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Mirrors > Home > ILE Home > Th. List > lemininf | GIF version |
Description: Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.) |
Ref | Expression |
---|---|
lemininf | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ, < ) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 999 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ) | |
2 | simp3 1000 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ) | |
3 | minmax 11251 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐵, 𝐶}, ℝ, < ) = -sup({-𝐵, -𝐶}, ℝ, < )) | |
4 | 1, 2, 3 | syl2anc 411 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → inf({𝐵, 𝐶}, ℝ, < ) = -sup({-𝐵, -𝐶}, ℝ, < )) |
5 | 4 | breq2d 4027 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ, < ) ↔ 𝐴 ≤ -sup({-𝐵, -𝐶}, ℝ, < ))) |
6 | 1 | renegcld 8350 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → -𝐵 ∈ ℝ) |
7 | 2 | renegcld 8350 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → -𝐶 ∈ ℝ) |
8 | maxcl 11232 | . . . 4 ⊢ ((-𝐵 ∈ ℝ ∧ -𝐶 ∈ ℝ) → sup({-𝐵, -𝐶}, ℝ, < ) ∈ ℝ) | |
9 | 6, 7, 8 | syl2anc 411 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → sup({-𝐵, -𝐶}, ℝ, < ) ∈ ℝ) |
10 | simp1 998 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ) | |
11 | lenegcon2 8437 | . . 3 ⊢ ((sup({-𝐵, -𝐶}, ℝ, < ) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (sup({-𝐵, -𝐶}, ℝ, < ) ≤ -𝐴 ↔ 𝐴 ≤ -sup({-𝐵, -𝐶}, ℝ, < ))) | |
12 | 9, 10, 11 | syl2anc 411 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({-𝐵, -𝐶}, ℝ, < ) ≤ -𝐴 ↔ 𝐴 ≤ -sup({-𝐵, -𝐶}, ℝ, < ))) |
13 | 10 | renegcld 8350 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → -𝐴 ∈ ℝ) |
14 | maxleastb 11236 | . . . 4 ⊢ ((-𝐵 ∈ ℝ ∧ -𝐶 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (sup({-𝐵, -𝐶}, ℝ, < ) ≤ -𝐴 ↔ (-𝐵 ≤ -𝐴 ∧ -𝐶 ≤ -𝐴))) | |
15 | 6, 7, 13, 14 | syl3anc 1248 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({-𝐵, -𝐶}, ℝ, < ) ≤ -𝐴 ↔ (-𝐵 ≤ -𝐴 ∧ -𝐶 ≤ -𝐴))) |
16 | 10, 1 | lenegd 8494 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) |
17 | 10, 2 | lenegd 8494 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐶 ↔ -𝐶 ≤ -𝐴)) |
18 | 16, 17 | anbi12d 473 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶) ↔ (-𝐵 ≤ -𝐴 ∧ -𝐶 ≤ -𝐴))) |
19 | 15, 18 | bitr4d 191 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({-𝐵, -𝐶}, ℝ, < ) ≤ -𝐴 ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) |
20 | 5, 12, 19 | 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 979 = wceq 1363 ∈ wcel 2158 {cpr 3605 class class class wbr 4015 supcsup 6994 infcinf 6995 ℝcr 7823 < clt 8005 ≤ cle 8006 -cneg 8142 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 ax-caucvg 7944 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-sup 6996 df-inf 6997 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-n0 9190 df-z 9267 df-uz 9542 df-rp 9667 df-seqfrec 10459 df-exp 10533 df-cj 10864 df-re 10865 df-im 10866 df-rsqrt 11020 df-abs 11021 |
This theorem is referenced by: mul0inf 11262 pc2dvds 12342 |
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