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Theorem xrmaxiflemval 10858
Description: Lemma for xrmaxif 10859. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
Hypothesis
Ref Expression
xrmaxiflemval.m  |-  M  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )
Assertion
Ref Expression
xrmaxiflemval  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( M  e.  RR*  /\  A. x  e.  { A ,  B }  -.  M  <  x  /\  A. x  e.  RR*  ( x  < 
M  ->  E. z  e.  { A ,  B } x  <  z ) ) )
Distinct variable groups:    x, A, z   
x, B, z
Allowed substitution hints:    M( x, z)

Proof of Theorem xrmaxiflemval
StepHypRef Expression
1 xrmaxiflemval.m . . 3  |-  M  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )
2 xrmaxiflemcl 10853 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  e.  RR* )
31, 2syl5eqel 2186 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  M  e.  RR* )
4 vex 2644 . . . . 5  |-  x  e. 
_V
54elpr 3495 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
6 xrmaxifle 10854 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
76, 1syl6breqr 3915 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  M )
8 xrlenlt 7701 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  M  e.  RR* )  ->  ( A  <_  M  <->  -.  M  <  A ) )
93, 8syldan 278 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  M  <->  -.  M  <  A ) )
107, 9mpbid 146 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  M  <  A )
11 breq2 3879 . . . . . . 7  |-  ( x  =  A  ->  ( M  <  x  <->  M  <  A ) )
1211notbid 633 . . . . . 6  |-  ( x  =  A  ->  ( -.  M  <  x  <->  -.  M  <  A ) )
1310, 12syl5ibrcom 156 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  =  A  ->  -.  M  <  x ) )
14 xrmaxifle 10854 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  B  <_  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) )
1514ancoms 266 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  <_  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) )
16 xrmaxiflemcom 10857 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) ) )
171, 16syl5eq 2144 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  M  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) )
1815, 17breqtrrd 3901 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  <_  M )
19 simpr 109 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
20 xrlenlt 7701 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  M  e.  RR* )  ->  ( B  <_  M  <->  -.  M  <  B ) )
2119, 3, 20syl2anc 406 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <_  M  <->  -.  M  <  B ) )
2218, 21mpbid 146 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  M  <  B )
23 breq2 3879 . . . . . . 7  |-  ( x  =  B  ->  ( M  <  x  <->  M  <  B ) )
2423notbid 633 . . . . . 6  |-  ( x  =  B  ->  ( -.  M  <  x  <->  -.  M  <  B ) )
2522, 24syl5ibrcom 156 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  =  B  ->  -.  M  <  x ) )
2613, 25jaod 678 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( x  =  A  \/  x  =  B )  ->  -.  M  <  x ) )
275, 26syl5bi 151 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  { A ,  B }  ->  -.  M  <  x ) )
2827ralrimiv 2463 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A. x  e.  { A ,  B }  -.  M  <  x
)
29 prid1g 3574 . . . . . . 7  |-  ( A  e.  RR*  ->  A  e. 
{ A ,  B } )
3029ad4antr 481 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  A )  ->  A  e.  { A ,  B } )
31 breq2 3879 . . . . . . 7  |-  ( z  =  A  ->  (
x  <  z  <->  x  <  A ) )
3231rspcev 2744 . . . . . 6  |-  ( ( A  e.  { A ,  B }  /\  x  <  A )  ->  E. z  e.  { A ,  B } x  <  z )
3330, 32sylancom 414 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  A )  ->  E. z  e.  { A ,  B } x  <  z )
34 prid2g 3575 . . . . . . 7  |-  ( B  e.  RR*  ->  B  e. 
{ A ,  B } )
3534ad4antlr 482 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  B )  ->  B  e.  { A ,  B } )
36 breq2 3879 . . . . . . 7  |-  ( z  =  B  ->  (
x  <  z  <->  x  <  B ) )
3736rspcev 2744 . . . . . 6  |-  ( ( B  e.  { A ,  B }  /\  x  <  B )  ->  E. z  e.  { A ,  B } x  <  z )
3835, 37sylancom 414 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  B )  ->  E. z  e.  { A ,  B } x  <  z )
39 simplll 503 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  A  e.  RR* )
40 simpllr 504 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  B  e.  RR* )
41 simplr 500 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  x  e.  RR* )
421breq2i 3883 . . . . . . . 8  |-  ( x  <  M  <->  x  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
4342biimpi 119 . . . . . . 7  |-  ( x  <  M  ->  x  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
4443adantl 273 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  x  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
4539, 40, 41, 44xrmaxiflemlub 10856 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  (
x  <  A  \/  x  <  B ) )
4633, 38, 45mpjaodan 753 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  E. z  e.  { A ,  B } x  <  z )
4746ex 114 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( x  <  M  ->  E. z  e.  { A ,  B }
x  <  z )
)
4847ralrimiva 2464 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A. x  e.  RR*  ( x  < 
M  ->  E. z  e.  { A ,  B } x  <  z ) )
493, 28, 483jca 1129 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( M  e.  RR*  /\  A. x  e.  { A ,  B }  -.  M  <  x  /\  A. x  e.  RR*  ( x  < 
M  ->  E. z  e.  { A ,  B } x  <  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 670    /\ w3a 930    = wceq 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   ifcif 3421   {cpr 3475   class class class wbr 3875   supcsup 6784   RRcr 7499   +oocpnf 7669   -oocmnf 7670   RR*cxr 7671    < clt 7672    <_ cle 7673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-sup 6786  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611
This theorem is referenced by:  xrmaxif  10859
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