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Theorem supminfxrrnmpt 44181
Description: The indexed supremum of a set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
supminfxrrnmpt.x 𝑥𝜑
supminfxrrnmpt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
Assertion
Ref Expression
supminfxrrnmpt (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem supminfxrrnmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 supminfxrrnmpt.x . . . 4 𝑥𝜑
2 eqid 2733 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supminfxrrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 43895 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
54supminfxr2 44179 . 2 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ))
6 xnegex 13187 . . . . . . . . . . . 12 -𝑒𝑦 ∈ V
72elrnmpt 5956 . . . . . . . . . . . 12 (-𝑒𝑦 ∈ V → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑒𝑦 = 𝐵))
86, 7ax-mp 5 . . . . . . . . . . 11 (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑒𝑦 = 𝐵)
98biimpi 215 . . . . . . . . . 10 (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 -𝑒𝑦 = 𝐵)
10 eqid 2733 . . . . . . . . . . 11 (𝑥𝐴 ↦ -𝑒𝐵) = (𝑥𝐴 ↦ -𝑒𝐵)
11 xnegneg 13193 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
1211eqcomd 2739 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ*𝑦 = -𝑒-𝑒𝑦)
1312adantr 482 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒-𝑒𝑦)
14 xnegeq 13186 . . . . . . . . . . . . . . . 16 (-𝑒𝑦 = 𝐵 → -𝑒-𝑒𝑦 = -𝑒𝐵)
1514adantl 483 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → -𝑒-𝑒𝑦 = -𝑒𝐵)
1613, 15eqtrd 2773 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒𝐵)
1716ex 414 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (-𝑒𝑦 = 𝐵𝑦 = -𝑒𝐵))
1817reximdv 3171 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (∃𝑥𝐴 -𝑒𝑦 = 𝐵 → ∃𝑥𝐴 𝑦 = -𝑒𝐵))
1918imp 408 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → ∃𝑥𝐴 𝑦 = -𝑒𝐵)
20 simpl 484 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ℝ*)
2110, 19, 20elrnmptd 5961 . . . . . . . . . 10 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
229, 21sylan2 594 . . . . . . . . 9 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
2322ex 414 . . . . . . . 8 (𝑦 ∈ ℝ* → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵)))
2423rgen 3064 . . . . . . 7 𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
25 rabss 4070 . . . . . . . 8 ({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝑒𝐵) ↔ ∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵)))
2625biimpri 227 . . . . . . 7 (∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵)) → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝑒𝐵))
2724, 26ax-mp 5 . . . . . 6 {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝑒𝐵)
2827a1i 11 . . . . 5 (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝑒𝐵))
29 nfcv 2904 . . . . . . . 8 𝑥-𝑒𝑦
30 nfmpt1 5257 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
3130nfrn 5952 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
3229, 31nfel 2918 . . . . . . 7 𝑥-𝑒𝑦 ∈ ran (𝑥𝐴𝐵)
33 nfcv 2904 . . . . . . 7 𝑥*
3432, 33nfrabw 3469 . . . . . 6 𝑥{𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}
35 xnegeq 13186 . . . . . . . 8 (𝑦 = -𝑒𝐵 → -𝑒𝑦 = -𝑒-𝑒𝐵)
3635eleq1d 2819 . . . . . . 7 (𝑦 = -𝑒𝐵 → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ -𝑒-𝑒𝐵 ∈ ran (𝑥𝐴𝐵)))
373xnegcld 13279 . . . . . . 7 ((𝜑𝑥𝐴) → -𝑒𝐵 ∈ ℝ*)
38 xnegneg 13193 . . . . . . . . 9 (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵)
393, 38syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝑒-𝑒𝐵 = 𝐵)
40 simpr 486 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
412, 40, 3elrnmpt1d 43932 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
4239, 41eqeltrd 2834 . . . . . . 7 ((𝜑𝑥𝐴) → -𝑒-𝑒𝐵 ∈ ran (𝑥𝐴𝐵))
4336, 37, 42elrabd 3686 . . . . . 6 ((𝜑𝑥𝐴) → -𝑒𝐵 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)})
441, 34, 10, 43rnmptssdf 43958 . . . . 5 (𝜑 → ran (𝑥𝐴 ↦ -𝑒𝐵) ⊆ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)})
4528, 44eqssd 4000 . . . 4 (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝑒𝐵))
4645infeq1d 9472 . . 3 (𝜑 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ) = inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
4746xnegeqd 44147 . 2 (𝜑 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
485, 47eqtrd 2773 1 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wnf 1786  wcel 2107  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  wss 3949  cmpt 5232  ran crn 5678  supcsup 9435  infcinf 9436  *cxr 11247   < clt 11248  -𝑒cxne 13089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-xneg 13092
This theorem is referenced by:  liminfvalxr  44499
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