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Theorem infnsuprnmpt 40264
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 2824 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 infnsuprnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 40192 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 infnsuprnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 40218 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 infnsuprnmpt.l . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
87rnmptlb 40253 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
9 infrenegsup 11335 . . 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 1496 . 2 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
11 eqid 2824 . . . . . . . . 9 (𝑥𝐴 ↦ -𝐵) = (𝑥𝐴 ↦ -𝐵)
12 rabidim2 40099 . . . . . . . . . . . 12 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → -𝑤 ∈ ran (𝑥𝐴𝐵))
1312adantl 475 . . . . . . . . . . 11 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → -𝑤 ∈ ran (𝑥𝐴𝐵))
14 negex 10598 . . . . . . . . . . . 12 -𝑤 ∈ V
152elrnmpt 5604 . . . . . . . . . . . 12 (-𝑤 ∈ V → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1614, 15ax-mp 5 . . . . . . . . . . 11 (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵)
1713, 16sylib 210 . . . . . . . . . 10 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → ∃𝑥𝐴 -𝑤 = 𝐵)
18 nfcv 2968 . . . . . . . . . . . . 13 𝑥𝑤
1918nfneg 10596 . . . . . . . . . . . . . . 15 𝑥-𝑤
20 nfmpt1 4969 . . . . . . . . . . . . . . . 16 𝑥(𝑥𝐴𝐵)
2120nfrn 5600 . . . . . . . . . . . . . . 15 𝑥ran (𝑥𝐴𝐵)
2219, 21nfel 2981 . . . . . . . . . . . . . 14 𝑥-𝑤 ∈ ran (𝑥𝐴𝐵)
23 nfcv 2968 . . . . . . . . . . . . . 14 𝑥
2422, 23nfrab 3333 . . . . . . . . . . . . 13 𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
2518, 24nfel 2981 . . . . . . . . . . . 12 𝑥 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
261, 25nfan 2004 . . . . . . . . . . 11 𝑥(𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
27 rabidim1 3327 . . . . . . . . . . . . 13 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → 𝑤 ∈ ℝ)
2827adantl 475 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ ℝ)
29 negeq 10592 . . . . . . . . . . . . . . . 16 (-𝑤 = 𝐵 → --𝑤 = -𝐵)
3029eqcomd 2830 . . . . . . . . . . . . . . 15 (-𝑤 = 𝐵 → -𝐵 = --𝑤)
31303ad2ant3 1171 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
32 simp1r 1261 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → 𝑤 ∈ ℝ)
33 recn 10341 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
3433negnegd 10703 . . . . . . . . . . . . . . 15 (𝑤 ∈ ℝ → --𝑤 = 𝑤)
3532, 34syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
3631, 35eqtr2d 2861 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
37363exp 1154 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ℝ) → (𝑥𝐴 → (-𝑤 = 𝐵𝑤 = -𝐵)))
3828, 37syldan 587 . . . . . . . . . . 11 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → (𝑥𝐴 → (-𝑤 = 𝐵𝑤 = -𝐵)))
3926, 38reximdai 3219 . . . . . . . . . 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 479 . . . . . . . . 9 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
4211, 40, 41elrnmptd 40173 . . . . . . . 8 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵))
4342ex 403 . . . . . . 7 (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
44 vex 3416 . . . . . . . . . . . . 13 𝑤 ∈ V
4511elrnmpt 5604 . . . . . . . . . . . . 13 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4644, 45ax-mp 5 . . . . . . . . . . . 12 (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) ↔ ∃𝑥𝐴 𝑤 = -𝐵)
4746biimpi 208 . . . . . . . . . . 11 (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) → ∃𝑥𝐴 𝑤 = -𝐵)
4847adantl 475 . . . . . . . . . 10 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → ∃𝑥𝐴 𝑤 = -𝐵)
4918, 23nfel 2981 . . . . . . . . . . . . 13 𝑥 𝑤 ∈ ℝ
5049, 22nfan 2004 . . . . . . . . . . . 12 𝑥(𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵))
51 simp3 1174 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑤 = -𝐵)
523renegcld 10780 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
53523adant3 1168 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝐵 ∈ ℝ)
5451, 53eqeltrd 2905 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑤 ∈ ℝ)
55 simp2 1173 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑥𝐴)
5651negeqd 10594 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 = --𝐵)
573recnd 10384 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
5857negnegd 10703 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → --𝐵 = 𝐵)
59583adant3 1168 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑤 = -𝐵) → --𝐵 = 𝐵)
6056, 59eqtrd 2860 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 = 𝐵)
61 rspe 3210 . . . . . . . . . . . . . . . 16 ((𝑥𝐴 ∧ -𝑤 = 𝐵) → ∃𝑥𝐴 -𝑤 = 𝐵)
6255, 60, 61syl2anc 581 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → ∃𝑥𝐴 -𝑤 = 𝐵)
6314a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 ∈ V)
642, 62, 63elrnmptd 40173 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 ∈ ran (𝑥𝐴𝐵))
6554, 64jca 509 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
66653exp 1154 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐴 → (𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))))
671, 50, 66rexlimd 3234 . . . . . . . . . . 11 (𝜑 → (∃𝑥𝐴 𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵))))
6867imp 397 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑥𝐴 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
6948, 68syldan 587 . . . . . . . . 9 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
70 rabid 3325 . . . . . . . . 9 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
7169, 70sylibr 226 . . . . . . . 8 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
7271ex 403 . . . . . . 7 (𝜑 → (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}))
7343, 72impbid 204 . . . . . 6 (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
7473alrimiv 2028 . . . . 5 (𝜑 → ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
75 nfrab1 3332 . . . . . 6 𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
76 nfcv 2968 . . . . . 6 𝑤ran (𝑥𝐴 ↦ -𝐵)
7775, 76dfcleqf 40071 . . . . 5 ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵) ↔ ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
7874, 77sylibr 226 . . . 4 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵))
7978supeq1d 8620 . . 3 (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
8079negeqd 10594 . 2 (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
81 eqidd 2825 . 2 (𝜑 → -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
8210, 80, 813eqtrd 2864 1 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113  wal 1656   = wceq 1658  wnf 1884  wcel 2166  wne 2998  wral 3116  wrex 3117  {crab 3120  Vcvv 3413  wss 3797  c0 4143   class class class wbr 4872  cmpt 4951  ran crn 5342  supcsup 8614  infcinf 8615  cr 10250   < clt 10390  cle 10391  -cneg 10585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208  ax-resscn 10308  ax-1cn 10309  ax-icn 10310  ax-addcl 10311  ax-addrcl 10312  ax-mulcl 10313  ax-mulrcl 10314  ax-mulcom 10315  ax-addass 10316  ax-mulass 10317  ax-distr 10318  ax-i2m1 10319  ax-1ne0 10320  ax-1rid 10321  ax-rnegex 10322  ax-rrecex 10323  ax-cnre 10324  ax-pre-lttri 10325  ax-pre-lttrn 10326  ax-pre-ltadd 10327  ax-pre-mulgt0 10328  ax-pre-sup 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-nel 3102  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-po 5262  df-so 5263  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-isom 6131  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-er 8008  df-en 8222  df-dom 8223  df-sdom 8224  df-sup 8616  df-inf 8617  df-pnf 10392  df-mnf 10393  df-xr 10394  df-ltxr 10395  df-le 10396  df-sub 10586  df-neg 10587
This theorem is referenced by:  smfinflem  41816
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