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Theorem infrnmptle 40148
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 1992 . 2 𝑦𝜑
2 nfv 1992 . 2 𝑧𝜑
3 infrnmptle.x . . 3 𝑥𝜑
4 eqid 2760 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
5 infrnmptle.b . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
63, 4, 5rnmptssd 39884 . 2 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
7 eqid 2760 . . 3 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
8 infrnmptle.c . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ*)
93, 7, 8rnmptssd 39884 . 2 (𝜑 → ran (𝑥𝐴𝐶) ⊆ ℝ*)
10 vex 3343 . . . . . 6 𝑦 ∈ V
117elrnmpt 5527 . . . . . 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 473 . . 3 ((𝜑𝑦 ∈ ran (𝑥𝐴𝐶)) → ∃𝑥𝐴 𝑦 = 𝐶)
15 nfmpt1 4899 . . . . . . 7 𝑥(𝑥𝐴𝐵)
1615nfrn 5523 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
17 nfv 1992 . . . . . 6 𝑥 𝑧𝑦
1816, 17nfrex 3145 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦
19 simpr 479 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
204elrnmpt1 5529 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
2119, 5, 20syl2anc 696 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
22213adant3 1127 . . . . . . 7 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵 ∈ ran (𝑥𝐴𝐵))
23 infrnmptle.l . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝐶)
24233adant3 1127 . . . . . . . 8 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵𝐶)
25 id 22 . . . . . . . . . 10 (𝑦 = 𝐶𝑦 = 𝐶)
2625eqcomd 2766 . . . . . . . . 9 (𝑦 = 𝐶𝐶 = 𝑦)
27263ad2ant3 1130 . . . . . . . 8 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐶 = 𝑦)
2824, 27breqtrd 4830 . . . . . . 7 ((𝜑𝑥𝐴𝑦 = 𝐶) → 𝐵𝑦)
29 breq1 4807 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
3029rspcev 3449 . . . . . . 7 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ 𝐵𝑦) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
3122, 28, 30syl2anc 696 . . . . . 6 ((𝜑𝑥𝐴𝑦 = 𝐶) → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
32313exp 1113 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)))
333, 18, 32rexlimd 3164 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
3433adantr 472 . . 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 40139 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 1072   = wceq 1632  wnf 1857  wcel 2139  wrex 3051  Vcvv 3340   class class class wbr 4804  cmpt 4881  ran crn 5267  infcinf 8512  *cxr 10265   < clt 10266  cle 10267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-sup 8513  df-inf 8514  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461
This theorem is referenced by:  limsupres  40440
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