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Theorem xrge0infssd 26059
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 11383 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
2 xrltso 11123 . . . . . 6  |-  <  Or  RR*
3 cnvso 5381 . . . . . 6  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
42, 3mpbi 208 . . . . 5  |-  `'  <  Or 
RR*
5 soss 4664 . . . . 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 3371 . . . 4  |-  ( ph  ->  C  C_  ( 0 [,] +oo ) )
11 xrge0infss 26058 . . . 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 26058 . . . 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 26009 . 2  |-  ( ph  ->  -.  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  ) `'  <  sup ( C , 
( 0 [,] +oo ) ,  `'  <  ) )
167, 14supcl 7713 . . . 4  |-  ( ph  ->  sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  e.  ( 0 [,] +oo ) )
171, 16sseldi 3359 . . 3  |-  ( ph  ->  sup ( B , 
( 0 [,] +oo ) ,  `'  <  )  e.  RR* )
187, 12supcl 7713 . . . 4  |-  ( ph  ->  sup ( C , 
( 0 [,] +oo ) ,  `'  <  )  e.  ( 0 [,] +oo ) )
191, 18sseldi 3359 . . 3  |-  ( ph  ->  sup ( C , 
( 0 [,] +oo ) ,  `'  <  )  e.  RR* )
20 brcnvg 5025 . . . . 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 9447 . . . 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 1756   A.wral 2720   E.wrex 2721    C_ wss 3333   class class class wbr 4297    Or wor 4645   `'ccnv 4844  (class class class)co 6096   supcsup 7695   0cc0 9287   +oocpnf 9420   RR*cxr 9422    < clt 9423    <_ cle 9424   [,]cicc 11308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-icc 11312
This theorem is referenced by:  omsmon  26716
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