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Theorem supminfxrrnmpt 40217
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 2771 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supminfxrrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 39905 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
54supminfxr2 40215 . 2 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ))
6 xnegex 12244 . . . . . . . . . . . 12 -𝑒𝑦 ∈ V
72elrnmpt 5510 . . . . . . . . . . . 12 (-𝑒𝑦 ∈ V → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑒𝑦 = 𝐵))
86, 7ax-mp 5 . . . . . . . . . . 11 (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑒𝑦 = 𝐵)
98biimpi 206 . . . . . . . . . 10 (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 -𝑒𝑦 = 𝐵)
10 eqid 2771 . . . . . . . . . . 11 (𝑥𝐴 ↦ -𝑒𝐵) = (𝑥𝐴 ↦ -𝑒𝐵)
11 xnegneg 12250 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
1211eqcomd 2777 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ*𝑦 = -𝑒-𝑒𝑦)
1312adantr 466 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒-𝑒𝑦)
14 xnegeq 12243 . . . . . . . . . . . . . . . 16 (-𝑒𝑦 = 𝐵 → -𝑒-𝑒𝑦 = -𝑒𝐵)
1514adantl 467 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → -𝑒-𝑒𝑦 = -𝑒𝐵)
1613, 15eqtrd 2805 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒𝐵)
1716ex 397 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (-𝑒𝑦 = 𝐵𝑦 = -𝑒𝐵))
1817reximdv 3164 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (∃𝑥𝐴 -𝑒𝑦 = 𝐵 → ∃𝑥𝐴 𝑦 = -𝑒𝐵))
1918imp 393 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → ∃𝑥𝐴 𝑦 = -𝑒𝐵)
20 simpl 468 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ℝ*)
2110, 19, 20elrnmptd 39886 . . . . . . . . . 10 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
229, 21sylan2 580 . . . . . . . . 9 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
2322ex 397 . . . . . . . 8 (𝑦 ∈ ℝ* → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵)))
2423rgen 3071 . . . . . . 7 𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
25 rabss 3828 . . . . . . . 8 ({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝑒𝐵) ↔ ∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵)))
2625biimpri 218 . . . . . . 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 2913 . . . . . . . 8 𝑥-𝑒𝑦
30 nfmpt1 4881 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
3130nfrn 5506 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
3229, 31nfel 2926 . . . . . . 7 𝑥-𝑒𝑦 ∈ ran (𝑥𝐴𝐵)
33 nfcv 2913 . . . . . . 7 𝑥*
3432, 33nfrab 3272 . . . . . 6 𝑥{𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}
35 xnegeq 12243 . . . . . . . 8 (𝑦 = -𝑒𝐵 → -𝑒𝑦 = -𝑒-𝑒𝐵)
3635eleq1d 2835 . . . . . . 7 (𝑦 = -𝑒𝐵 → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ -𝑒-𝑒𝐵 ∈ ran (𝑥𝐴𝐵)))
373xnegcld 12335 . . . . . . 7 ((𝜑𝑥𝐴) → -𝑒𝐵 ∈ ℝ*)
38 xnegneg 12250 . . . . . . . . 9 (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵)
393, 38syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝑒-𝑒𝐵 = 𝐵)
40 simpr 471 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
412, 40, 3elrnmpt1d 39953 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
4239, 41eqeltrd 2850 . . . . . . 7 ((𝜑𝑥𝐴) → -𝑒-𝑒𝐵 ∈ ran (𝑥𝐴𝐵))
4336, 37, 42elrabd 3517 . . . . . 6 ((𝜑𝑥𝐴) → -𝑒𝐵 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)})
441, 34, 10, 43rnmptssdf 39987 . . . . 5 (𝜑 → ran (𝑥𝐴 ↦ -𝑒𝐵) ⊆ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)})
4528, 44eqssd 3769 . . . 4 (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝑒𝐵))
4645infeq1d 8539 . . 3 (𝜑 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ) = inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
4746xnegeqd 40180 . 2 (𝜑 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
485, 47eqtrd 2805 1 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wnf 1856  wcel 2145  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  wss 3723  cmpt 4863  ran crn 5250  supcsup 8502  infcinf 8503  *cxr 10275   < clt 10276  -𝑒cxne 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8504  df-inf 8505  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-xneg 12151
This theorem is referenced by:  liminfvalxr  40533
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