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Theorem infrpgernmpt 40110
Description: The infimum of a non empty, 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 1956 . . 3 𝑤𝜑
2 infrpgernmpt.x . . . 4 𝑥𝜑
3 eqid 2724 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
4 infrpgernmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
52, 3, 4rnmptssd 39801 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
6 infrpgernmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
72, 4, 3, 6rnmptn0 39829 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
8 infrpgernmpt.y . . . . 5 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
9 breq1 4763 . . . . . . 7 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
109ralbidv 3088 . . . . . 6 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
1110cbvrexv 3275 . . . . 5 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
128, 11sylib 208 . . . 4 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
1312rnmptlb 39869 . . 3 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
14 infrpgernmpt.c . . 3 (𝜑𝐶 ∈ ℝ+)
151, 5, 7, 13, 14infrpge 39982 . 2 (𝜑 → ∃𝑤 ∈ ran (𝑥𝐴𝐵)𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
16 simpll 807 . . . 4 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → 𝜑)
17 simpr 479 . . . 4 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
18 vex 3307 . . . . . . 7 𝑤 ∈ V
193elrnmpt 5479 . . . . . . 7 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵))
2018, 19ax-mp 5 . . . . . 6 (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵)
2120biimpi 206 . . . . 5 (𝑤 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑤 = 𝐵)
2221ad2antlr 765 . . . 4 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥𝐴 𝑤 = 𝐵)
23 nfcv 2866 . . . . . . . 8 𝑥𝑤
24 nfcv 2866 . . . . . . . 8 𝑥
25 nfmpt1 4855 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐵)
2625nfrn 5475 . . . . . . . . . 10 𝑥ran (𝑥𝐴𝐵)
27 nfcv 2866 . . . . . . . . . 10 𝑥*
28 nfcv 2866 . . . . . . . . . 10 𝑥 <
2926, 27, 28nfinf 8504 . . . . . . . . 9 𝑥inf(ran (𝑥𝐴𝐵), ℝ*, < )
30 nfcv 2866 . . . . . . . . 9 𝑥 +𝑒
31 nfcv 2866 . . . . . . . . 9 𝑥𝐶
3229, 30, 31nfov 6791 . . . . . . . 8 𝑥(inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)
3323, 24, 32nfbr 4807 . . . . . . 7 𝑥 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)
342, 33nfan 1941 . . . . . 6 𝑥(𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
35 id 22 . . . . . . . . . . . 12 (𝑤 = 𝐵𝑤 = 𝐵)
3635eqcomd 2730 . . . . . . . . . . 11 (𝑤 = 𝐵𝐵 = 𝑤)
3736adantl 473 . . . . . . . . . 10 ((𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 = 𝑤)
38 simpl 474 . . . . . . . . . 10 ((𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
3937, 38eqbrtrd 4782 . . . . . . . . 9 ((𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
4039ex 449 . . . . . . . 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 473 . . . . . 6 ((𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → (𝑥𝐴 → (𝑤 = 𝐵𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))))
4334, 42reximdai 3114 . . . . 5 ((𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → (∃𝑥𝐴 𝑤 = 𝐵 → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)))
4443imp 444 . . . 4 (((𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) ∧ ∃𝑥𝐴 𝑤 = 𝐵) → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
4516, 17, 22, 44syl21anc 1438 . . 3 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
4645rexlimdva2 39755 . 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 196  wa 383   = wceq 1596  wnf 1821  wcel 2103  wne 2896  wral 3014  wrex 3015  Vcvv 3304  c0 4023   class class class wbr 4760  cmpt 4837  ran crn 5219  (class class class)co 6765  infcinf 8463  cr 10048  *cxr 10186   < clt 10187  cle 10188  +crp 11946   +𝑒 cxad 12058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-po 5139  df-so 5140  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-sup 8464  df-inf 8465  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-rp 11947  df-xadd 12061
This theorem is referenced by:  limsupgtlem  40429
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