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Theorem infxrgelbrnmpt 45404
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 2735 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 infxrgelbrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
41, 2, 3rnmptssd 45139 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ*)
5 infxrgelbrnmpt.c . . 3 (𝜑𝐶 ∈ ℝ*)
6 infxrgelb 13374 . . 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 5256 . . . . . . 7 𝑥(𝑥𝐴𝐵)
98nfrn 5966 . . . . . 6 𝑥ran (𝑥𝐴𝐵)
10 nfv 1912 . . . . . 6 𝑥 𝐶𝑧
119, 10nfralw 3309 . . . . 5 𝑥𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧
121, 11nfan 1897 . . . 4 𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧)
13 simpr 484 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥𝐴)
142elrnmpt1 5974 . . . . . . . 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 5152 . . . . . . 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 3255 . . 3 ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝐶𝑧) → ∀𝑥𝐴 𝐶𝐵)
23 vex 3482 . . . . . . . . 9 𝑧 ∈ V
242elrnmpt 5972 . . . . . . . . 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 3282 . . . . . . . 8 𝑥𝑥𝐴 𝐶𝐵
29 rspa 3246 . . . . . . . . . 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 3264 . . . . . . 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 3144 . . . 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 1537  wnf 1780  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963   class class class wbr 5148  cmpt 5231  ran crn 5690  infcinf 9479  *cxr 11292   < clt 11293  cle 11294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493
This theorem is referenced by: (None)
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