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Theorem supminfxrrnmpt 44991
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 2725 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supminfxrrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 44708 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
54supminfxr2 44989 . 2 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ))
6 xnegex 13222 . . . . . . . . . . . 12 -𝑒𝑦 ∈ V
72elrnmpt 5958 . . . . . . . . . . . 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 2725 . . . . . . . . . . 11 (𝑥𝐴 ↦ -𝑒𝐵) = (𝑥𝐴 ↦ -𝑒𝐵)
11 xnegneg 13228 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
1211eqcomd 2731 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ*𝑦 = -𝑒-𝑒𝑦)
1312adantr 479 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒-𝑒𝑦)
14 xnegeq 13221 . . . . . . . . . . . . . . . 16 (-𝑒𝑦 = 𝐵 → -𝑒-𝑒𝑦 = -𝑒𝐵)
1514adantl 480 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → -𝑒-𝑒𝑦 = -𝑒𝐵)
1613, 15eqtrd 2765 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒𝐵)
1716ex 411 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (-𝑒𝑦 = 𝐵𝑦 = -𝑒𝐵))
1817reximdv 3159 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (∃𝑥𝐴 -𝑒𝑦 = 𝐵 → ∃𝑥𝐴 𝑦 = -𝑒𝐵))
1918imp 405 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → ∃𝑥𝐴 𝑦 = -𝑒𝐵)
20 simpl 481 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ℝ*)
2110, 19, 20elrnmptd 5963 . . . . . . . . . 10 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
229, 21sylan2 591 . . . . . . . . 9 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
2322ex 411 . . . . . . . 8 (𝑦 ∈ ℝ* → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵)))
2423rgen 3052 . . . . . . 7 𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
25 rabss 4065 . . . . . . . 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 2891 . . . . . . . 8 𝑥-𝑒𝑦
30 nfmpt1 5257 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
3130nfrn 5954 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
3229, 31nfel 2906 . . . . . . 7 𝑥-𝑒𝑦 ∈ ran (𝑥𝐴𝐵)
33 nfcv 2891 . . . . . . 7 𝑥*
3432, 33nfrabw 3456 . . . . . 6 𝑥{𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}
35 xnegeq 13221 . . . . . . . 8 (𝑦 = -𝑒𝐵 → -𝑒𝑦 = -𝑒-𝑒𝐵)
3635eleq1d 2810 . . . . . . 7 (𝑦 = -𝑒𝐵 → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ -𝑒-𝑒𝐵 ∈ ran (𝑥𝐴𝐵)))
373xnegcld 13314 . . . . . . 7 ((𝜑𝑥𝐴) → -𝑒𝐵 ∈ ℝ*)
38 xnegneg 13228 . . . . . . . . 9 (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵)
393, 38syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝑒-𝑒𝐵 = 𝐵)
40 simpr 483 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
412, 40, 3elrnmpt1d 5964 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
4239, 41eqeltrd 2825 . . . . . . 7 ((𝜑𝑥𝐴) → -𝑒-𝑒𝐵 ∈ ran (𝑥𝐴𝐵))
4336, 37, 42elrabd 3681 . . . . . 6 ((𝜑𝑥𝐴) → -𝑒𝐵 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)})
441, 34, 10, 43rnmptssdf 44768 . . . . 5 (𝜑 → ran (𝑥𝐴 ↦ -𝑒𝐵) ⊆ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)})
4528, 44eqssd 3994 . . . 4 (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝑒𝐵))
4645infeq1d 9502 . . 3 (𝜑 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ) = inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
4746xnegeqd 44957 . 2 (𝜑 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
485, 47eqtrd 2765 1 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wnf 1777  wcel 2098  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  wss 3944  cmpt 5232  ran crn 5679  supcsup 9465  infcinf 9466  *cxr 11279   < clt 11280  -𝑒cxne 13124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-xneg 13127
This theorem is referenced by:  liminfvalxr  45309
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