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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 11002 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → inf({𝐵, 𝐶}, ℝ*, < ) = -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < )) | |
2 | 1 | breq2d 3911 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ*, < ) ↔ 𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
3 | 2 | 3adant1 984 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ*, < ) ↔ 𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
4 | xnegcl 9583 | . . . . . 6 ⊢ (𝐵 ∈ ℝ* → -𝑒𝐵 ∈ ℝ*) | |
5 | 4 | 3ad2ant2 988 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐵 ∈ ℝ*) |
6 | xnegcl 9583 | . . . . . 6 ⊢ (𝐶 ∈ ℝ* → -𝑒𝐶 ∈ ℝ*) | |
7 | 6 | 3ad2ant3 989 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐶 ∈ ℝ*) |
8 | xrmaxcl 10989 | . . . . 5 ⊢ ((-𝑒𝐵 ∈ ℝ* ∧ -𝑒𝐶 ∈ ℝ*) → sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ∈ ℝ*) | |
9 | 5, 7, 8 | syl2anc 408 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ∈ ℝ*) |
10 | xnegcl 9583 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) | |
11 | 10 | 3ad2ant1 987 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐴 ∈ ℝ*) |
12 | xleneg 9588 | . . . 4 ⊢ ((sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ∈ ℝ* ∧ -𝑒𝐴 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ -𝑒-𝑒𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) | |
13 | 9, 11, 12 | syl2anc 408 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ -𝑒-𝑒𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
14 | xnegneg 9584 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) | |
15 | 14 | 3ad2ant1 987 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒-𝑒𝐴 = 𝐴) |
16 | 15 | breq1d 3909 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (-𝑒-𝑒𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ↔ 𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
17 | 13, 16 | bitrd 187 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ 𝐴 ≤ -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ))) |
18 | xrmaxlesup 10996 | . . . 4 ⊢ ((-𝑒𝐵 ∈ ℝ* ∧ -𝑒𝐶 ∈ ℝ* ∧ -𝑒𝐴 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ (-𝑒𝐵 ≤ -𝑒𝐴 ∧ -𝑒𝐶 ≤ -𝑒𝐴))) | |
19 | 5, 7, 11, 18 | syl3anc 1201 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ (-𝑒𝐵 ≤ -𝑒𝐴 ∧ -𝑒𝐶 ≤ -𝑒𝐴))) |
20 | xleneg 9588 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴)) | |
21 | 20 | 3adant3 986 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴)) |
22 | xleneg 9588 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐶 ↔ -𝑒𝐶 ≤ -𝑒𝐴)) | |
23 | 22 | 3adant2 985 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐶 ↔ -𝑒𝐶 ≤ -𝑒𝐴)) |
24 | 21, 23 | anbi12d 464 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶) ↔ (-𝑒𝐵 ≤ -𝑒𝐴 ∧ -𝑒𝐶 ≤ -𝑒𝐴))) |
25 | 19, 24 | bitr4d 190 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ≤ -𝑒𝐴 ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
26 | 3, 17, 25 | 3bitr2d 215 | 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 103 ↔ wb 104 ∧ w3a 947 = wceq 1316 ∈ wcel 1465 {cpr 3498 class class class wbr 3899 supcsup 6837 infcinf 6838 ℝ*cxr 7767 < clt 7768 ≤ cle 7769 -𝑒cxne 9524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-isom 5102 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-sup 6839 df-inf 6840 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-n0 8946 df-z 9023 df-uz 9295 df-rp 9410 df-xneg 9527 df-seqfrec 10187 df-exp 10261 df-cj 10582 df-re 10583 df-im 10584 df-rsqrt 10738 df-abs 10739 |
This theorem is referenced by: xrbdtri 11013 bdxmet 12597 bdmet 12598 |
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