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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 11272 | . . . 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 4015 | . 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 9854 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐵 ∈ ℝ*) |
6 | simp3 999 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐶 ∈ ℝ*) | |
7 | 6 | xnegcld 9854 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐶 ∈ ℝ*) |
8 | simp1 997 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐴 ∈ ℝ*) | |
9 | 8 | xnegcld 9854 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒𝐴 ∈ ℝ*) |
10 | xrmaxltsup 11265 | . . . 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 11259 | . . . . . . 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 9854 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → -𝑒sup({-𝑒𝐵, -𝑒𝐶}, ℝ*, < ) ∈ ℝ*) |
15 | xltneg 9835 | . . . . 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 9832 | . . . . . 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 4013 | . . . 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 9835 | . . . . 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 9835 | . . . . 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 3593 class class class wbr 4003 supcsup 6980 infcinf 6981 ℝ*cxr 7990 < clt 7991 -𝑒cxne 9768 |
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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-sup 6982 df-inf 6983 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-n0 9176 df-z 9253 df-uz 9528 df-rp 9653 df-xneg 9771 df-seqfrec 10445 df-exp 10519 df-cj 10850 df-re 10851 df-im 10852 df-rsqrt 11006 df-abs 11007 |
This theorem is referenced by: xrminrpcl 11281 iooinsup 11284 blininf 13860 bdxmet 13937 bdmopn 13940 |
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