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Theorem infxrnegsupex 10871
Description: The infimum of a set of extended reals 𝐴 is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)
Hypotheses
Ref Expression
infxrnegsupex.ex (𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
infxrnegsupex.ss (𝜑𝐴 ⊆ ℝ*)
Assertion
Ref Expression
infxrnegsupex (𝜑 → inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem infxrnegsupex
Dummy variables 𝑓 𝑔 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrlttri3 9424 . . . . 5 ((𝑓 ∈ ℝ*𝑔 ∈ ℝ*) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
21adantl 273 . . . 4 ((𝜑 ∧ (𝑓 ∈ ℝ*𝑔 ∈ ℝ*)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
3 infxrnegsupex.ex . . . 4 (𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
42, 3infclti 6825 . . 3 (𝜑 → inf(𝐴, ℝ*, < ) ∈ ℝ*)
5 xnegneg 9457 . . 3 (inf(𝐴, ℝ*, < ) ∈ ℝ* → -𝑒-𝑒inf(𝐴, ℝ*, < ) = inf(𝐴, ℝ*, < ))
64, 5syl 14 . 2 (𝜑 → -𝑒-𝑒inf(𝐴, ℝ*, < ) = inf(𝐴, ℝ*, < ))
7 xnegeq 9451 . . . . . . . . 9 (𝑤 = 𝑧 → -𝑒𝑤 = -𝑒𝑧)
87cbvmptv 3964 . . . . . . . 8 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑧 ∈ ℝ* ↦ -𝑒𝑧)
98mptpreima 4968 . . . . . . 7 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴) = {𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}
10 eqid 2100 . . . . . . . . . 10 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤)
1110xrnegiso 10870 . . . . . . . . 9 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*) ∧ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤))
1211simpri 112 . . . . . . . 8 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤)
1312imaeq1i 4814 . . . . . . 7 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴)
149, 13eqtr3i 2122 . . . . . 6 {𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴} = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴)
1514supeq1i 6790 . . . . 5 sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ) = sup(((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴), ℝ*, < )
1611simpli 110 . . . . . . . . 9 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*)
17 isocnv 5644 . . . . . . . . 9 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*) → (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*))
1816, 17ax-mp 7 . . . . . . . 8 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*)
19 isoeq1 5634 . . . . . . . . 9 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤) → ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*) ↔ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*)))
2012, 19ax-mp 7 . . . . . . . 8 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*) ↔ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*))
2118, 20mpbi 144 . . . . . . 7 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*)
2221a1i 9 . . . . . 6 (𝜑 → (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*))
23 infxrnegsupex.ss . . . . . 6 (𝜑𝐴 ⊆ ℝ*)
243cnvinfex 6820 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
252cnvti 6821 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ℝ*𝑔 ∈ ℝ*)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
2622, 23, 24, 25supisoti 6812 . . . . 5 (𝜑 → sup(((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴), ℝ*, < ) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, < )))
2715, 26syl5eq 2144 . . . 4 (𝜑 → sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, < )))
28 df-inf 6787 . . . . . . 7 inf(𝐴, ℝ*, < ) = sup(𝐴, ℝ*, < )
2928eqcomi 2104 . . . . . 6 sup(𝐴, ℝ*, < ) = inf(𝐴, ℝ*, < )
3029fveq2i 5356 . . . . 5 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, < )) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘inf(𝐴, ℝ*, < ))
31 eqidd 2101 . . . . . 6 (𝜑 → (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤))
32 xnegeq 9451 . . . . . . 7 (𝑤 = inf(𝐴, ℝ*, < ) → -𝑒𝑤 = -𝑒inf(𝐴, ℝ*, < ))
3332adantl 273 . . . . . 6 ((𝜑𝑤 = inf(𝐴, ℝ*, < )) → -𝑒𝑤 = -𝑒inf(𝐴, ℝ*, < ))
344xnegcld 9479 . . . . . 6 (𝜑 → -𝑒inf(𝐴, ℝ*, < ) ∈ ℝ*)
3531, 33, 4, 34fvmptd 5434 . . . . 5 (𝜑 → ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘inf(𝐴, ℝ*, < )) = -𝑒inf(𝐴, ℝ*, < ))
3630, 35syl5eq 2144 . . . 4 (𝜑 → ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, < )) = -𝑒inf(𝐴, ℝ*, < ))
3727, 36eqtr2d 2133 . . 3 (𝜑 → -𝑒inf(𝐴, ℝ*, < ) = sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ))
38 xnegeq 9451 . . 3 (-𝑒inf(𝐴, ℝ*, < ) = sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ) → -𝑒-𝑒inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ))
3937, 38syl 14 . 2 (𝜑 → -𝑒-𝑒inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ))
406, 39eqtr3d 2134 1 (𝜑 → inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104   = wceq 1299  wcel 1448  wral 2375  wrex 2376  {crab 2379  wss 3021   class class class wbr 3875  cmpt 3929  ccnv 4476  cima 4480  cfv 5059   Isom wiso 5060  supcsup 6784  infcinf 6785  *cxr 7671   < clt 7672  -𝑒cxne 9397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-apti 7610  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-sub 7806  df-neg 7807  df-xneg 9400
This theorem is referenced by:  xrminmax  10873
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