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Theorem infrnmptle 42448
 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 1915 . 2 𝑦𝜑
2 nfv 1915 . 2 𝑧𝜑
3 infrnmptle.x . . 3 𝑥𝜑
4 eqid 2758 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
5 infrnmptle.b . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
63, 4, 5rnmptssd 42216 . 2 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
7 eqid 2758 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
8 infrnmptle.c . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ*)
93, 7, 8rnmptssd 42216 . 2 (𝜑 → ran (𝑥𝐴𝐶) ⊆ ℝ*)
10 vex 3413 . . . . . 6 𝑦 ∈ V
117elrnmpt 5797 . . . . . 6 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝐴𝐶) ↔ ∃𝑥𝐴 𝑦 = 𝐶))
1210, 11ax-mp 5 . . . . 5 (𝑦 ∈ ran (𝑥𝐴𝐶) ↔ ∃𝑥𝐴 𝑦 = 𝐶)
1312biimpi 219 . . . 4 (𝑦 ∈ ran (𝑥𝐴𝐶) → ∃𝑥𝐴 𝑦 = 𝐶)
1413adantl 485 . . 3 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐶)) → ∃𝑥𝐴 𝑦 = 𝐶)
15 nfmpt1 5130 . . . . . . 7 𝑥(𝑥𝐴𝐵)
1615nfrn 5793 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
17 nfv 1915 . . . . . 6 𝑥 𝑧𝑦
1816, 17nfrex 3233 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
19 simpr 488 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
204elrnmpt1 5799 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2119, 5, 20syl2anc 587 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
22213adant3 1129 . . . . . . 7 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵 ∈ ran (𝑥𝐴𝐵))
23 infrnmptle.l . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝐶)
24233adant3 1129 . . . . . . . 8 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵𝐶)
25 id 22 . . . . . . . . . 10 (𝑦 = 𝐶𝑦 = 𝐶)
2625eqcomd 2764 . . . . . . . . 9 (𝑦 = 𝐶𝐶 = 𝑦)
27263ad2ant3 1132 . . . . . . . 8 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐶 = 𝑦)
2824, 27breqtrd 5058 . . . . . . 7 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵𝑦)
29 breq1 5035 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
3029rspcev 3541 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ 𝐵𝑦) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
3122, 28, 30syl2anc 587 . . . . . 6 ((𝜑𝑥𝐴𝑦 = 𝐶) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
32313exp 1116 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)))
333, 18, 32rexlimd 3241 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
3433adantr 484 . . 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 42439 1 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ inf(ran (𝑥𝐴𝐶), ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∃wrex 3071  Vcvv 3409   class class class wbr 5032   ↦ cmpt 5112  ran crn 5525  infcinf 8938  ℝ*cxr 10712   < clt 10713   ≤ cle 10714 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-po 5443  df-so 5444  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-sup 8939  df-inf 8940  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911 This theorem is referenced by:  limsupres  42735
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