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Theorem supxrleubrnmptf 42077
Description: The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
supxrleubrnmptf.x 𝑥𝜑
supxrleubrnmptf.a 𝑥𝐴
supxrleubrnmptf.n 𝑥𝐶
supxrleubrnmptf.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
supxrleubrnmptf.c (𝜑𝐶 ∈ ℝ*)
Assertion
Ref Expression
supxrleubrnmptf (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))

Proof of Theorem supxrleubrnmptf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 supxrleubrnmptf.a . . . . . . 7 𝑥𝐴
2 nfcv 2958 . . . . . . 7 𝑦𝐴
3 nfcv 2958 . . . . . . 7 𝑦𝐵
4 nfcsb1v 3855 . . . . . . 7 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3845 . . . . . . 7 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
61, 2, 3, 4, 5cbvmptf 5132 . . . . . 6 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
76rneqi 5775 . . . . 5 ran (𝑥𝐴𝐵) = ran (𝑦𝐴𝑦 / 𝑥𝐵)
87supeq1i 8899 . . . 4 sup(ran (𝑥𝐴𝐵), ℝ*, < ) = sup(ran (𝑦𝐴𝑦 / 𝑥𝐵), ℝ*, < )
98breq1i 5040 . . 3 (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ sup(ran (𝑦𝐴𝑦 / 𝑥𝐵), ℝ*, < ) ≤ 𝐶)
109a1i 11 . 2 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ sup(ran (𝑦𝐴𝑦 / 𝑥𝐵), ℝ*, < ) ≤ 𝐶))
11 nfv 1915 . . 3 𝑦𝜑
12 supxrleubrnmptf.x . . . . . 6 𝑥𝜑
131nfcri 2946 . . . . . 6 𝑥 𝑦𝐴
1412, 13nfan 1900 . . . . 5 𝑥(𝜑𝑦𝐴)
154nfel1 2974 . . . . 5 𝑥𝑦 / 𝑥𝐵 ∈ ℝ*
1614, 15nfim 1897 . . . 4 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ∈ ℝ*)
17 eleq1w 2875 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1817anbi2d 631 . . . . 5 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
195eleq1d 2877 . . . . 5 (𝑥 = 𝑦 → (𝐵 ∈ ℝ*𝑦 / 𝑥𝐵 ∈ ℝ*))
2018, 19imbi12d 348 . . . 4 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*) ↔ ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ∈ ℝ*)))
21 supxrleubrnmptf.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
2216, 20, 21chvarfv 2241 . . 3 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ∈ ℝ*)
23 supxrleubrnmptf.c . . 3 (𝜑𝐶 ∈ ℝ*)
2411, 22, 23supxrleubrnmpt 42030 . 2 (𝜑 → (sup(ran (𝑦𝐴𝑦 / 𝑥𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶))
25 nfcv 2958 . . . . 5 𝑥
26 supxrleubrnmptf.n . . . . 5 𝑥𝐶
274, 25, 26nfbr 5080 . . . 4 𝑥𝑦 / 𝑥𝐵𝐶
28 nfv 1915 . . . 4 𝑦 𝐵𝐶
29 eqcom 2808 . . . . . . . 8 (𝑥 = 𝑦𝑦 = 𝑥)
3029imbi1i 353 . . . . . . 7 ((𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵) ↔ (𝑦 = 𝑥𝐵 = 𝑦 / 𝑥𝐵))
31 eqcom 2808 . . . . . . . 8 (𝐵 = 𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵 = 𝐵)
3231imbi2i 339 . . . . . . 7 ((𝑦 = 𝑥𝐵 = 𝑦 / 𝑥𝐵) ↔ (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵))
3330, 32bitri 278 . . . . . 6 ((𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵) ↔ (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵))
345, 33mpbi 233 . . . . 5 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
3534breq1d 5043 . . . 4 (𝑦 = 𝑥 → (𝑦 / 𝑥𝐵𝐶𝐵𝐶))
362, 1, 27, 28, 35cbvralfw 3385 . . 3 (∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
3736a1i 11 . 2 (𝜑 → (∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
3810, 24, 373bitrd 308 1 (𝜑 → (sup(ran (𝑥𝐴𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2112  wnfc 2939  wral 3109  csb 3831   class class class wbr 5033  cmpt 5113  ran crn 5524  supcsup 8892  *cxr 10667   < clt 10668  cle 10669
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866
This theorem is referenced by:  liminflelimsuplem  42404
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