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Theorem infnsuprnmpt 44254
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 2731 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 infnsuprnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 44195 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 infnsuprnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 6244 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 infnsuprnmpt.l . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
87rnmptlb 44247 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
9 infrenegsup 12202 . . 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 1370 . 2 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
11 eqid 2731 . . . . . . . . 9 (𝑥𝐴 ↦ -𝐵) = (𝑥𝐴 ↦ -𝐵)
12 rabidim2 44094 . . . . . . . . . . . 12 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → -𝑤 ∈ ran (𝑥𝐴𝐵))
1312adantl 481 . . . . . . . . . . 11 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → -𝑤 ∈ ran (𝑥𝐴𝐵))
14 negex 11463 . . . . . . . . . . . 12 -𝑤 ∈ V
152elrnmpt 5956 . . . . . . . . . . . 12 (-𝑤 ∈ V → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1614, 15ax-mp 5 . . . . . . . . . . 11 (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵)
1713, 16sylib 217 . . . . . . . . . 10 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → ∃𝑥𝐴 -𝑤 = 𝐵)
18 nfcv 2902 . . . . . . . . . . . . 13 𝑥𝑤
1918nfneg 11461 . . . . . . . . . . . . . . 15 𝑥-𝑤
20 nfmpt1 5257 . . . . . . . . . . . . . . . 16 𝑥(𝑥𝐴𝐵)
2120nfrn 5952 . . . . . . . . . . . . . . 15 𝑥ran (𝑥𝐴𝐵)
2219, 21nfel 2916 . . . . . . . . . . . . . 14 𝑥-𝑤 ∈ ran (𝑥𝐴𝐵)
23 nfcv 2902 . . . . . . . . . . . . . 14 𝑥
2422, 23nfrabw 3467 . . . . . . . . . . . . 13 𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
2518, 24nfel 2916 . . . . . . . . . . . 12 𝑥 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
261, 25nfan 1901 . . . . . . . . . . 11 𝑥(𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
27 rabidim1 3452 . . . . . . . . . . . . 13 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → 𝑤 ∈ ℝ)
2827adantl 481 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ ℝ)
29 negeq 11457 . . . . . . . . . . . . . . . 16 (-𝑤 = 𝐵 → --𝑤 = -𝐵)
3029eqcomd 2737 . . . . . . . . . . . . . . 15 (-𝑤 = 𝐵 → -𝐵 = --𝑤)
31303ad2ant3 1134 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
32 simp1r 1197 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → 𝑤 ∈ ℝ)
33 recn 11203 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
3433negnegd 11567 . . . . . . . . . . . . . . 15 (𝑤 ∈ ℝ → --𝑤 = 𝑤)
3532, 34syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
3631, 35eqtr2d 2772 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
37363exp 1118 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ℝ) → (𝑥𝐴 → (-𝑤 = 𝐵𝑤 = -𝐵)))
3828, 37syldan 590 . . . . . . . . . . 11 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → (𝑥𝐴 → (-𝑤 = 𝐵𝑤 = -𝐵)))
3926, 38reximdai 3257 . . . . . . . . . 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 484 . . . . . . . . 9 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
4211, 40, 41elrnmptd 5961 . . . . . . . 8 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵))
4342ex 412 . . . . . . 7 (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
44 vex 3477 . . . . . . . . . . . . 13 𝑤 ∈ V
4511elrnmpt 5956 . . . . . . . . . . . . 13 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4644, 45ax-mp 5 . . . . . . . . . . . 12 (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) ↔ ∃𝑥𝐴 𝑤 = -𝐵)
4746biimpi 215 . . . . . . . . . . 11 (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) → ∃𝑥𝐴 𝑤 = -𝐵)
4847adantl 481 . . . . . . . . . 10 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → ∃𝑥𝐴 𝑤 = -𝐵)
4918, 23nfel 2916 . . . . . . . . . . . . 13 𝑥 𝑤 ∈ ℝ
5049, 22nfan 1901 . . . . . . . . . . . 12 𝑥(𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵))
51 simp3 1137 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑤 = -𝐵)
523renegcld 11646 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
53523adant3 1131 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝐵 ∈ ℝ)
5451, 53eqeltrd 2832 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑤 ∈ ℝ)
55 simp2 1136 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑥𝐴)
5651negeqd 11459 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 = --𝐵)
573recnd 11247 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
5857negnegd 11567 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → --𝐵 = 𝐵)
59583adant3 1131 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑤 = -𝐵) → --𝐵 = 𝐵)
6056, 59eqtrd 2771 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 = 𝐵)
61 rspe 3245 . . . . . . . . . . . . . . . 16 ((𝑥𝐴 ∧ -𝑤 = 𝐵) → ∃𝑥𝐴 -𝑤 = 𝐵)
6255, 60, 61syl2anc 583 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → ∃𝑥𝐴 -𝑤 = 𝐵)
6314a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 ∈ V)
642, 62, 63elrnmptd 5961 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 ∈ ran (𝑥𝐴𝐵))
6554, 64jca 511 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
66653exp 1118 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐴 → (𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))))
671, 50, 66rexlimd 3262 . . . . . . . . . . 11 (𝜑 → (∃𝑥𝐴 𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵))))
6867imp 406 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑥𝐴 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
6948, 68syldan 590 . . . . . . . . 9 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
70 rabid 3451 . . . . . . . . 9 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
7169, 70sylibr 233 . . . . . . . 8 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
7271ex 412 . . . . . . 7 (𝜑 → (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}))
7343, 72impbid 211 . . . . . 6 (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
7473alrimiv 1929 . . . . 5 (𝜑 → ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
75 nfrab1 3450 . . . . . 6 𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
76 nfcv 2902 . . . . . 6 𝑤ran (𝑥𝐴 ↦ -𝐵)
7775, 76cleqf 2933 . . . . 5 ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵) ↔ ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
7874, 77sylibr 233 . . . 4 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵))
7978supeq1d 9444 . . 3 (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
8079negeqd 11459 . 2 (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
81 eqidd 2732 . 2 (𝜑 → -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
8210, 80, 813eqtrd 2775 1 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086  wal 1538   = wceq 1540  wnf 1784  wcel 2105  wne 2939  wral 3060  wrex 3069  {crab 3431  Vcvv 3473  wss 3949  c0 4323   class class class wbr 5149  cmpt 5232  ran crn 5678  supcsup 9438  infcinf 9439  cr 11112   < clt 11253  cle 11254  -cneg 11450
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9440  df-inf 9441  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452
This theorem is referenced by:  smfinflem  45833
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