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Theorem infrenegsupex 9624
Description: The infimum of a set of reals 𝐴 is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)
Hypotheses
Ref Expression
infrenegsupex.ex (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
infrenegsupex.ss (𝜑𝐴 ⊆ ℝ)
Assertion
Ref Expression
infrenegsupex (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧𝐴}, ℝ, < ))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem infrenegsupex
Dummy variables 𝑓 𝑔 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lttri3 8067 . . . . . 6 ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
21adantl 277 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
3 infrenegsupex.ex . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
42, 3infclti 7052 . . . 4 (𝜑 → inf(𝐴, ℝ, < ) ∈ ℝ)
54recnd 8016 . . 3 (𝜑 → inf(𝐴, ℝ, < ) ∈ ℂ)
65negnegd 8289 . 2 (𝜑 → --inf(𝐴, ℝ, < ) = inf(𝐴, ℝ, < ))
7 negeq 8180 . . . . . . . . 9 (𝑤 = 𝑧 → -𝑤 = -𝑧)
87cbvmptv 4114 . . . . . . . 8 (𝑤 ∈ ℝ ↦ -𝑤) = (𝑧 ∈ ℝ ↦ -𝑧)
98mptpreima 5140 . . . . . . 7 ((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴) = {𝑧 ∈ ℝ ∣ -𝑧𝐴}
10 eqid 2189 . . . . . . . . . 10 (𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤)
1110negiso 8942 . . . . . . . . 9 ((𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ) ∧ (𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤))
1211simpri 113 . . . . . . . 8 (𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤)
1312imaeq1i 4985 . . . . . . 7 ((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴) = ((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴)
149, 13eqtr3i 2212 . . . . . 6 {𝑧 ∈ ℝ ∣ -𝑧𝐴} = ((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴)
1514supeq1i 7017 . . . . 5 sup({𝑧 ∈ ℝ ∣ -𝑧𝐴}, ℝ, < ) = sup(((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴), ℝ, < )
1611simpli 111 . . . . . . . . 9 (𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ)
17 isocnv 5833 . . . . . . . . 9 ((𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ) → (𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ))
1816, 17ax-mp 5 . . . . . . . 8 (𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ)
19 isoeq1 5823 . . . . . . . . 9 ((𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤) → ((𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ) ↔ (𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ)))
2012, 19ax-mp 5 . . . . . . . 8 ((𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ) ↔ (𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ))
2118, 20mpbi 145 . . . . . . 7 (𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ)
2221a1i 9 . . . . . 6 (𝜑 → (𝑤 ∈ ℝ ↦ -𝑤) Isom < , < (ℝ, ℝ))
23 infrenegsupex.ss . . . . . 6 (𝜑𝐴 ⊆ ℝ)
243cnvinfex 7047 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
252cnvti 7048 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
2622, 23, 24, 25supisoti 7039 . . . . 5 (𝜑 → sup(((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴), ℝ, < ) = ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, < )))
2715, 26eqtrid 2234 . . . 4 (𝜑 → sup({𝑧 ∈ ℝ ∣ -𝑧𝐴}, ℝ, < ) = ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, < )))
28 df-inf 7014 . . . . . . 7 inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, < )
2928eqcomi 2193 . . . . . 6 sup(𝐴, ℝ, < ) = inf(𝐴, ℝ, < )
3029fveq2i 5537 . . . . 5 ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, < )) = ((𝑤 ∈ ℝ ↦ -𝑤)‘inf(𝐴, ℝ, < ))
31 eqidd 2190 . . . . . 6 (𝜑 → (𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤))
32 negeq 8180 . . . . . . 7 (𝑤 = inf(𝐴, ℝ, < ) → -𝑤 = -inf(𝐴, ℝ, < ))
3332adantl 277 . . . . . 6 ((𝜑𝑤 = inf(𝐴, ℝ, < )) → -𝑤 = -inf(𝐴, ℝ, < ))
345negcld 8285 . . . . . 6 (𝜑 → -inf(𝐴, ℝ, < ) ∈ ℂ)
3531, 33, 4, 34fvmptd 5618 . . . . 5 (𝜑 → ((𝑤 ∈ ℝ ↦ -𝑤)‘inf(𝐴, ℝ, < )) = -inf(𝐴, ℝ, < ))
3630, 35eqtrid 2234 . . . 4 (𝜑 → ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, < )) = -inf(𝐴, ℝ, < ))
3727, 36eqtr2d 2223 . . 3 (𝜑 → -inf(𝐴, ℝ, < ) = sup({𝑧 ∈ ℝ ∣ -𝑧𝐴}, ℝ, < ))
3837negeqd 8182 . 2 (𝜑 → --inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧𝐴}, ℝ, < ))
396, 38eqtr3d 2224 1 (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧𝐴}, ℝ, < ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wral 2468  wrex 2469  {crab 2472  wss 3144   class class class wbr 4018  cmpt 4079  ccnv 4643  cima 4647  cfv 5235   Isom wiso 5236  supcsup 7011  infcinf 7012  cc 7839  cr 7840   < clt 8022  -cneg 8159
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-apti 7956  ax-pre-ltadd 7957
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-sup 7013  df-inf 7014  df-pnf 8024  df-mnf 8025  df-ltxr 8027  df-sub 8160  df-neg 8161
This theorem is referenced by:  supminfex  9627  minmax  11270  infssuzcldc  11984
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