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Theorem supminfxrrnmpt 45482
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 2737 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supminfxrrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 45201 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
54supminfxr2 45480 . 2 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ))
6 xnegex 13250 . . . . . . . . . . . 12 -𝑒𝑦 ∈ V
72elrnmpt 5969 . . . . . . . . . . . 12 (-𝑒𝑦 ∈ V → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑒𝑦 = 𝐵))
86, 7ax-mp 5 . . . . . . . . . . 11 (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑒𝑦 = 𝐵)
98biimpi 216 . . . . . . . . . 10 (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 -𝑒𝑦 = 𝐵)
10 eqid 2737 . . . . . . . . . . 11 (𝑥𝐴 ↦ -𝑒𝐵) = (𝑥𝐴 ↦ -𝑒𝐵)
11 xnegneg 13256 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
1211eqcomd 2743 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ*𝑦 = -𝑒-𝑒𝑦)
1312adantr 480 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒-𝑒𝑦)
14 xnegeq 13249 . . . . . . . . . . . . . . . 16 (-𝑒𝑦 = 𝐵 → -𝑒-𝑒𝑦 = -𝑒𝐵)
1514adantl 481 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → -𝑒-𝑒𝑦 = -𝑒𝐵)
1613, 15eqtrd 2777 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒𝐵)
1716ex 412 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (-𝑒𝑦 = 𝐵𝑦 = -𝑒𝐵))
1817reximdv 3170 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (∃𝑥𝐴 -𝑒𝑦 = 𝐵 → ∃𝑥𝐴 𝑦 = -𝑒𝐵))
1918imp 406 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → ∃𝑥𝐴 𝑦 = -𝑒𝐵)
20 simpl 482 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ℝ*)
2110, 19, 20elrnmptd 5974 . . . . . . . . . 10 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
229, 21sylan2 593 . . . . . . . . 9 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
2322ex 412 . . . . . . . 8 (𝑦 ∈ ℝ* → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵)))
2423rgen 3063 . . . . . . 7 𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
25 rabss 4072 . . . . . . . 8 ({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝑒𝐵) ↔ ∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵)))
2625biimpri 228 . . . . . . 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 2905 . . . . . . . 8 𝑥-𝑒𝑦
30 nfmpt1 5250 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
3130nfrn 5963 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
3229, 31nfel 2920 . . . . . . 7 𝑥-𝑒𝑦 ∈ ran (𝑥𝐴𝐵)
33 nfcv 2905 . . . . . . 7 𝑥*
3432, 33nfrabw 3475 . . . . . 6 𝑥{𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}
35 xnegeq 13249 . . . . . . . 8 (𝑦 = -𝑒𝐵 → -𝑒𝑦 = -𝑒-𝑒𝐵)
3635eleq1d 2826 . . . . . . 7 (𝑦 = -𝑒𝐵 → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ -𝑒-𝑒𝐵 ∈ ran (𝑥𝐴𝐵)))
373xnegcld 13342 . . . . . . 7 ((𝜑𝑥𝐴) → -𝑒𝐵 ∈ ℝ*)
38 xnegneg 13256 . . . . . . . . 9 (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵)
393, 38syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝑒-𝑒𝐵 = 𝐵)
40 simpr 484 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
412, 40, 3elrnmpt1d 5975 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
4239, 41eqeltrd 2841 . . . . . . 7 ((𝜑𝑥𝐴) → -𝑒-𝑒𝐵 ∈ ran (𝑥𝐴𝐵))
4336, 37, 42elrabd 3694 . . . . . 6 ((𝜑𝑥𝐴) → -𝑒𝐵 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)})
441, 34, 10, 43rnmptssdf 45261 . . . . 5 (𝜑 → ran (𝑥𝐴 ↦ -𝑒𝐵) ⊆ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)})
4528, 44eqssd 4001 . . . 4 (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝑒𝐵))
4645infeq1d 9517 . . 3 (𝜑 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ) = inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
4746xnegeqd 45448 . 2 (𝜑 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
485, 47eqtrd 2777 1 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wnf 1783  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  cmpt 5225  ran crn 5686  supcsup 9480  infcinf 9481  *cxr 11294   < clt 11295  -𝑒cxne 13151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-xneg 13154
This theorem is referenced by:  liminfvalxr  45798
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