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

Theorem infrpgernmpt 42470
 Description: The infimum of a nonempty, bounded below, indexed subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
infrpgernmpt.x 𝑥𝜑
infrpgernmpt.a (𝜑𝐴 ≠ ∅)
infrpgernmpt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
infrpgernmpt.y (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
infrpgernmpt.c (𝜑𝐶 ∈ ℝ+)
Assertion
Ref Expression
infrpgernmpt (𝜑 → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem infrpgernmpt
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . 3 𝑤𝜑
2 infrpgernmpt.x . . . 4 𝑥𝜑
3 eqid 2758 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
4 infrpgernmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
52, 3, 4rnmptssd 42194 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
6 infrpgernmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
72, 4, 3, 6rnmptn0 6073 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
8 infrpgernmpt.y . . . . 5 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
9 breq1 5035 . . . . . . 7 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
109ralbidv 3126 . . . . . 6 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
1110cbvrexvw 3362 . . . . 5 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
128, 11sylib 221 . . . 4 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
1312rnmptlb 42248 . . 3 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
14 infrpgernmpt.c . . 3 (𝜑𝐶 ∈ ℝ+)
151, 5, 7, 13, 14infrpge 42351 . 2 (𝜑 → ∃𝑤 ∈ ran (𝑥𝐴𝐵)𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
16 simpll 766 . . . 4 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → 𝜑)
17 simpr 488 . . . 4 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
18 vex 3413 . . . . . . 7 𝑤 ∈ V
193elrnmpt 5797 . . . . . . 7 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵))
2018, 19ax-mp 5 . . . . . 6 (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵)
2120biimpi 219 . . . . 5 (𝑤 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑤 = 𝐵)
2221ad2antlr 726 . . . 4 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥𝐴 𝑤 = 𝐵)
23 nfcv 2919 . . . . . . . 8 𝑥𝑤
24 nfcv 2919 . . . . . . . 8 𝑥
25 nfmpt1 5130 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐵)
2625nfrn 5793 . . . . . . . . . 10 𝑥ran (𝑥𝐴𝐵)
27 nfcv 2919 . . . . . . . . . 10 𝑥*
28 nfcv 2919 . . . . . . . . . 10 𝑥 <
2926, 27, 28nfinf 8979 . . . . . . . . 9 𝑥inf(ran (𝑥𝐴𝐵), ℝ*, < )
30 nfcv 2919 . . . . . . . . 9 𝑥 +𝑒
31 nfcv 2919 . . . . . . . . 9 𝑥𝐶
3229, 30, 31nfov 7180 . . . . . . . 8 𝑥(inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)
3323, 24, 32nfbr 5079 . . . . . . 7 𝑥 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)
342, 33nfan 1900 . . . . . 6 𝑥(𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
35 id 22 . . . . . . . . . . . 12 (𝑤 = 𝐵𝑤 = 𝐵)
3635eqcomd 2764 . . . . . . . . . . 11 (𝑤 = 𝐵𝐵 = 𝑤)
3736adantl 485 . . . . . . . . . 10 ((𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 = 𝑤)
38 simpl 486 . . . . . . . . . 10 ((𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
3937, 38eqbrtrd 5054 . . . . . . . . 9 ((𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
4039ex 416 . . . . . . . 8 (𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) → (𝑤 = 𝐵𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)))
4140a1d 25 . . . . . . 7 (𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) → (𝑥𝐴 → (𝑤 = 𝐵𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))))
4241adantl 485 . . . . . 6 ((𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → (𝑥𝐴 → (𝑤 = 𝐵𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))))
4334, 42reximdai 3235 . . . . 5 ((𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → (∃𝑥𝐴 𝑤 = 𝐵 → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)))
4443imp 410 . . . 4 (((𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) ∧ ∃𝑥𝐴 𝑤 = 𝐵) → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
4516, 17, 22, 44syl21anc 836 . . 3 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
4645rexlimdva2 3211 . 2 (𝜑 → (∃𝑤 ∈ ran (𝑥𝐴𝐵)𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)))
4715, 46mpd 15 1 (𝜑 → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  Vcvv 3409  ∅c0 4225   class class class wbr 5032   ↦ cmpt 5112  ran crn 5525  (class class class)co 7150  infcinf 8938  ℝcr 10574  ℝ*cxr 10712   < clt 10713   ≤ cle 10714  ℝ+crp 12430   +𝑒 cxad 12546 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-po 5443  df-so 5444  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-sup 8939  df-inf 8940  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-rp 12431  df-xadd 12549 This theorem is referenced by:  limsupgtlem  42785
 Copyright terms: Public domain W3C validator