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Theorem infrpgernmpt 41308
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 1892 . . 3 𝑤𝜑
2 infrpgernmpt.x . . . 4 𝑥𝜑
3 eqid 2795 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
4 infrpgernmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
52, 3, 4rnmptssd 41023 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
6 infrpgernmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
72, 4, 3, 6rnmptn0 41049 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
8 infrpgernmpt.y . . . . 5 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
9 breq1 4969 . . . . . . 7 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
109ralbidv 3164 . . . . . 6 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
1110cbvrexv 3404 . . . . 5 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
128, 11sylib 219 . . . 4 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
1312rnmptlb 41080 . . 3 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
14 infrpgernmpt.c . . 3 (𝜑𝐶 ∈ ℝ+)
151, 5, 7, 13, 14infrpge 41185 . 2 (𝜑 → ∃𝑤 ∈ ran (𝑥𝐴𝐵)𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
16 simpll 763 . . . 4 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → 𝜑)
17 simpr 485 . . . 4 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
18 vex 3440 . . . . . . 7 𝑤 ∈ V
193elrnmpt 5715 . . . . . . 7 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵))
2018, 19ax-mp 5 . . . . . 6 (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵)
2120biimpi 217 . . . . 5 (𝑤 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑤 = 𝐵)
2221ad2antlr 723 . . . 4 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥𝐴 𝑤 = 𝐵)
23 nfcv 2949 . . . . . . . 8 𝑥𝑤
24 nfcv 2949 . . . . . . . 8 𝑥
25 nfmpt1 5063 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐵)
2625nfrn 5711 . . . . . . . . . 10 𝑥ran (𝑥𝐴𝐵)
27 nfcv 2949 . . . . . . . . . 10 𝑥*
28 nfcv 2949 . . . . . . . . . 10 𝑥 <
2926, 27, 28nfinf 8797 . . . . . . . . 9 𝑥inf(ran (𝑥𝐴𝐵), ℝ*, < )
30 nfcv 2949 . . . . . . . . 9 𝑥 +𝑒
31 nfcv 2949 . . . . . . . . 9 𝑥𝐶
3229, 30, 31nfov 7051 . . . . . . . 8 𝑥(inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)
3323, 24, 32nfbr 5013 . . . . . . 7 𝑥 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)
342, 33nfan 1881 . . . . . 6 𝑥(𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
35 id 22 . . . . . . . . . . . 12 (𝑤 = 𝐵𝑤 = 𝐵)
3635eqcomd 2801 . . . . . . . . . . 11 (𝑤 = 𝐵𝐵 = 𝑤)
3736adantl 482 . . . . . . . . . 10 ((𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 = 𝑤)
38 simpl 483 . . . . . . . . . 10 ((𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
3937, 38eqbrtrd 4988 . . . . . . . . 9 ((𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
4039ex 413 . . . . . . . 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 482 . . . . . 6 ((𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → (𝑥𝐴 → (𝑤 = 𝐵𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))))
4334, 42reximdai 3272 . . . . 5 ((𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → (∃𝑥𝐴 𝑤 = 𝐵 → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)))
4443imp 407 . . . 4 (((𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) ∧ ∃𝑥𝐴 𝑤 = 𝐵) → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
4516, 17, 22, 44syl21anc 834 . . 3 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
4645rexlimdva2 3250 . 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 207  wa 396   = wceq 1522  wnf 1765  wcel 2081  wne 2984  wral 3105  wrex 3106  Vcvv 3437  c0 4215   class class class wbr 4966  cmpt 5045  ran crn 5449  (class class class)co 7021  infcinf 8756  cr 10387  *cxr 10525   < clt 10526  cle 10527  +crp 12244   +𝑒 cxad 12360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465  ax-pre-sup 10466
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-po 5367  df-so 5368  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-1st 7550  df-2nd 7551  df-er 8144  df-en 8363  df-dom 8364  df-sdom 8365  df-sup 8757  df-inf 8758  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-rp 12245  df-xadd 12363
This theorem is referenced by:  limsupgtlem  41625
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