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Mirrors > Home > ILE Home > Th. List > xrltmininf | GIF version |
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.) |
Ref | Expression |
---|---|
xrltmininf | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < inf({𝐵, 𝐶}, ℝ*, < ) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrminmax 11273 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → inf({𝐵, 𝐶}, ℝ*, < ) = -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < )) | |
2 | 1 | 3adant1 1015 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → inf({𝐵, 𝐶}, ℝ*, < ) = -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < )) |
3 | 2 | breq2d 4016 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < inf({𝐵, 𝐶}, ℝ*, < ) ↔ 𝐴 < -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
4 | simp2 998 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐵 ∈ ℝ*) | |
5 | 4 | xnegcld 9855 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐵 ∈ ℝ*) |
6 | simp3 999 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐶 ∈ ℝ*) | |
7 | 6 | xnegcld 9855 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐶 ∈ ℝ*) |
8 | simp1 997 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐴 ∈ ℝ*) | |
9 | 8 | xnegcld 9855 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐴 ∈ ℝ*) |
10 | xrmaxltsup 11266 | . . . 4 ⊢ ((-𝑒𝐵 ∈ ℝ* ∧ -𝑒𝐶 ∈ ℝ* ∧ -𝑒𝐴 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) < -𝑒𝐴 ↔ (-𝑒𝐵 < -𝑒𝐴 ∧ -𝑒𝐶 < -𝑒𝐴))) | |
11 | 5, 7, 9, 10 | syl3anc 1238 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) < -𝑒𝐴 ↔ (-𝑒𝐵 < -𝑒𝐴 ∧ -𝑒𝐶 < -𝑒𝐴))) |
12 | xrmaxcl 11260 | . . . . . . 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 9855 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ∈ ℝ*) |
15 | xltneg 9836 | . . . . 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 9833 | . . . . . 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 4014 | . . . 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 9836 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴)) | |
22 | 21 | 3adant3 1017 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴)) |
23 | xltneg 9836 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < 𝐶 ↔ -𝑒𝐶 < -𝑒𝐴)) | |
24 | 23 | 3adant2 1016 | . . . 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 978 = wceq 1353 ∈ wcel 2148 {cpr 3594 class class class wbr 4004 supcsup 6981 infcinf 6982 ℝ*cxr 7991 < clt 7992 -𝑒cxne 9769 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-sup 6983 df-inf 6984 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-rp 9654 df-xneg 9772 df-seqfrec 10446 df-exp 10520 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 |
This theorem is referenced by: xrminrpcl 11282 iooinsup 11285 blininf 13927 bdxmet 14004 bdmopn 14007 |
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