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Theorem infxrgelbrnmpt 39996
 Description: The infimum of an indexed set of extended reals is greater than or equal to a lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
infxrgelbrnmpt.x 𝑥𝜑
infxrgelbrnmpt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
infxrgelbrnmpt.c (𝜑𝐶 ∈ ℝ*)
Assertion
Ref Expression
infxrgelbrnmpt (𝜑 → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑥𝐴 𝐶𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem infxrgelbrnmpt
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 infxrgelbrnmpt.x . . . 4 𝑥𝜑
2 eqid 2651 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 infxrgelbrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 39699 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
5 infxrgelbrnmpt.c . . 3 (𝜑𝐶 ∈ ℝ*)
6 infxrgelb 12203 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ*𝐶 ∈ ℝ*) → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧))
74, 5, 6syl2anc 694 . 2 (𝜑 → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧))
8 nfmpt1 4780 . . . . . . 7 𝑥(𝑥𝐴𝐵)
98nfrn 5400 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
10 nfv 1883 . . . . . 6 𝑥 𝐶𝑧
119, 10nfral 2974 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧
121, 11nfan 1868 . . . 4 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
13 simpr 476 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥𝐴)
142elrnmpt1 5406 . . . . . . . 8 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1513, 3, 14syl2anc 694 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1615adantlr 751 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
17 simplr 807 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
18 breq2 4689 . . . . . . 7 (𝑧 = 𝐵 → (𝐶𝑧𝐶𝐵))
1918rspcva 3338 . . . . . 6 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → 𝐶𝐵)
2016, 17, 19syl2anc 694 . . . . 5 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) ∧ 𝑥𝐴) → 𝐶𝐵)
2120ex 449 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → (𝑥𝐴𝐶𝐵))
2212, 21ralrimi 2986 . . 3 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → ∀𝑥𝐴 𝐶𝐵)
23 vex 3234 . . . . . . . . 9 𝑧 ∈ V
242elrnmpt 5404 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
2523, 24ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
2625biimpi 206 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
2726adantl 481 . . . . . 6 ((∀𝑥𝐴 𝐶𝐵𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
28 nfra1 2970 . . . . . . . 8 𝑥𝑥𝐴 𝐶𝐵
29 rspa 2959 . . . . . . . . . 10 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶𝐵)
3018biimprcd 240 . . . . . . . . . 10 (𝐶𝐵 → (𝑧 = 𝐵𝐶𝑧))
3129, 30syl 17 . . . . . . . . 9 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → (𝑧 = 𝐵𝐶𝑧))
3231ex 449 . . . . . . . 8 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝐶𝑧)))
3328, 10, 32rexlimd 3055 . . . . . . 7 (∀𝑥𝐴 𝐶𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝐶𝑧))
3433adantr 480 . . . . . 6 ((∀𝑥𝐴 𝐶𝐵𝑧 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 𝑧 = 𝐵𝐶𝑧))
3527, 34mpd 15 . . . . 5 ((∀𝑥𝐴 𝐶𝐵𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝐶𝑧)
3635ralrimiva 2995 . . . 4 (∀𝑥𝐴 𝐶𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
3736adantl 481 . . 3 ((𝜑 ∧ ∀𝑥𝐴 𝐶𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
3822, 37impbida 895 . 2 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧 ↔ ∀𝑥𝐴 𝐶𝐵))
397, 38bitrd 268 1 (𝜑 → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑥𝐴 𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  Ⅎwnf 1748   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607   class class class wbr 4685   ↦ cmpt 4762  ran crn 5144  infcinf 8388  ℝ*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: (None)
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