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Theorem xrge0infssd 27277
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 11607 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
2 xrltso 11347 . . . . . 6  |-  <  Or  RR*
3 cnvso 5546 . . . . . 6  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
42, 3mpbi 208 . . . . 5  |-  `'  <  Or 
RR*
5 soss 4818 . . . . 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 3514 . . . 4  |-  ( ph  ->  C  C_  ( 0 [,] +oo ) )
11 xrge0infss 27276 . . . 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 16 . . 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 27276 . . . 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 16 . . 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 27227 . 2  |-  ( ph  ->  -.  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) `'  <  sup ( C , 
( 0 [,] +oo ) ,  `'  <  ) )
167, 14supcl 7918 . . . 4  |-  ( ph  ->  sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  e.  ( 0 [,] +oo ) )
171, 16sseldi 3502 . . 3  |-  ( ph  ->  sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  e.  RR* )
187, 12supcl 7918 . . . 4  |-  ( ph  ->  sup ( C , 
( 0 [,] +oo ) ,  `'  <  )  e.  ( 0 [,] +oo ) )
191, 18sseldi 3502 . . 3  |-  ( ph  ->  sup ( C , 
( 0 [,] +oo ) ,  `'  <  )  e.  RR* )
20 brcnvg 5183 . . . . 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 294 . . . 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 9652 . . . 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 256 . . 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 661 . 2  |-  ( ph  ->  ( -.  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) `'  <  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  )  <->  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  )  <_  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
2515, 24mpbid 210 1  |-  ( ph  ->  sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  <_  sup ( C , 
( 0 [,] +oo ) ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   class class class wbr 4447    Or wor 4799   `'ccnv 4998  (class class class)co 6284   supcsup 7900   0cc0 9492   +oocpnf 9625   RR*cxr 9627    < clt 9628    <_ cle 9629   [,]cicc 11532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-icc 11536
This theorem is referenced by:  omsmon  27935
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