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Mirrors > Home > ILE Home > Th. List > xrlemininf | GIF version |
Description: Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 4-May-2023.) |
Ref | Expression |
---|---|
xrlemininf | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ*, < ) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrminmax 11240 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → inf({𝐵, 𝐶}, ℝ*, < ) = -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < )) | |
2 | 1 | breq2d 4010 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ*, < ) ↔ 𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
3 | 2 | 3adant1 1015 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ*, < ) ↔ 𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
4 | xnegcl 9801 | . . . . . 6 ⊢ (𝐵 ∈ ℝ* → -𝑒𝐵 ∈ ℝ*) | |
5 | 4 | 3ad2ant2 1019 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐵 ∈ ℝ*) |
6 | xnegcl 9801 | . . . . . 6 ⊢ (𝐶 ∈ ℝ* → -𝑒𝐶 ∈ ℝ*) | |
7 | 6 | 3ad2ant3 1020 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐶 ∈ ℝ*) |
8 | xrmaxcl 11227 | . . . . 5 ⊢ ((-𝑒𝐵 ∈ ℝ* ∧ -𝑒𝐶 ∈ ℝ*) → sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ∈ ℝ*) | |
9 | 5, 7, 8 | syl2anc 411 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ∈ ℝ*) |
10 | xnegcl 9801 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) | |
11 | 10 | 3ad2ant1 1018 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐴 ∈ ℝ*) |
12 | xleneg 9806 | . . . 4 ⊢ ((sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ∈ ℝ* ∧ -𝑒𝐴 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ -𝑒-𝑒𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) | |
13 | 9, 11, 12 | syl2anc 411 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ -𝑒-𝑒𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
14 | xnegneg 9802 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) | |
15 | 14 | 3ad2ant1 1018 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒-𝑒𝐴 = 𝐴) |
16 | 15 | breq1d 4008 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (-𝑒-𝑒𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ↔ 𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
17 | 13, 16 | bitrd 188 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ 𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
18 | xrmaxlesup 11234 | . . . 4 ⊢ ((-𝑒𝐵 ∈ ℝ* ∧ -𝑒𝐶 ∈ ℝ* ∧ -𝑒𝐴 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ (-𝑒𝐵 ≤ -𝑒𝐴 ∧ -𝑒𝐶 ≤ -𝑒𝐴))) | |
19 | 5, 7, 11, 18 | syl3anc 1238 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ (-𝑒𝐵 ≤ -𝑒𝐴 ∧ -𝑒𝐶 ≤ -𝑒𝐴))) |
20 | xleneg 9806 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴)) | |
21 | 20 | 3adant3 1017 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴)) |
22 | xleneg 9806 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐶 ↔ -𝑒𝐶 ≤ -𝑒𝐴)) | |
23 | 22 | 3adant2 1016 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐶 ↔ -𝑒𝐶 ≤ -𝑒𝐴)) |
24 | 21, 23 | anbi12d 473 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶) ↔ (-𝑒𝐵 ≤ -𝑒𝐴 ∧ -𝑒𝐶 ≤ -𝑒𝐴))) |
25 | 19, 24 | bitr4d 191 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
26 | 3, 17, 25 | 3bitr2d 216 | 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 2146 {cpr 3590 class class class wbr 3998 supcsup 6971 infcinf 6972 ℝ*cxr 7965 < clt 7966 ≤ cle 7967 -𝑒cxne 9738 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 |
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 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-isom 5217 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-sup 6973 df-inf 6974 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-inn 8891 df-2 8949 df-3 8950 df-4 8951 df-n0 9148 df-z 9225 df-uz 9500 df-rp 9623 df-xneg 9741 df-seqfrec 10414 df-exp 10488 df-cj 10818 df-re 10819 df-im 10820 df-rsqrt 10974 df-abs 10975 |
This theorem is referenced by: xrbdtri 11251 bdxmet 13494 bdmet 13495 |
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