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Theorem xrge0infssd 28182
Description: Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019.)
Hypotheses
Ref Expression
xrge0infssd.1  |-  ( ph  ->  C  C_  B )
xrge0infssd.2  |-  ( ph  ->  B  C_  ( 0 [,] +oo ) )
Assertion
Ref Expression
xrge0infssd  |-  ( ph  ->  sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  <_  sup ( C , 
( 0 [,] +oo ) ,  `'  <  ) )

Proof of Theorem xrge0infssd
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iccssxr 11717 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
2 xrltso 11440 . . . . . 6  |-  <  Or  RR*
3 cnvso 5395 . . . . . 6  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
42, 3mpbi 211 . . . . 5  |-  `'  <  Or 
RR*
5 soss 4793 . . . . 5  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( `'  <  Or 
RR*  ->  `'  <  Or  ( 0 [,] +oo ) ) )
61, 4, 5mp2 9 . . . 4  |-  `'  <  Or  ( 0 [,] +oo )
76a1i 11 . . 3  |-  ( ph  ->  `'  <  Or  ( 0 [,] +oo ) )
8 xrge0infssd.1 . . 3  |-  ( ph  ->  C  C_  B )
9 xrge0infssd.2 . . 3  |-  ( ph  ->  B  C_  ( 0 [,] +oo ) )
108, 9sstrd 3480 . . . 4  |-  ( ph  ->  C  C_  ( 0 [,] +oo ) )
11 xrge0infss 28181 . . . 4  |-  ( C 
C_  ( 0 [,] +oo )  ->  E. x  e.  ( 0 [,] +oo ) ( A. y  e.  C  -.  x `'  <  y  /\  A. y  e.  ( 0 [,] +oo ) ( y `'  <  x  ->  E. z  e.  C  y `'  <  z ) ) )
1210, 11syl 17 . . 3  |-  ( ph  ->  E. x  e.  ( 0 [,] +oo )
( A. y  e.  C  -.  x `'  <  y  /\  A. y  e.  ( 0 [,] +oo ) ( y `'  <  x  ->  E. z  e.  C  y `'  <  z ) ) )
13 xrge0infss 28181 . . . 4  |-  ( B 
C_  ( 0 [,] +oo )  ->  E. x  e.  ( 0 [,] +oo ) ( A. y  e.  B  -.  x `'  <  y  /\  A. y  e.  ( 0 [,] +oo ) ( y `'  <  x  ->  E. z  e.  B  y `'  <  z ) ) )
149, 13syl 17 . . 3  |-  ( ph  ->  E. x  e.  ( 0 [,] +oo )
( A. y  e.  B  -.  x `'  <  y  /\  A. y  e.  ( 0 [,] +oo ) ( y `'  <  x  ->  E. z  e.  B  y `'  <  z ) ) )
157, 8, 9, 12, 14supssd 28130 . 2  |-  ( ph  ->  -.  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) `'  <  sup ( C , 
( 0 [,] +oo ) ,  `'  <  ) )
167, 14supcl 7978 . . . 4  |-  ( ph  ->  sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  e.  ( 0 [,] +oo ) )
171, 16sseldi 3468 . . 3  |-  ( ph  ->  sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  e.  RR* )
187, 12supcl 7978 . . . 4  |-  ( ph  ->  sup ( C , 
( 0 [,] +oo ) ,  `'  <  )  e.  ( 0 [,] +oo ) )
191, 18sseldi 3468 . . 3  |-  ( ph  ->  sup ( C , 
( 0 [,] +oo ) ,  `'  <  )  e.  RR* )
20 brcnvg 5035 . . . . 5  |-  ( ( sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  e.  RR*  /\  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  e.  RR* )  ->  ( sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) `'  <  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  <->  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  <  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
2120notbid 295 . . . 4  |-  ( ( sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  e.  RR*  /\  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  e.  RR* )  ->  ( -.  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) `'  <  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  <->  -.  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  <  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
22 xrlenlt 9698 . . . 4  |-  ( ( sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  e.  RR*  /\  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  e.  RR* )  ->  ( sup ( B ,  ( 0 [,] +oo ) ,  `'  <  )  <_  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  <->  -.  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  <  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
2321, 22bitr4d 259 . . 3  |-  ( ( sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  e.  RR*  /\  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  e.  RR* )  ->  ( -.  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) `'  <  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  <->  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  )  <_  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
2417, 19, 23syl2anc 665 . 2  |-  ( ph  ->  ( -.  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) `'  <  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  <->  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  )  <_  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
2515, 24mpbid 213 1  |-  ( ph  ->  sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  <_  sup ( C , 
( 0 [,] +oo ) ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    e. wcel 1870   A.wral 2782   E.wrex 2783    C_ wss 3442   class class class wbr 4426    Or wor 4774   `'ccnv 4853  (class class class)co 6305   supcsup 7960   0cc0 9538   +oocpnf 9671   RR*cxr 9673    < clt 9674    <_ cle 9675   [,]cicc 11638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-icc 11642
This theorem is referenced by:  omsmon  28959
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