Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  infnsuprnmpt Structured version   Visualization version   GIF version

Theorem infnsuprnmpt 39287
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 2621 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 infnsuprnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 39207 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 infnsuprnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 39235 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 infnsuprnmpt.l . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
87rnmptlb 39275 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
9 infrenegsup 11003 . . 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 1325 . 2 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
11 eqid 2621 . . . . . . . . 9 (𝑥𝐴 ↦ -𝐵) = (𝑥𝐴 ↦ -𝐵)
12 rabidim2 39110 . . . . . . . . . . . 12 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → -𝑤 ∈ ran (𝑥𝐴𝐵))
1312adantl 482 . . . . . . . . . . 11 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → -𝑤 ∈ ran (𝑥𝐴𝐵))
14 negex 10276 . . . . . . . . . . . 12 -𝑤 ∈ V
152elrnmpt 5370 . . . . . . . . . . . 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 2763 . . . . . . . . . . . . 13 𝑥𝑤
1918nfneg 10274 . . . . . . . . . . . . . . 15 𝑥-𝑤
20 nfmpt1 4745 . . . . . . . . . . . . . . . 16 𝑥(𝑥𝐴𝐵)
2120nfrn 5366 . . . . . . . . . . . . . . 15 𝑥ran (𝑥𝐴𝐵)
2219, 21nfel 2776 . . . . . . . . . . . . . 14 𝑥-𝑤 ∈ ran (𝑥𝐴𝐵)
23 nfcv 2763 . . . . . . . . . . . . . 14 𝑥
2422, 23nfrab 3121 . . . . . . . . . . . . 13 𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
2518, 24nfel 2776 . . . . . . . . . . . 12 𝑥 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
261, 25nfan 1827 . . . . . . . . . . 11 𝑥(𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
27 rabidim1 3115 . . . . . . . . . . . . 13 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → 𝑤 ∈ ℝ)
2827adantl 482 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ ℝ)
29 negeq 10270 . . . . . . . . . . . . . . . 16 (-𝑤 = 𝐵 → --𝑤 = -𝐵)
3029eqcomd 2627 . . . . . . . . . . . . . . 15 (-𝑤 = 𝐵 → -𝐵 = --𝑤)
31303ad2ant3 1083 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
32 simp1r 1085 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → 𝑤 ∈ ℝ)
33 recn 10023 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
3433negnegd 10380 . . . . . . . . . . . . . . 15 (𝑤 ∈ ℝ → --𝑤 = 𝑤)
3532, 34syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
3631, 35eqtr2d 2656 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
37363exp 1263 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ℝ) → (𝑥𝐴 → (-𝑤 = 𝐵𝑤 = -𝐵)))
3828, 37syldan 487 . . . . . . . . . . 11 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → (𝑥𝐴 → (-𝑤 = 𝐵𝑤 = -𝐵)))
3926, 38reximdai 3011 . . . . . . . . . 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 477 . . . . . . . . 9 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
4211, 40, 41elrnmptd 39188 . . . . . . . 8 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵))
4342ex 450 . . . . . . 7 (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
44 vex 3201 . . . . . . . . . . . . 13 𝑤 ∈ V
4511elrnmpt 5370 . . . . . . . . . . . . 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 482 . . . . . . . . . 10 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → ∃𝑥𝐴 𝑤 = -𝐵)
4918, 23nfel 2776 . . . . . . . . . . . . 13 𝑥 𝑤 ∈ ℝ
5049, 22nfan 1827 . . . . . . . . . . . 12 𝑥(𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵))
51 simp3 1062 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑤 = -𝐵)
523renegcld 10454 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
53523adant3 1080 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝐵 ∈ ℝ)
5451, 53eqeltrd 2700 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑤 ∈ ℝ)
55 simp2 1061 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑥𝐴)
5651negeqd 10272 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 = --𝐵)
573recnd 10065 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
5857negnegd 10380 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → --𝐵 = 𝐵)
59583adant3 1080 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑤 = -𝐵) → --𝐵 = 𝐵)
6056, 59eqtrd 2655 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 = 𝐵)
61 rspe 3002 . . . . . . . . . . . . . . . 16 ((𝑥𝐴 ∧ -𝑤 = 𝐵) → ∃𝑥𝐴 -𝑤 = 𝐵)
6255, 60, 61syl2anc 693 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → ∃𝑥𝐴 -𝑤 = 𝐵)
6314a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 ∈ V)
642, 62, 63elrnmptd 39188 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 ∈ ran (𝑥𝐴𝐵))
6554, 64jca 554 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
66653exp 1263 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐴 → (𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))))
671, 50, 66rexlimd 3024 . . . . . . . . . . 11 (𝜑 → (∃𝑥𝐴 𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵))))
6867imp 445 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑥𝐴 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
6948, 68syldan 487 . . . . . . . . 9 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
70 rabid 3114 . . . . . . . . 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 450 . . . . . . 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 1854 . . . . 5 (𝜑 → ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
75 nfrab1 3120 . . . . . 6 𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
76 nfcv 2763 . . . . . 6 𝑤ran (𝑥𝐴 ↦ -𝐵)
7775, 76dfcleqf 39081 . . . . 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 8349 . . 3 (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
8079negeqd 10272 . 2 (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
81 eqidd 2622 . 2 (𝜑 → -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
8210, 80, 813eqtrd 2659 1 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037  wal 1480   = wceq 1482  wnf 1707  wcel 1989  wne 2793  wral 2911  wrex 2912  {crab 2915  Vcvv 3198  wss 3572  c0 3913   class class class wbr 4651  cmpt 4727  ran crn 5113  supcsup 8343  infcinf 8344  cr 9932   < clt 10071  cle 10072  -cneg 10264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-pre-sup 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-po 5033  df-so 5034  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-sup 8345  df-inf 8346  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266
This theorem is referenced by:  smfinflem  40792
  Copyright terms: Public domain W3C validator