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Theorem infrpgernmpt 43690
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 1917 . . 3 𝑤𝜑
2 infrpgernmpt.x . . . 4 𝑥𝜑
3 eqid 2736 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
4 infrpgernmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
52, 3, 4rnmptssd 43406 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
6 infrpgernmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
72, 4, 3, 6rnmptn0 6196 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
8 infrpgernmpt.y . . . . 5 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)
9 breq1 5108 . . . . . . 7 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
109ralbidv 3174 . . . . . 6 (𝑦 = 𝑤 → (∀𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴 𝑤𝐵))
1110cbvrexvw 3226 . . . . 5 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
128, 11sylib 217 . . . 4 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥𝐴 𝑤𝐵)
1312rnmptlb 43460 . . 3 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑤𝑧)
14 infrpgernmpt.c . . 3 (𝜑𝐶 ∈ ℝ+)
151, 5, 7, 13, 14infrpge 43575 . 2 (𝜑 → ∃𝑤 ∈ ran (𝑥𝐴𝐵)𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
16 simpll 765 . . . 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 3449 . . . . . . 7 𝑤 ∈ V
193elrnmpt 5911 . . . . . . 7 (𝑤 ∈ V → (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵))
2018, 19ax-mp 5 . . . . . 6 (𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑤 = 𝐵)
2120biimpi 215 . . . . 5 (𝑤 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑤 = 𝐵)
2221ad2antlr 725 . . . 4 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥𝐴 𝑤 = 𝐵)
23 nfcv 2907 . . . . . . . 8 𝑥𝑤
24 nfcv 2907 . . . . . . . 8 𝑥
25 nfmpt1 5213 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐵)
2625nfrn 5907 . . . . . . . . . 10 𝑥ran (𝑥𝐴𝐵)
27 nfcv 2907 . . . . . . . . . 10 𝑥*
28 nfcv 2907 . . . . . . . . . 10 𝑥 <
2926, 27, 28nfinf 9418 . . . . . . . . 9 𝑥inf(ran (𝑥𝐴𝐵), ℝ*, < )
30 nfcv 2907 . . . . . . . . 9 𝑥 +𝑒
31 nfcv 2907 . . . . . . . . 9 𝑥𝐶
3229, 30, 31nfov 7387 . . . . . . . 8 𝑥(inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)
3323, 24, 32nfbr 5152 . . . . . . 7 𝑥 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)
342, 33nfan 1902 . . . . . 6 𝑥(𝜑𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
35 id 22 . . . . . . . . . . . 12 (𝑤 = 𝐵𝑤 = 𝐵)
3635eqcomd 2742 . . . . . . . . . . 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 5127 . . . . . . . . 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 3244 . . . . 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 836 . . 3 (((𝜑𝑤 ∈ ran (𝑥𝐴𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥𝐴 𝐵 ≤ (inf(ran (𝑥𝐴𝐵), ℝ*, < ) +𝑒 𝐶))
4645rexlimdva2 3154 . 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 205  wa 396   = wceq 1541  wnf 1785  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  c0 4282   class class class wbr 5105  cmpt 5188  ran crn 5634  (class class class)co 7357  infcinf 9377  cr 11050  *cxr 11188   < clt 11189  cle 11190  +crp 12915   +𝑒 cxad 13031
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-rp 12916  df-xadd 13034
This theorem is referenced by:  limsupgtlem  44008
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