Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  infxrgelbrnmpt Structured version   Visualization version   GIF version

Theorem infxrgelbrnmpt 45465
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 2737 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 infxrgelbrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 45201 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
5 infxrgelbrnmpt.c . . 3 (𝜑𝐶 ∈ ℝ*)
6 infxrgelb 13377 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ*𝐶 ∈ ℝ*) → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧))
74, 5, 6syl2anc 584 . 2 (𝜑 → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧))
8 nfmpt1 5250 . . . . . . 7 𝑥(𝑥𝐴𝐵)
98nfrn 5963 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
10 nfv 1914 . . . . . 6 𝑥 𝐶𝑧
119, 10nfralw 3311 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧
121, 11nfan 1899 . . . 4 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
13 simpr 484 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥𝐴)
142elrnmpt1 5971 . . . . . . . 8 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1513, 3, 14syl2anc 584 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1615adantlr 715 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
17 simplr 769 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
18 breq2 5147 . . . . . . 7 (𝑧 = 𝐵 → (𝐶𝑧𝐶𝐵))
1918rspcva 3620 . . . . . 6 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → 𝐶𝐵)
2016, 17, 19syl2anc 584 . . . . 5 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) ∧ 𝑥𝐴) → 𝐶𝐵)
2120ex 412 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → (𝑥𝐴𝐶𝐵))
2212, 21ralrimi 3257 . . 3 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → ∀𝑥𝐴 𝐶𝐵)
23 vex 3484 . . . . . . . . 9 𝑧 ∈ V
242elrnmpt 5969 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
2523, 24ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
2625biimpi 216 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
2726adantl 481 . . . . . 6 ((∀𝑥𝐴 𝐶𝐵𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
28 nfra1 3284 . . . . . . . 8 𝑥𝑥𝐴 𝐶𝐵
29 rspa 3248 . . . . . . . . . 10 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶𝐵)
3018biimprcd 250 . . . . . . . . . 10 (𝐶𝐵 → (𝑧 = 𝐵𝐶𝑧))
3129, 30syl 17 . . . . . . . . 9 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → (𝑧 = 𝐵𝐶𝑧))
3231ex 412 . . . . . . . 8 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝐶𝑧)))
3328, 10, 32rexlimd 3266 . . . . . . 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 3146 . . . 4 (∀𝑥𝐴 𝐶𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
3736adantl 481 . . 3 ((𝜑 ∧ ∀𝑥𝐴 𝐶𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
3822, 37impbida 801 . 2 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧 ↔ ∀𝑥𝐴 𝐶𝐵))
397, 38bitrd 279 1 (𝜑 → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑥𝐴 𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wnf 1783  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951   class class class wbr 5143  cmpt 5225  ran crn 5686  infcinf 9481  *cxr 11294   < clt 11295  cle 11296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator