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Theorem lemininf 11731

Description: Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.)

**Assertion**
Ref	Expression
lemininf	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ, < ) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶)))

**Proof of Theorem lemininf**
Step	Hyp	Ref	Expression
1		simp2 1022	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ)
2		simp3 1023	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ)
3		minmax 11727	. . . 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 4094	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ, < ) ↔ 𝐴 ≤ -sup({-𝐵, -𝐶}, ℝ, < )))
6	1	renegcld 8514	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → -𝐵 ∈ ℝ)
7	2	renegcld 8514	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → -𝐶 ∈ ℝ)
8		maxcl 11707	. . . 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 1021	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
11		lenegcon2 8602	. . 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 8514	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → -𝐴 ∈ ℝ)
14		maxleastb 11711	. . . 4 ⊢ ((-𝐵 ∈ ℝ ∧ -𝐶 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (sup({-𝐵, -𝐶}, ℝ, < ) ≤ -𝐴 ↔ (-𝐵 ≤ -𝐴 ∧ -𝐶 ≤ -𝐴)))
15	6, 7, 13, 14	syl3anc 1271	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({-𝐵, -𝐶}, ℝ, < ) ≤ -𝐴 ↔ (-𝐵 ≤ -𝐴 ∧ -𝐶 ≤ -𝐴)))
16	10, 1	lenegd 8659	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴))
17	10, 2	lenegd 8659	. . . 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 1002 = wceq 1395 ∈ wcel 2200 {cpr 3667 class class class wbr 4082 supcsup 7137 infcinf 7138 ℝcr 7986 < clt 8169 ≤ cle 8170 -cneg 8306

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4521 ax-setind 4626 ax-iinf 4677 ax-cnex 8078 ax-resscn 8079 ax-1cn 8080 ax-1re 8081 ax-icn 8082 ax-addcl 8083 ax-addrcl 8084 ax-mulcl 8085 ax-mulrcl 8086 ax-addcom 8087 ax-mulcom 8088 ax-addass 8089 ax-mulass 8090 ax-distr 8091 ax-i2m1 8092 ax-0lt1 8093 ax-1rid 8094 ax-0id 8095 ax-rnegex 8096 ax-precex 8097 ax-cnre 8098 ax-pre-ltirr 8099 ax-pre-ltwlin 8100 ax-pre-lttrn 8101 ax-pre-apti 8102 ax-pre-ltadd 8103 ax-pre-mulgt0 8104 ax-pre-mulext 8105 ax-arch 8106 ax-caucvg 8107

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4381 df-po 4384 df-iso 4385 df-iord 4454 df-on 4456 df-ilim 4457 df-suc 4459 df-iom 4680 df-xp 4722 df-rel 4723 df-cnv 4724 df-co 4725 df-dm 4726 df-rn 4727 df-res 4728 df-ima 4729 df-iota 5274 df-fun 5316 df-fn 5317 df-f 5318 df-f1 5319 df-fo 5320 df-f1o 5321 df-fv 5322 df-isom 5323 df-riota 5947 df-ov 5997 df-oprab 5998 df-mpo 5999 df-1st 6276 df-2nd 6277 df-recs 6441 df-frec 6527 df-sup 7139 df-inf 7140 df-pnf 8171 df-mnf 8172 df-xr 8173 df-ltxr 8174 df-le 8175 df-sub 8307 df-neg 8308 df-reap 8710 df-ap 8717 df-div 8808 df-inn 9099 df-2 9157 df-3 9158 df-4 9159 df-n0 9358 df-z 9435 df-uz 9711 df-rp 9838 df-seqfrec 10657 df-exp 10748 df-cj 11339 df-re 11340 df-im 11341 df-rsqrt 11495 df-abs 11496

This theorem is referenced by: mul0inf 11738 pc2dvds 12839

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