xrlemininf - Intuitionistic Logic Explorer

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Theorem xrlemininf 11278

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 11272	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
2	1	breq2d 4015	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ^, < ) ↔ 𝐴 ≤ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
3	2	3adant1 1015	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ^, < ) ↔ 𝐴 ≤ -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
4		xnegcl 9831	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
5	4	3ad2ant2 1019	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
6		xnegcl 9831	. . . . . 6 ⊢ (𝐶 ∈ ℝ^* → -_𝑒𝐶 ∈ ℝ^*)
7	6	3ad2ant3 1020	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
8		xrmaxcl 11259	. . . . 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 9831	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
11	10	3ad2ant1 1018	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
12		xleneg 9836	. . . 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 9832	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
15	14	3ad2ant1 1018	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒-_𝑒𝐴 = 𝐴)
16	15	breq1d 4013	. . 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 11266	. . . 4 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^* ∧ -_𝑒𝐴 ∈ ℝ^) → (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ≤ -_𝑒𝐴 ↔ (-_𝑒𝐵 ≤ -_𝑒𝐴 ∧ -_𝑒𝐶 ≤ -_𝑒𝐴)))
19	5, 7, 11, 18	syl3anc 1238	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ≤ -_𝑒𝐴 ↔ (-_𝑒𝐵 ≤ -_𝑒𝐴 ∧ -_𝑒𝐶 ≤ -_𝑒𝐴)))
20		xleneg 9836	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ -_𝑒𝐵 ≤ -_𝑒𝐴))
21	20	3adant3 1017	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ -_𝑒𝐵 ≤ -_𝑒𝐴))
22		xleneg 9836	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐶 ↔ -_𝑒𝐶 ≤ -_𝑒𝐴))
23	22	3adant2 1016	. . . 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 978 = wceq 1353 ∈ wcel 2148 {cpr 3593 class class class wbr 4003 supcsup 6980 infcinf 6981 ℝ^*cxr 7990 < clt 7991 ≤ cle 7992 -_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: xrbdtri 11283 bdxmet 13937 bdmet 13938

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