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Theorem supminfxrrnmpt 41897
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 2820 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supminfxrrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 41608 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
54supminfxr2 41895 . 2 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ))
6 xnegex 12576 . . . . . . . . . . . 12 -𝑒𝑦 ∈ V
72elrnmpt 5800 . . . . . . . . . . . 12 (-𝑒𝑦 ∈ V → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑒𝑦 = 𝐵))
86, 7ax-mp 5 . . . . . . . . . . 11 (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑒𝑦 = 𝐵)
98biimpi 218 . . . . . . . . . 10 (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 -𝑒𝑦 = 𝐵)
10 eqid 2820 . . . . . . . . . . 11 (𝑥𝐴 ↦ -𝑒𝐵) = (𝑥𝐴 ↦ -𝑒𝐵)
11 xnegneg 12582 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
1211eqcomd 2826 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ*𝑦 = -𝑒-𝑒𝑦)
1312adantr 483 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒-𝑒𝑦)
14 xnegeq 12575 . . . . . . . . . . . . . . . 16 (-𝑒𝑦 = 𝐵 → -𝑒-𝑒𝑦 = -𝑒𝐵)
1514adantl 484 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → -𝑒-𝑒𝑦 = -𝑒𝐵)
1613, 15eqtrd 2855 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒𝐵)
1716ex 415 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (-𝑒𝑦 = 𝐵𝑦 = -𝑒𝐵))
1817reximdv 3258 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (∃𝑥𝐴 -𝑒𝑦 = 𝐵 → ∃𝑥𝐴 𝑦 = -𝑒𝐵))
1918imp 409 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → ∃𝑥𝐴 𝑦 = -𝑒𝐵)
20 simpl 485 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ℝ*)
2110, 19, 20elrnmptd 41590 . . . . . . . . . 10 ((𝑦 ∈ ℝ* ∧ ∃𝑥𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
229, 21sylan2 594 . . . . . . . . 9 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
2322ex 415 . . . . . . . 8 (𝑦 ∈ ℝ* → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵)))
2423rgen 3135 . . . . . . 7 𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵))
25 rabss 4023 . . . . . . . 8 ({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝑒𝐵) ↔ ∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) → 𝑦 ∈ ran (𝑥𝐴 ↦ -𝑒𝐵)))
2625biimpri 230 . . . . . . 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 2973 . . . . . . . 8 𝑥-𝑒𝑦
30 nfmpt1 5136 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
3130nfrn 5796 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
3229, 31nfel 2987 . . . . . . 7 𝑥-𝑒𝑦 ∈ ran (𝑥𝐴𝐵)
33 nfcv 2973 . . . . . . 7 𝑥*
3432, 33nfrabw 3369 . . . . . 6 𝑥{𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}
35 xnegeq 12575 . . . . . . . 8 (𝑦 = -𝑒𝐵 → -𝑒𝑦 = -𝑒-𝑒𝐵)
3635eleq1d 2895 . . . . . . 7 (𝑦 = -𝑒𝐵 → (-𝑒𝑦 ∈ ran (𝑥𝐴𝐵) ↔ -𝑒-𝑒𝐵 ∈ ran (𝑥𝐴𝐵)))
373xnegcld 12668 . . . . . . 7 ((𝜑𝑥𝐴) → -𝑒𝐵 ∈ ℝ*)
38 xnegneg 12582 . . . . . . . . 9 (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵)
393, 38syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝑒-𝑒𝐵 = 𝐵)
40 simpr 487 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
412, 40, 3elrnmpt1d 41650 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
4239, 41eqeltrd 2911 . . . . . . 7 ((𝜑𝑥𝐴) → -𝑒-𝑒𝐵 ∈ ran (𝑥𝐴𝐵))
4336, 37, 42elrabd 3658 . . . . . 6 ((𝜑𝑥𝐴) → -𝑒𝐵 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)})
441, 34, 10, 43rnmptssdf 41676 . . . . 5 (𝜑 → ran (𝑥𝐴 ↦ -𝑒𝐵) ⊆ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)})
4528, 44eqssd 3959 . . . 4 (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝑒𝐵))
4645infeq1d 8915 . . 3 (𝜑 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ) = inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
4746xnegeqd 41861 . 2 (𝜑 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥𝐴𝐵)}, ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
485, 47eqtrd 2855 1 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥𝐴 ↦ -𝑒𝐵), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wnf 1784  wcel 2114  wral 3125  wrex 3126  {crab 3129  Vcvv 3470  wss 3909  cmpt 5118  ran crn 5528  supcsup 8878  infcinf 8879  *cxr 10648   < clt 10649  -𝑒cxne 12479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435  ax-cnex 10567  ax-resscn 10568  ax-1cn 10569  ax-icn 10570  ax-addcl 10571  ax-addrcl 10572  ax-mulcl 10573  ax-mulrcl 10574  ax-mulcom 10575  ax-addass 10576  ax-mulass 10577  ax-distr 10578  ax-i2m1 10579  ax-1ne0 10580  ax-1rid 10581  ax-rnegex 10582  ax-rrecex 10583  ax-cnre 10584  ax-pre-lttri 10585  ax-pre-lttrn 10586  ax-pre-ltadd 10587  ax-pre-mulgt0 10588  ax-pre-sup 10589
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-po 5446  df-so 5447  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-isom 6336  df-riota 7087  df-ov 7132  df-oprab 7133  df-mpo 7134  df-er 8263  df-en 8484  df-dom 8485  df-sdom 8486  df-sup 8880  df-inf 8881  df-pnf 10651  df-mnf 10652  df-xr 10653  df-ltxr 10654  df-le 10655  df-sub 10846  df-neg 10847  df-xneg 12482
This theorem is referenced by:  liminfvalxr  42212
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