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Theorem infrnmptle 43557
Description: An indexed infimum of extended reals is smaller than another indexed infimum of extended reals, when every indexed element is smaller than the corresponding one. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
infrnmptle.x 𝑥𝜑
infrnmptle.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
infrnmptle.c ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ*)
infrnmptle.l ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
infrnmptle (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ inf(ran (𝑥𝐴𝐶), ℝ*, < ))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem infrnmptle
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1917 . 2 𝑦𝜑
2 nfv 1917 . 2 𝑧𝜑
3 infrnmptle.x . . 3 𝑥𝜑
4 eqid 2736 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
5 infrnmptle.b . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
63, 4, 5rnmptssd 43315 . 2 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
7 eqid 2736 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
8 infrnmptle.c . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ*)
93, 7, 8rnmptssd 43315 . 2 (𝜑 → ran (𝑥𝐴𝐶) ⊆ ℝ*)
10 vex 3447 . . . . . 6 𝑦 ∈ V
117elrnmpt 5909 . . . . . 6 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝐴𝐶) ↔ ∃𝑥𝐴 𝑦 = 𝐶))
1210, 11ax-mp 5 . . . . 5 (𝑦 ∈ ran (𝑥𝐴𝐶) ↔ ∃𝑥𝐴 𝑦 = 𝐶)
1312biimpi 215 . . . 4 (𝑦 ∈ ran (𝑥𝐴𝐶) → ∃𝑥𝐴 𝑦 = 𝐶)
1413adantl 482 . . 3 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐶)) → ∃𝑥𝐴 𝑦 = 𝐶)
15 nfmpt1 5211 . . . . . . 7 𝑥(𝑥𝐴𝐵)
1615nfrn 5905 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
17 nfv 1917 . . . . . 6 𝑥 𝑧𝑦
1816, 17nfrexw 3294 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
19 simpr 485 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
204elrnmpt1 5911 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2119, 5, 20syl2anc 584 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
22213adant3 1132 . . . . . . 7 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵 ∈ ran (𝑥𝐴𝐵))
23 infrnmptle.l . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝐶)
24233adant3 1132 . . . . . . . 8 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵𝐶)
25 id 22 . . . . . . . . . 10 (𝑦 = 𝐶𝑦 = 𝐶)
2625eqcomd 2742 . . . . . . . . 9 (𝑦 = 𝐶𝐶 = 𝑦)
27263ad2ant3 1135 . . . . . . . 8 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐶 = 𝑦)
2824, 27breqtrd 5129 . . . . . . 7 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵𝑦)
29 breq1 5106 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
3029rspcev 3579 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ 𝐵𝑦) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
3122, 28, 30syl2anc 584 . . . . . 6 ((𝜑𝑥𝐴𝑦 = 𝐶) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
32313exp 1119 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)))
333, 18, 32rexlimd 3247 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
3433adantr 481 . . 3 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐶)) → (∃𝑥𝐴 𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
3514, 34mpd 15 . 2 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐶)) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
361, 2, 6, 9, 35infleinf2 43548 1 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ inf(ran (𝑥𝐴𝐶), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wnf 1785  wcel 2106  wrex 3071  Vcvv 3443   class class class wbr 5103  cmpt 5186  ran crn 5632  infcinf 9335  *cxr 11146   < clt 11147  cle 11148
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 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087
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 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-po 5543  df-so 5544  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-er 8606  df-en 8842  df-dom 8843  df-sdom 8844  df-sup 9336  df-inf 9337  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346
This theorem is referenced by:  limsupres  43841
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