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Theorem infxrnegsupex 11285
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 9811 . . . . 5 ((𝑓 ∈ ℝ*𝑔 ∈ ℝ*) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
21adantl 277 . . . 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 7036 . . 3 (𝜑 → inf(𝐴, ℝ*, < ) ∈ ℝ*)
5 xnegneg 9847 . . 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 9841 . . . . . . . . 9 (𝑤 = 𝑧 → -𝑒𝑤 = -𝑒𝑧)
87cbvmptv 4111 . . . . . . . 8 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑧 ∈ ℝ* ↦ -𝑒𝑧)
98mptpreima 5134 . . . . . . 7 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴) = {𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}
10 eqid 2187 . . . . . . . . . 10 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤)
1110xrnegiso 11284 . . . . . . . . 9 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*) ∧ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤))
1211simpri 113 . . . . . . . 8 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤)
1312imaeq1i 4979 . . . . . . 7 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴)
149, 13eqtr3i 2210 . . . . . 6 {𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴} = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴)
1514supeq1i 7001 . . . . 5 sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ) = sup(((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴), ℝ*, < )
1611simpli 111 . . . . . . . . 9 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*)
17 isocnv 5825 . . . . . . . . 9 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*) → (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*))
1816, 17ax-mp 5 . . . . . . . 8 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*)
19 isoeq1 5815 . . . . . . . . 9 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤) → ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*) ↔ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*)))
2012, 19ax-mp 5 . . . . . . . 8 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*) ↔ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*))
2118, 20mpbi 145 . . . . . . 7 (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*)
2221a1i 9 . . . . . 6 (𝜑 → (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , < (ℝ*, ℝ*))
23 infxrnegsupex.ss . . . . . 6 (𝜑𝐴 ⊆ ℝ*)
243cnvinfex 7031 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
252cnvti 7032 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ℝ*𝑔 ∈ ℝ*)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
2622, 23, 24, 25supisoti 7023 . . . . 5 (𝜑 → sup(((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴), ℝ*, < ) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, < )))
2715, 26eqtrid 2232 . . . 4 (𝜑 → sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, < )))
28 df-inf 6998 . . . . . . 7 inf(𝐴, ℝ*, < ) = sup(𝐴, ℝ*, < )
2928eqcomi 2191 . . . . . 6 sup(𝐴, ℝ*, < ) = inf(𝐴, ℝ*, < )
3029fveq2i 5530 . . . . 5 ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, < )) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘inf(𝐴, ℝ*, < ))
31 eqidd 2188 . . . . . 6 (𝜑 → (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤))
32 xnegeq 9841 . . . . . . 7 (𝑤 = inf(𝐴, ℝ*, < ) → -𝑒𝑤 = -𝑒inf(𝐴, ℝ*, < ))
3332adantl 277 . . . . . 6 ((𝜑𝑤 = inf(𝐴, ℝ*, < )) → -𝑒𝑤 = -𝑒inf(𝐴, ℝ*, < ))
344xnegcld 9869 . . . . . 6 (𝜑 → -𝑒inf(𝐴, ℝ*, < ) ∈ ℝ*)
3531, 33, 4, 34fvmptd 5610 . . . . 5 (𝜑 → ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘inf(𝐴, ℝ*, < )) = -𝑒inf(𝐴, ℝ*, < ))
3630, 35eqtrid 2232 . . . 4 (𝜑 → ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, < )) = -𝑒inf(𝐴, ℝ*, < ))
3727, 36eqtr2d 2221 . . 3 (𝜑 → -𝑒inf(𝐴, ℝ*, < ) = sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ))
38 xnegeq 9841 . . 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 2222 1 (𝜑 → inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧𝐴}, ℝ*, < ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158  wral 2465  wrex 2466  {crab 2469  wss 3141   class class class wbr 4015  cmpt 4076  ccnv 4637  cima 4641  cfv 5228   Isom wiso 5229  supcsup 6995  infcinf 6996  *cxr 8005   < clt 8006  -𝑒cxne 9783
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-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-distr 7929  ax-i2m1 7930  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-apti 7940  ax-pre-ltadd 7941
This theorem depends on definitions:  df-bi 117  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-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  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-sup 6997  df-inf 6998  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-sub 8144  df-neg 8145  df-xneg 9786
This theorem is referenced by:  xrminmax  11287
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