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Theorem infxrgelbrnmpt 41950
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 2824 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 infxrgelbrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 41680 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
5 infxrgelbrnmpt.c . . 3 (𝜑𝐶 ∈ ℝ*)
6 infxrgelb 12716 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ*𝐶 ∈ ℝ*) → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧))
74, 5, 6syl2anc 587 . 2 (𝜑 → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧))
8 nfmpt1 5147 . . . . . . 7 𝑥(𝑥𝐴𝐵)
98nfrn 5807 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
10 nfv 1916 . . . . . 6 𝑥 𝐶𝑧
119, 10nfralw 3219 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧
121, 11nfan 1901 . . . 4 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
13 simpr 488 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥𝐴)
142elrnmpt1 5813 . . . . . . . 8 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1513, 3, 14syl2anc 587 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1615adantlr 714 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) ∧ 𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
17 simplr 768 . . . . . 6 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) ∧ 𝑥𝐴) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
18 breq2 5053 . . . . . . 7 (𝑧 = 𝐵 → (𝐶𝑧𝐶𝐵))
1918rspcva 3606 . . . . . 6 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → 𝐶𝐵)
2016, 17, 19syl2anc 587 . . . . 5 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) ∧ 𝑥𝐴) → 𝐶𝐵)
2120ex 416 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → (𝑥𝐴𝐶𝐵))
2212, 21ralrimi 3210 . . 3 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → ∀𝑥𝐴 𝐶𝐵)
23 vex 3482 . . . . . . . . 9 𝑧 ∈ V
242elrnmpt 5811 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
2523, 24ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
2625biimpi 219 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
2726adantl 485 . . . . . 6 ((∀𝑥𝐴 𝐶𝐵𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
28 nfra1 3213 . . . . . . . 8 𝑥𝑥𝐴 𝐶𝐵
29 rspa 3200 . . . . . . . . . 10 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶𝐵)
3018biimprcd 253 . . . . . . . . . 10 (𝐶𝐵 → (𝑧 = 𝐵𝐶𝑧))
3129, 30syl 17 . . . . . . . . 9 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → (𝑧 = 𝐵𝐶𝑧))
3231ex 416 . . . . . . . 8 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝐶𝑧)))
3328, 10, 32rexlimd 3309 . . . . . . 7 (∀𝑥𝐴 𝐶𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝐶𝑧))
3433adantr 484 . . . . . 6 ((∀𝑥𝐴 𝐶𝐵𝑧 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 𝑧 = 𝐵𝐶𝑧))
3527, 34mpd 15 . . . . 5 ((∀𝑥𝐴 𝐶𝐵𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝐶𝑧)
3635ralrimiva 3176 . . . 4 (∀𝑥𝐴 𝐶𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
3736adantl 485 . . 3 ((𝜑 ∧ ∀𝑥𝐴 𝐶𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
3822, 37impbida 800 . 2 (𝜑 → (∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧 ↔ ∀𝑥𝐴 𝐶𝐵))
397, 38bitrd 282 1 (𝜑 → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑥𝐴 𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2115  wral 3132  wrex 3133  Vcvv 3479  wss 3918   class class class wbr 5049  cmpt 5129  ran crn 5539  infcinf 8891  *cxr 10661   < clt 10662  cle 10663
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-cnex 10580  ax-resscn 10581  ax-1cn 10582  ax-icn 10583  ax-addcl 10584  ax-addrcl 10585  ax-mulcl 10586  ax-mulrcl 10587  ax-mulcom 10588  ax-addass 10589  ax-mulass 10590  ax-distr 10591  ax-i2m1 10592  ax-1ne0 10593  ax-1rid 10594  ax-rnegex 10595  ax-rrecex 10596  ax-cnre 10597  ax-pre-lttri 10598  ax-pre-lttrn 10599  ax-pre-ltadd 10600  ax-pre-mulgt0 10601  ax-pre-sup 10602
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-po 5457  df-so 5458  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-sup 8892  df-inf 8893  df-pnf 10664  df-mnf 10665  df-xr 10666  df-ltxr 10667  df-le 10668  df-sub 10859  df-neg 10860
This theorem is referenced by: (None)
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