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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  Ⅎwnf 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  ℝcr 10048  ℝ*cxr 10186   < clt 10187   ≤ cle 10188  ℝ+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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