xrlemininf - Intuitionistic Logic Explorer

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Theorem xrlemininf 11414

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.)

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

**Proof of Theorem xrlemininf**
Step	Hyp	Ref	Expression
1		xrminmax 11408	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
2	1	breq2d 4041	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ^, < ) ↔ 𝐴 ≤ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
3	2	3adant1 1017	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ^, < ) ↔ 𝐴 ≤ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
4		xnegcl 9898	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
5	4	3ad2ant2 1021	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
6		xnegcl 9898	. . . . . 6 ⊢ (𝐶 ∈ ℝ^* → -_𝑒𝐶 ∈ ℝ^*)
7	6	3ad2ant3 1022	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
8		xrmaxcl 11395	. . . . 5 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
9	5, 7, 8	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
10		xnegcl 9898	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
11	10	3ad2ant1 1020	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
12		xleneg 9903	. . . 4 ⊢ ((sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^ ∧ -_𝑒𝐴 ∈ ℝ^) → (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ≤ -_𝑒𝐴 ↔ -_𝑒-_𝑒𝐴 ≤ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
13	9, 11, 12	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ≤ -_𝑒𝐴 ↔ -_𝑒-_𝑒𝐴 ≤ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
14		xnegneg 9899	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
15	14	3ad2ant1 1020	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒-_𝑒𝐴 = 𝐴)
16	15	breq1d 4039	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (-_𝑒-_𝑒𝐴 ≤ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ↔ 𝐴 ≤ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
17	13, 16	bitrd 188	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ≤ -_𝑒𝐴 ↔ 𝐴 ≤ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
18		xrmaxlesup 11402	. . . 4 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^* ∧ -_𝑒𝐴 ∈ ℝ^) → (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ≤ -_𝑒𝐴 ↔ (-_𝑒𝐵 ≤ -_𝑒𝐴 ∧ -_𝑒𝐶 ≤ -_𝑒𝐴)))
19	5, 7, 11, 18	syl3anc 1249	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ≤ -_𝑒𝐴 ↔ (-_𝑒𝐵 ≤ -_𝑒𝐴 ∧ -_𝑒𝐶 ≤ -_𝑒𝐴)))
20		xleneg 9903	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ -_𝑒𝐵 ≤ -_𝑒𝐴))
21	20	3adant3 1019	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ -_𝑒𝐵 ≤ -_𝑒𝐴))
22		xleneg 9903	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐶 ↔ -_𝑒𝐶 ≤ -_𝑒𝐴))
23	22	3adant2 1018	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐶 ↔ -_𝑒𝐶 ≤ -_𝑒𝐴))
24	21, 23	anbi12d 473	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶) ↔ (-_𝑒𝐵 ≤ -_𝑒𝐴 ∧ -_𝑒𝐶 ≤ -_𝑒𝐴)))
25	19, 24	bitr4d 191	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ≤ -_𝑒𝐴 ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶)))
26	3, 17, 25	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 980 = wceq 1364 ∈ wcel 2164 {cpr 3619 class class class wbr 4029 supcsup 7041 infcinf 7042 ℝ^*cxr 8053 < clt 8054 ≤ cle 8055 -_𝑒cxne 9835

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-sup 7043 df-inf 7044 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-rp 9720 df-xneg 9838 df-seqfrec 10519 df-exp 10610 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143

This theorem is referenced by: xrbdtri 11419 bdxmet 14669 bdmet 14670

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