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Theorem infxrgelbrnmpt 43696
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 43423 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
5 infxrgelbrnmpt.c . . 3 (𝜑𝐶 ∈ ℝ*)
6 infxrgelb 13255 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ*𝐶 ∈ ℝ*) → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧))
74, 5, 6syl2anc 585 . 2 (𝜑 → (𝐶 ≤ inf(ran (𝑥𝐴𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧))
8 nfmpt1 5214 . . . . . . 7 𝑥(𝑥𝐴𝐵)
98nfrn 5908 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
10 nfv 1918 . . . . . 6 𝑥 𝐶𝑧
119, 10nfralw 3295 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧
121, 11nfan 1903 . . . 4 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
13 simpr 486 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥𝐴)
142elrnmpt1 5914 . . . . . . . 8 ((𝑥𝐴𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥𝐴𝐵))
1513, 3, 14syl2anc 585 . . . . . . 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 5110 . . . . . . 7 (𝑧 = 𝐵 → (𝐶𝑧𝐶𝐵))
1918rspcva 3580 . . . . . 6 ((𝐵 ∈ ran (𝑥𝐴𝐵) ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → 𝐶𝐵)
2016, 17, 19syl2anc 585 . . . . 5 (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) ∧ 𝑥𝐴) → 𝐶𝐵)
2120ex 414 . . . 4 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → (𝑥𝐴𝐶𝐵))
2212, 21ralrimi 3241 . . 3 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → ∀𝑥𝐴 𝐶𝐵)
23 vex 3450 . . . . . . . . 9 𝑧 ∈ V
242elrnmpt 5912 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
2523, 24ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
2625biimpi 215 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
2726adantl 483 . . . . . 6 ((∀𝑥𝐴 𝐶𝐵𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
28 nfra1 3268 . . . . . . . 8 𝑥𝑥𝐴 𝐶𝐵
29 rspa 3232 . . . . . . . . . 10 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶𝐵)
3018biimprcd 250 . . . . . . . . . 10 (𝐶𝐵 → (𝑧 = 𝐵𝐶𝑧))
3129, 30syl 17 . . . . . . . . 9 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → (𝑧 = 𝐵𝐶𝑧))
3231ex 414 . . . . . . . 8 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴 → (𝑧 = 𝐵𝐶𝑧)))
3328, 10, 32rexlimd 3250 . . . . . . 7 (∀𝑥𝐴 𝐶𝐵 → (∃𝑥𝐴 𝑧 = 𝐵𝐶𝑧))
3433adantr 482 . . . . . 6 ((∀𝑥𝐴 𝐶𝐵𝑧 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 𝑧 = 𝐵𝐶𝑧))
3527, 34mpd 15 . . . . 5 ((∀𝑥𝐴 𝐶𝐵𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝐶𝑧)
3635ralrimiva 3144 . . . 4 (∀𝑥𝐴 𝐶𝐵 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
3736adantl 483 . . 3 ((𝜑 ∧ ∀𝑥𝐴 𝐶𝐵) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
3822, 37impbida 800 . 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 205  wa 397   = wceq 1542  wnf 1786  wcel 2107  wral 3065  wrex 3074  Vcvv 3446  wss 3911   class class class wbr 5106  cmpt 5189  ran crn 5635  infcinf 9378  *cxr 11189   < clt 11190  cle 11191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-sup 9379  df-inf 9380  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389
This theorem is referenced by: (None)
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