xrltmininf - Intuitionistic Logic Explorer

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

Theorem xrltmininf 11891

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 11886	. . . 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 4105	. 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 10133	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
6		simp3 1026	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
7	6	xnegcld 10133	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
8		simp1 1024	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
9	8	xnegcld 10133	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
10		xrmaxltsup 11879	. . . 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 11873	. . . . . . 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 10133	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
15		xltneg 10114	. . . . 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 10111	. . . . . 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 4103	. . . 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 10114	. . . . 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 10114	. . . . 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 2202 {cpr 3674 class class class wbr 4093 supcsup 7224 infcinf 7225 ℝ^*cxr 8256 < clt 8257 -_𝑒cxne 10047

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-sup 7226 df-inf 7227 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-reap 8798 df-ap 8805 df-div 8896 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-n0 9446 df-z 9523 df-uz 9799 df-rp 9932 df-xneg 10050 df-seqfrec 10754 df-exp 10845 df-cj 11463 df-re 11464 df-im 11465 df-rsqrt 11619 df-abs 11620

This theorem is referenced by: xrminrpcl 11895 iooinsup 11898 blininf 15215 bdxmet 15292 bdmopn 15295

W3C validator