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Theorem infnsuprnmpt 39983
Description: The indexed infimum of real numbers is the negative of the indexed supremum of the negative values. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
infnsuprnmpt.x 𝑥𝜑
infnsuprnmpt.a (𝜑𝐴 ≠ ∅)
infnsuprnmpt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
infnsuprnmpt.l (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
Assertion
Ref Expression
infnsuprnmpt (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem infnsuprnmpt
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 infnsuprnmpt.x . . . 4 𝑥𝜑
2 eqid 2771 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 infnsuprnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 39905 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 infnsuprnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 39931 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 infnsuprnmpt.l . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
87rnmptlb 39971 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
9 infrenegsup 11208 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ ∧ ran (𝑥𝐴𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧) → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
104, 6, 8, 9syl3anc 1476 . 2 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
11 eqid 2771 . . . . . . . . 9 (𝑥𝐴 ↦ -𝐵) = (𝑥𝐴 ↦ -𝐵)
12 rabidim2 39805 . . . . . . . . . . . 12 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → -𝑤 ∈ ran (𝑥𝐴𝐵))
1312adantl 467 . . . . . . . . . . 11 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → -𝑤 ∈ ran (𝑥𝐴𝐵))
14 negex 10481 . . . . . . . . . . . 12 -𝑤 ∈ V
152elrnmpt 5510 . . . . . . . . . . . 12 (-𝑤 ∈ V → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1614, 15ax-mp 5 . . . . . . . . . . 11 (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵)
1713, 16sylib 208 . . . . . . . . . 10 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → ∃𝑥𝐴 -𝑤 = 𝐵)
18 nfcv 2913 . . . . . . . . . . . . 13 𝑥𝑤
1918nfneg 10479 . . . . . . . . . . . . . . 15 𝑥-𝑤
20 nfmpt1 4881 . . . . . . . . . . . . . . . 16 𝑥(𝑥𝐴𝐵)
2120nfrn 5506 . . . . . . . . . . . . . . 15 𝑥ran (𝑥𝐴𝐵)
2219, 21nfel 2926 . . . . . . . . . . . . . 14 𝑥-𝑤 ∈ ran (𝑥𝐴𝐵)
23 nfcv 2913 . . . . . . . . . . . . . 14 𝑥
2422, 23nfrab 3272 . . . . . . . . . . . . 13 𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
2518, 24nfel 2926 . . . . . . . . . . . 12 𝑥 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
261, 25nfan 1980 . . . . . . . . . . 11 𝑥(𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
27 rabidim1 3266 . . . . . . . . . . . . 13 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → 𝑤 ∈ ℝ)
2827adantl 467 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ ℝ)
29 negeq 10475 . . . . . . . . . . . . . . . 16 (-𝑤 = 𝐵 → --𝑤 = -𝐵)
3029eqcomd 2777 . . . . . . . . . . . . . . 15 (-𝑤 = 𝐵 → -𝐵 = --𝑤)
31303ad2ant3 1129 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
32 simp1r 1240 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → 𝑤 ∈ ℝ)
33 recn 10228 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
3433negnegd 10585 . . . . . . . . . . . . . . 15 (𝑤 ∈ ℝ → --𝑤 = 𝑤)
3532, 34syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
3631, 35eqtr2d 2806 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
37363exp 1112 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ℝ) → (𝑥𝐴 → (-𝑤 = 𝐵𝑤 = -𝐵)))
3828, 37syldan 579 . . . . . . . . . . 11 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → (𝑥𝐴 → (-𝑤 = 𝐵𝑤 = -𝐵)))
3926, 38reximdai 3160 . . . . . . . . . 10 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → (∃𝑥𝐴 -𝑤 = 𝐵 → ∃𝑥𝐴 𝑤 = -𝐵))
4017, 39mpd 15 . . . . . . . . 9 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → ∃𝑥𝐴 𝑤 = -𝐵)
41 simpr 471 . . . . . . . . 9 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
4211, 40, 41elrnmptd 39886 . . . . . . . 8 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵))
4342ex 397 . . . . . . 7 (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
44 vex 3354 . . . . . . . . . . . . 13 𝑤 ∈ V
4511elrnmpt 5510 . . . . . . . . . . . . 13 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4644, 45ax-mp 5 . . . . . . . . . . . 12 (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) ↔ ∃𝑥𝐴 𝑤 = -𝐵)
4746biimpi 206 . . . . . . . . . . 11 (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) → ∃𝑥𝐴 𝑤 = -𝐵)
4847adantl 467 . . . . . . . . . 10 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → ∃𝑥𝐴 𝑤 = -𝐵)
4918, 23nfel 2926 . . . . . . . . . . . . 13 𝑥 𝑤 ∈ ℝ
5049, 22nfan 1980 . . . . . . . . . . . 12 𝑥(𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵))
51 simp3 1132 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑤 = -𝐵)
523renegcld 10659 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
53523adant3 1126 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝐵 ∈ ℝ)
5451, 53eqeltrd 2850 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑤 ∈ ℝ)
55 simp2 1131 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑥𝐴)
5651negeqd 10477 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 = --𝐵)
573recnd 10270 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
5857negnegd 10585 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → --𝐵 = 𝐵)
59583adant3 1126 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑤 = -𝐵) → --𝐵 = 𝐵)
6056, 59eqtrd 2805 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 = 𝐵)
61 rspe 3151 . . . . . . . . . . . . . . . 16 ((𝑥𝐴 ∧ -𝑤 = 𝐵) → ∃𝑥𝐴 -𝑤 = 𝐵)
6255, 60, 61syl2anc 573 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → ∃𝑥𝐴 -𝑤 = 𝐵)
6314a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 ∈ V)
642, 62, 63elrnmptd 39886 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 ∈ ran (𝑥𝐴𝐵))
6554, 64jca 501 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
66653exp 1112 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐴 → (𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))))
671, 50, 66rexlimd 3174 . . . . . . . . . . 11 (𝜑 → (∃𝑥𝐴 𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵))))
6867imp 393 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑥𝐴 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
6948, 68syldan 579 . . . . . . . . 9 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
70 rabid 3264 . . . . . . . . 9 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
7169, 70sylibr 224 . . . . . . . 8 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
7271ex 397 . . . . . . 7 (𝜑 → (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}))
7343, 72impbid 202 . . . . . 6 (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
7473alrimiv 2007 . . . . 5 (𝜑 → ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
75 nfrab1 3271 . . . . . 6 𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
76 nfcv 2913 . . . . . 6 𝑤ran (𝑥𝐴 ↦ -𝐵)
7775, 76dfcleqf 39776 . . . . 5 ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵) ↔ ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
7874, 77sylibr 224 . . . 4 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵))
7978supeq1d 8508 . . 3 (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
8079negeqd 10477 . 2 (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
81 eqidd 2772 . 2 (𝜑 → -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
8210, 80, 813eqtrd 2809 1 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wnf 1856  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  wss 3723  c0 4063   class class class wbr 4786  cmpt 4863  ran crn 5250  supcsup 8502  infcinf 8503  cr 10137   < clt 10276  cle 10277  -cneg 10469
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-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 835  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
This theorem is referenced by:  smfinflem  41543
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