 10256	. . . 4 ⊢ ((-_e𝐴 ∈ ℝ^* ∧ sup({-_e𝐵, -_e𝐶}, ℝ^, < ) ∈ ℝ^) → (-_e𝐴 < sup({-_e𝐵, -_e𝐶}, ℝ^, < ) ↔ -_esup({-_e𝐵, -_e𝐶}, ℝ^, < ) < -_e-_e𝐴))
18	6, 16, 17	syl2anc 415	. . 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 1029	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
21		simp1 1028	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
22		xltneg 10256	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ -_e𝐴 < -_e𝐵))
23	20, 21, 22	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ -_e𝐴 < -_e𝐵))
24		simp3 1030	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
25		xltneg 10256	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ -_e𝐴 < -_e𝐶))
26	24, 21, 25	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ -_e𝐴 < -_e𝐶))
27	23, 26	orbi12d 805	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 105 ∨ wo 720 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 {cpr 3710 class class class wbr 4131 supcsup 7329 infcinf 7330 ℝ^*cxr 8366 < clt 8367 -_ecxne 10189

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-sup 7331 df-inf 7332 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-n0 9576 df-z 9657 df-uz 9939 df-rp 10073 df-xneg 10192 df-seqfrec 10907 df-exp 10998 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788

This theorem is used by: bdbl 15741

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