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Theorem xrmaxiflemval 11271
Description: Lemma for xrmaxif 11272. 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 11266 . . 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, 2eqeltrid 2274 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  M  e.  RR* )
4 vex 2752 . . . . 5  |-  x  e. 
_V
54elpr 3625 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
6 xrmaxifle 11267 . . . . . . . 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, 1breqtrrdi 4057 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  M )
8 xrlenlt 8035 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  M  e.  RR* )  ->  ( A  <_  M  <->  -.  M  <  A ) )
93, 8syldan 282 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  M  <->  -.  M  <  A ) )
107, 9mpbid 147 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  M  <  A )
11 breq2 4019 . . . . . . 7  |-  ( x  =  A  ->  ( M  <  x  <->  M  <  A ) )
1211notbid 668 . . . . . 6  |-  ( x  =  A  ->  ( -.  M  <  x  <->  -.  M  <  A ) )
1310, 12syl5ibrcom 157 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  =  A  ->  -.  M  <  x ) )
14 xrmaxifle 11267 . . . . . . . . 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 268 . . . . . . . 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 11270 . . . . . . . . 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, 16eqtrid 2232 . . . . . . . 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 4043 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  <_  M )
19 simpr 110 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
20 xrlenlt 8035 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  M  e.  RR* )  ->  ( B  <_  M  <->  -.  M  <  B ) )
2119, 3, 20syl2anc 411 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <_  M  <->  -.  M  <  B ) )
2218, 21mpbid 147 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  M  <  B )
23 breq2 4019 . . . . . . 7  |-  ( x  =  B  ->  ( M  <  x  <->  M  <  B ) )
2423notbid 668 . . . . . 6  |-  ( x  =  B  ->  ( -.  M  <  x  <->  -.  M  <  B ) )
2522, 24syl5ibrcom 157 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  =  B  ->  -.  M  <  x ) )
2613, 25jaod 718 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( x  =  A  \/  x  =  B )  ->  -.  M  <  x ) )
275, 26biimtrid 152 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  { A ,  B }  ->  -.  M  <  x ) )
2827ralrimiv 2559 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A. x  e.  { A ,  B }  -.  M  <  x
)
29 prid1g 3708 . . . . . . 7  |-  ( A  e.  RR*  ->  A  e. 
{ A ,  B } )
3029ad4antr 494 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  A )  ->  A  e.  { A ,  B } )
31 breq2 4019 . . . . . . 7  |-  ( z  =  A  ->  (
x  <  z  <->  x  <  A ) )
3231rspcev 2853 . . . . . 6  |-  ( ( A  e.  { A ,  B }  /\  x  <  A )  ->  E. z  e.  { A ,  B } x  <  z )
3330, 32sylancom 420 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  A )  ->  E. z  e.  { A ,  B } x  <  z )
34 prid2g 3709 . . . . . . 7  |-  ( B  e.  RR*  ->  B  e. 
{ A ,  B } )
3534ad4antlr 495 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  B )  ->  B  e.  { A ,  B } )
36 breq2 4019 . . . . . . 7  |-  ( z  =  B  ->  (
x  <  z  <->  x  <  B ) )
3736rspcev 2853 . . . . . 6  |-  ( ( B  e.  { A ,  B }  /\  x  <  B )  ->  E. z  e.  { A ,  B } x  <  z )
3835, 37sylancom 420 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  x  e.  RR* )  /\  x  <  M )  /\  x  <  B )  ->  E. z  e.  { A ,  B } x  <  z )
39 simplll 533 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  A  e.  RR* )
40 simpllr 534 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  B  e.  RR* )
41 simplr 528 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  x  e.  RR* )
421breq2i 4023 . . . . . . . 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 120 . . . . . . 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 277 . . . . . 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 11269 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  (
x  <  A  \/  x  <  B ) )
4633, 38, 45mpjaodan 799 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e. 
RR* )  /\  x  <  M )  ->  E. z  e.  { A ,  B } x  <  z )
4746ex 115 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( x  <  M  ->  E. z  e.  { A ,  B }
x  <  z )
)
4847ralrimiva 2560 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A. x  e.  RR*  ( x  < 
M  ->  E. z  e.  { A ,  B } x  <  z ) )
493, 28, 483jca 1178 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 104    <-> wb 105    \/ wo 709    /\ w3a 979    = wceq 1363    e. wcel 2158   A.wral 2465   E.wrex 2466   ifcif 3546   {cpr 3605   class class class wbr 4015   supcsup 6994   RRcr 7823   +oocpnf 8002   -oocmnf 8003   RR*cxr 8004    < clt 8005    <_ cle 8006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-sup 6996  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-rp 9667  df-seqfrec 10459  df-exp 10533  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021
This theorem is referenced by:  xrmaxif  11272
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