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Mirrors > Home > ILE Home > Th. List > xrminrpcl | GIF version |
Description: The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.) |
Ref | Expression |
---|---|
xrminrpcl | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → inf({𝐴, 𝐵}, ℝ*, < ) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpxr 9588 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*) | |
2 | rpxr 9588 | . . . 4 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ*) | |
3 | xrminmax 11192 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → inf({𝐴, 𝐵}, ℝ*, < ) = -𝑒sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < )) | |
4 | 1, 2, 3 | syl2an 287 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → inf({𝐴, 𝐵}, ℝ*, < ) = -𝑒sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < )) |
5 | rpre 9587 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
6 | rexneg 9757 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
7 | renegcl 8150 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
8 | 6, 7 | eqeltrd 2241 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ) |
9 | 5, 8 | syl 14 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → -𝑒𝐴 ∈ ℝ) |
10 | rpre 9587 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
11 | rexneg 9757 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → -𝑒𝐵 = -𝐵) | |
12 | renegcl 8150 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
13 | 11, 12 | eqeltrd 2241 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → -𝑒𝐵 ∈ ℝ) |
14 | 10, 13 | syl 14 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → -𝑒𝐵 ∈ ℝ) |
15 | xrmaxrecl 11182 | . . . . . . 7 ⊢ ((-𝑒𝐴 ∈ ℝ ∧ -𝑒𝐵 ∈ ℝ) → sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < ) = sup({-𝑒𝐴, -𝑒𝐵}, ℝ, < )) | |
16 | 9, 14, 15 | syl2an 287 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < ) = sup({-𝑒𝐴, -𝑒𝐵}, ℝ, < )) |
17 | maxcl 11138 | . . . . . . 7 ⊢ ((-𝑒𝐴 ∈ ℝ ∧ -𝑒𝐵 ∈ ℝ) → sup({-𝑒𝐴, -𝑒𝐵}, ℝ, < ) ∈ ℝ) | |
18 | 9, 14, 17 | syl2an 287 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → sup({-𝑒𝐴, -𝑒𝐵}, ℝ, < ) ∈ ℝ) |
19 | 16, 18 | eqeltrd 2241 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < ) ∈ ℝ) |
20 | rexneg 9757 | . . . . 5 ⊢ (sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < ) ∈ ℝ → -𝑒sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < ) = -sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < )) | |
21 | 19, 20 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → -𝑒sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < ) = -sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < )) |
22 | 19 | renegcld 8269 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → -sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < ) ∈ ℝ) |
23 | 21, 22 | eqeltrd 2241 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → -𝑒sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < ) ∈ ℝ) |
24 | 4, 23 | eqeltrd 2241 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → inf({𝐴, 𝐵}, ℝ*, < ) ∈ ℝ) |
25 | rpgt0 9592 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
26 | rpgt0 9592 | . . . 4 ⊢ (𝐵 ∈ ℝ+ → 0 < 𝐵) | |
27 | 25, 26 | anim12i 336 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (0 < 𝐴 ∧ 0 < 𝐵)) |
28 | 0xr 7936 | . . . 4 ⊢ 0 ∈ ℝ* | |
29 | xrltmininf 11197 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 < inf({𝐴, 𝐵}, ℝ*, < ) ↔ (0 < 𝐴 ∧ 0 < 𝐵))) | |
30 | 28, 1, 2, 29 | mp3an3an 1332 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (0 < inf({𝐴, 𝐵}, ℝ*, < ) ↔ (0 < 𝐴 ∧ 0 < 𝐵))) |
31 | 27, 30 | mpbird 166 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → 0 < inf({𝐴, 𝐵}, ℝ*, < )) |
32 | 24, 31 | elrpd 9620 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → inf({𝐴, 𝐵}, ℝ*, < ) ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 {cpr 3571 class class class wbr 3976 supcsup 6938 infcinf 6939 ℝcr 7743 0cc0 7744 ℝ*cxr 7923 < clt 7924 -cneg 8061 ℝ+crp 9580 -𝑒cxne 9696 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-rp 9581 df-xneg 9699 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 |
This theorem is referenced by: blin2 12979 xmettx 13057 |
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