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Theorem infrnmptle 39963
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 1883 . 2 𝑦𝜑
2 nfv 1883 . 2 𝑧𝜑
3 infrnmptle.x . . 3 𝑥𝜑
4 eqid 2651 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
5 infrnmptle.b . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
63, 4, 5rnmptssd 39699 . 2 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
7 eqid 2651 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
8 infrnmptle.c . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ*)
93, 7, 8rnmptssd 39699 . 2 (𝜑 → ran (𝑥𝐴𝐶) ⊆ ℝ*)
10 vex 3234 . . . . . 6 𝑦 ∈ V
117elrnmpt 5404 . . . . . 6 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝐴𝐶) ↔ ∃𝑥𝐴 𝑦 = 𝐶))
1210, 11ax-mp 5 . . . . 5 (𝑦 ∈ ran (𝑥𝐴𝐶) ↔ ∃𝑥𝐴 𝑦 = 𝐶)
1312biimpi 206 . . . 4 (𝑦 ∈ ran (𝑥𝐴𝐶) → ∃𝑥𝐴 𝑦 = 𝐶)
1413adantl 481 . . 3 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐶)) → ∃𝑥𝐴 𝑦 = 𝐶)
15 nfmpt1 4780 . . . . . . 7 𝑥(𝑥𝐴𝐵)
1615nfrn 5400 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
17 nfv 1883 . . . . . 6 𝑥 𝑧𝑦
1816, 17nfrex 3036 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
19 simpr 476 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
204elrnmpt1 5406 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2119, 5, 20syl2anc 694 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
22213adant3 1101 . . . . . . 7 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵 ∈ ran (𝑥𝐴𝐵))
23 infrnmptle.l . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝐶)
24233adant3 1101 . . . . . . . 8 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵𝐶)
25 id 22 . . . . . . . . . 10 (𝑦 = 𝐶𝑦 = 𝐶)
2625eqcomd 2657 . . . . . . . . 9 (𝑦 = 𝐶𝐶 = 𝑦)
27263ad2ant3 1104 . . . . . . . 8 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐶 = 𝑦)
2824, 27breqtrd 4711 . . . . . . 7 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵𝑦)
29 breq1 4688 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
3029rspcev 3340 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ 𝐵𝑦) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
3122, 28, 30syl2anc 694 . . . . . 6 ((𝜑𝑥𝐴𝑦 = 𝐶) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
32313exp 1283 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)))
333, 18, 32rexlimd 3055 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
3433adantr 480 . . 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 39954 1 (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ inf(ran (𝑥𝐴𝐶), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wnf 1748  wcel 2030  wrex 2942  Vcvv 3231   class class class wbr 4685  cmpt 4762  ran crn 5144  infcinf 8388  *cxr 10111   < clt 10112  cle 10113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307
This theorem is referenced by:  limsupres  40255
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