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 45194
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 2734 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 infnsuprnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 45138 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 infnsuprnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 6265 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 infnsuprnmpt.l . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
87rnmptlb 45187 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)
9 infrenegsup 12248 . . 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 2734 . . . . . . . . 9 (𝑥𝐴 ↦ -𝐵) = (𝑥𝐴 ↦ -𝐵)
12 rabidim2 45041 . . . . . . . . . . . 12 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → -𝑤 ∈ ran (𝑥𝐴𝐵))
1312adantl 481 . . . . . . . . . . 11 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → -𝑤 ∈ ran (𝑥𝐴𝐵))
14 negex 11503 . . . . . . . . . . . 12 -𝑤 ∈ V
152elrnmpt 5971 . . . . . . . . . . . 12 (-𝑤 ∈ V → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1614, 15ax-mp 5 . . . . . . . . . . 11 (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵)
1713, 16sylib 218 . . . . . . . . . 10 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → ∃𝑥𝐴 -𝑤 = 𝐵)
18 nfcv 2902 . . . . . . . . . . . . 13 𝑥𝑤
1918nfneg 11501 . . . . . . . . . . . . . . 15 𝑥-𝑤
20 nfmpt1 5255 . . . . . . . . . . . . . . . 16 𝑥(𝑥𝐴𝐵)
2120nfrn 5965 . . . . . . . . . . . . . . 15 𝑥ran (𝑥𝐴𝐵)
2219, 21nfel 2917 . . . . . . . . . . . . . 14 𝑥-𝑤 ∈ ran (𝑥𝐴𝐵)
23 nfcv 2902 . . . . . . . . . . . . . 14 𝑥
2422, 23nfrabw 3472 . . . . . . . . . . . . 13 𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
2518, 24nfel 2917 . . . . . . . . . . . 12 𝑥 𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
261, 25nfan 1896 . . . . . . . . . . 11 𝑥(𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
27 rabidim1 3455 . . . . . . . . . . . . 13 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} → 𝑤 ∈ ℝ)
2827adantl 481 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → 𝑤 ∈ ℝ)
29 negeq 11497 . . . . . . . . . . . . . . . 16 (-𝑤 = 𝐵 → --𝑤 = -𝐵)
3029eqcomd 2740 . . . . . . . . . . . . . . 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 11242 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
3433negnegd 11608 . . . . . . . . . . . . . . 15 (𝑤 ∈ ℝ → --𝑤 = 𝑤)
3532, 34syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
3631, 35eqtr2d 2775 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴 ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
37363exp 1118 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ℝ) → (𝑥𝐴 → (-𝑤 = 𝐵𝑤 = -𝐵)))
3828, 37syldan 591 . . . . . . . . . . 11 ((𝜑𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}) → (𝑥𝐴 → (-𝑤 = 𝐵𝑤 = -𝐵)))
3926, 38reximdai 3258 . . . . . . . . . 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 5976 . . . . . . . 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 3481 . . . . . . . . . . . . 13 𝑤 ∈ V
4511elrnmpt 5971 . . . . . . . . . . . . 13 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4644, 45ax-mp 5 . . . . . . . . . . . 12 (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) ↔ ∃𝑥𝐴 𝑤 = -𝐵)
4746biimpi 216 . . . . . . . . . . 11 (𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵) → ∃𝑥𝐴 𝑤 = -𝐵)
4847adantl 481 . . . . . . . . . 10 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → ∃𝑥𝐴 𝑤 = -𝐵)
4918, 23nfel 2917 . . . . . . . . . . . . 13 𝑥 𝑤 ∈ ℝ
5049, 22nfan 1896 . . . . . . . . . . . 12 𝑥(𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵))
51 simp3 1137 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑤 = -𝐵)
523renegcld 11687 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
53523adant3 1131 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝐵 ∈ ℝ)
5451, 53eqeltrd 2838 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑤 ∈ ℝ)
55 simp2 1136 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑤 = -𝐵) → 𝑥𝐴)
5651negeqd 11499 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 = --𝐵)
573recnd 11286 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
5857negnegd 11608 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → --𝐵 = 𝐵)
59583adant3 1131 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴𝑤 = -𝐵) → --𝐵 = 𝐵)
6056, 59eqtrd 2774 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 = 𝐵)
61 rspe 3246 . . . . . . . . . . . . . . . 16 ((𝑥𝐴 ∧ -𝑤 = 𝐵) → ∃𝑥𝐴 -𝑤 = 𝐵)
6255, 60, 61syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → ∃𝑥𝐴 -𝑤 = 𝐵)
6314a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑤 = -𝐵) → -𝑤 ∈ V)
642, 62, 63elrnmptd 5976 . . . . . . . . . . . . . 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 3263 . . . . . . . . . . 11 (𝜑 → (∃𝑥𝐴 𝑤 = -𝐵 → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵))))
6867imp 406 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑥𝐴 𝑤 = -𝐵) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
6948, 68syldan 591 . . . . . . . . 9 ((𝜑𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)) → (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
70 rabid 3454 . . . . . . . . 9 (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ (𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)))
7169, 70sylibr 234 . . . . . . . 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 212 . . . . . 6 (𝜑 → (𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
7473alrimiv 1924 . . . . 5 (𝜑 → ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
75 nfrab1 3453 . . . . . 6 𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
76 nfcv 2902 . . . . . 6 𝑤ran (𝑥𝐴 ↦ -𝐵)
7775, 76cleqf 2931 . . . . 5 ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵) ↔ ∀𝑤(𝑤 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ↔ 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
7874, 77sylibr 234 . . . 4 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵))
7978supeq1d 9483 . . 3 (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
8079negeqd 11499 . 2 (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
81 eqidd 2735 . 2 (𝜑 → -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
8210, 80, 813eqtrd 2778 1 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1534   = wceq 1536  wnf 1779  wcel 2105  wne 2937  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  wss 3962  c0 4338   class class class wbr 5147  cmpt 5230  ran crn 5689  supcsup 9477  infcinf 9478  cr 11151   < clt 11292  cle 11293  -cneg 11490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492
This theorem is referenced by:  smfinflem  46772
  Copyright terms: Public domain W3C validator