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Theorem xrmaxiflemlub 10903
Description: Lemma for xrmaxif 10906. A least upper bound for  { A ,  B }. (Contributed by Jim Kingdon, 28-Apr-2023.)
Hypotheses
Ref Expression
xrmaxiflemlub.a  |-  ( ph  ->  A  e.  RR* )
xrmaxiflemlub.b  |-  ( ph  ->  B  e.  RR* )
xrmaxiflemlub.c  |-  ( ph  ->  C  e.  RR* )
xrmaxiflemlub.clt  |-  ( ph  ->  C  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )
Assertion
Ref Expression
xrmaxiflemlub  |-  ( ph  ->  ( C  <  A  \/  C  <  B ) )

Proof of Theorem xrmaxiflemlub
StepHypRef Expression
1 xrmaxiflemlub.clt . . 3  |-  ( ph  ->  C  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )
2 xrmaxiflemlub.c . . . 4  |-  ( ph  ->  C  e.  RR* )
3 xrmaxiflemlub.a . . . . 5  |-  ( ph  ->  A  e.  RR* )
4 xrmaxiflemlub.b . . . . 5  |-  ( ph  ->  B  e.  RR* )
5 xrmaxiflemcl 10900 . . . . 5  |-  ( ( 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* )
63, 4, 5syl2anc 406 . . . 4  |-  ( ph  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  e.  RR* )
7 xrltso 9469 . . . . 5  |-  <  Or  RR*
8 sowlin 4200 . . . . 5  |-  ( (  <  Or  RR*  /\  ( C  e.  RR*  /\  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  e.  RR*  /\  A  e.  RR* ) )  -> 
( C  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  ->  ( C  <  A  \/  A  < 
if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) ) ) )
97, 8mpan 418 . . . 4  |-  ( ( C  e.  RR*  /\  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  e.  RR*  /\  A  e.  RR* )  ->  ( C  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )  ->  ( C  < 
A  \/  A  < 
if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) ) ) )
102, 6, 3, 9syl3anc 1197 . . 3  |-  ( ph  ->  ( C  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  ->  ( C  <  A  \/  A  < 
if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) ) ) )
111, 10mpd 13 . 2  |-  ( ph  ->  ( C  <  A  \/  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) ) )
121adantr 272 . . . . 5  |-  ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  ->  C  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
133adantr 272 . . . . . 6  |-  ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  ->  A  e.  RR* )
144adantr 272 . . . . . 6  |-  ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  ->  B  e.  RR* )
15 simplr 502 . . . . . . . . 9  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  B  = +oo )  ->  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )
16 simpr 109 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  B  = +oo )  ->  B  = +oo )
1716iftrued 3445 . . . . . . . . 9  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  = +oo )
1815, 17breqtrd 3917 . . . . . . . 8  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  B  = +oo )  ->  A  < +oo )
1918, 16breqtrrd 3919 . . . . . . 7  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  B  = +oo )  ->  A  <  B )
20 simplr 502 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  ->  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
21 simpr 109 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  ->  -.  B  = +oo )
2221iffalsed 3448 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )  =  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )
2320, 22breqtrd 3917 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  ->  A  <  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )
2423adantr 272 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  <  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )
25 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  B  = -oo )
2625iftrued 3445 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) )  =  A )
2724, 26breqtrd 3917 . . . . . . . . 9  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  <  A )
28 xrltnr 9453 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  -.  A  <  A )
293, 28syl 14 . . . . . . . . . 10  |-  ( ph  ->  -.  A  <  A
)
3029ad3antrrr 481 . . . . . . . . 9  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  -.  A  <  A )
3127, 30pm2.21dd 592 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  <  B )
3223adantr 272 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  <  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )
33 simpr 109 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  B  = -oo )
3433iffalsed 3448 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -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 ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )
3532, 34breqtrd 3917 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  <  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) )
3635adantr 272 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  <  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )
37 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  = +oo )
3837iftrued 3445 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )  = +oo )
3936, 38breqtrd 3917 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  < +oo )
40 nltpnft 9484 . . . . . . . . . . . . 13  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
413, 40syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  = +oo  <->  -.  A  < +oo ) )
4241ad4antr 483 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  ( A  = +oo  <->  -.  A  < +oo ) )
4337, 42mpbid 146 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  -.  A  < +oo )
4439, 43pm2.21dd 592 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  <  B
)
4535adantr 272 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  A  <  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )
46 simpr 109 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  -.  A  = +oo )
4746iffalsed 3448 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )  =  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )
4845, 47breqtrd 3917 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  A  <  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) )
4948adantr 272 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  A  <  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )
50 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  A  = -oo )
5150iftrued 3445 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
)  =  B )
5249, 51breqtrd 3917 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  A  <  B )
5329ad5antr 485 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  <  A
)
54 simp-5l 515 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ph )
55 simp-4r 514 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = +oo )
5654, 55jca 302 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( ph  /\  -.  B  = +oo )
)
57 simpllr 506 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = -oo )
5856, 57jca 302 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo ) )
59 simplr 502 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = +oo )
6058, 59jca 302 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )
)
61 simpr 109 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = -oo )
62 simplr 502 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = +oo )
63 simpr 109 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = -oo )
64 elxr 9450 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
653, 64sylib 121 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
6665ad4antr 483 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
6762, 63, 66ecase23d 1309 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  e.  RR )
6860, 61, 67syl2anc 406 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  e.  RR )
69 simp-4r 514 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = +oo )
70 simpllr 506 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = -oo )
71 elxr 9450 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
724, 71sylib 121 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
7372ad4antr 483 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
7469, 70, 73ecase23d 1309 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  B  e.  RR )
7560, 61, 74syl2anc 406 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  B  e.  RR )
7648adantr 272 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  <  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) )
7761iffalsed 3448 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) )  =  sup ( { A ,  B } ,  RR ,  <  ) )
7876, 77breqtrd 3917 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  <  sup ( { A ,  B } ,  RR ,  <  )
)
79 maxleastlt 10873 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  e.  RR  /\  A  <  sup ( { A ,  B } ,  RR ,  <  ) ) )  -> 
( A  <  A  \/  A  <  B ) )
8068, 75, 68, 78, 79syl22anc 1198 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( A  <  A  \/  A  <  B ) )
8180orcomd 701 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( A  <  B  \/  A  <  A ) )
8253, 81ecased 1308 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  <  B )
83 xrmnfdc 9513 . . . . . . . . . . . 12  |-  ( A  e.  RR*  -> DECID  A  = -oo )
84 exmiddc 804 . . . . . . . . . . . 12  |-  (DECID  A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo ) )
853, 83, 843syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( A  = -oo  \/  -.  A  = -oo ) )
8685ad4antr 483 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  ( A  = -oo  \/  -.  A  = -oo ) )
8752, 82, 86mpjaodan 770 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  A  <  B
)
88 xrpnfdc 9512 . . . . . . . . . . 11  |-  ( A  e.  RR*  -> DECID  A  = +oo )
89 exmiddc 804 . . . . . . . . . . 11  |-  (DECID  A  = +oo  ->  ( A  = +oo  \/  -.  A  = +oo ) )
903, 88, 893syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( A  = +oo  \/  -.  A  = +oo ) )
9190ad3antrrr 481 . . . . . . . . 9  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  = +oo  \/  -.  A  = +oo ) )
9244, 87, 91mpjaodan 770 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) ) )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  <  B )
93 xrmnfdc 9513 . . . . . . . . . 10  |-  ( B  e.  RR*  -> DECID  B  = -oo )
94 exmiddc 804 . . . . . . . . . 10  |-  (DECID  B  = -oo  ->  ( B  = -oo  \/  -.  B  = -oo ) )
954, 93, 943syl 17 . . . . . . . . 9  |-  ( ph  ->  ( B  = -oo  \/  -.  B  = -oo ) )
9695ad2antrr 477 . . . . . . . 8  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  ->  ( B  = -oo  \/  -.  B  = -oo ) )
9731, 92, 96mpjaodan 770 . . . . . . 7  |-  ( ( ( ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  /\  -.  B  = +oo )  ->  A  <  B
)
98 xrpnfdc 9512 . . . . . . . 8  |-  ( B  e.  RR*  -> DECID  B  = +oo )
99 exmiddc 804 . . . . . . . 8  |-  (DECID  B  = +oo  ->  ( B  = +oo  \/  -.  B  = +oo ) )
10014, 98, 993syl 17 . . . . . . 7  |-  ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  ->  ( B  = +oo  \/  -.  B  = +oo )
)
10119, 97, 100mpjaodan 770 . . . . . 6  |-  ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  ->  A  <  B )
10213, 14, 101xrmaxiflemab 10902 . . . . 5  |-  ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  =  B )
10312, 102breqtrd 3917 . . . 4  |-  ( (
ph  /\  A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  ->  C  <  B )
104103ex 114 . . 3  |-  ( ph  ->  ( A  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  ->  C  <  B ) )
105104orim2d 760 . 2  |-  ( ph  ->  ( ( C  < 
A  \/  A  < 
if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )  -> 
( C  <  A  \/  C  <  B ) ) )
10611, 105mpd 13 1  |-  ( ph  ->  ( C  <  A  \/  C  <  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680  DECID wdc 802    \/ w3o 942    /\ w3a 943    = wceq 1312    e. wcel 1461   ifcif 3438   {cpr 3492   class class class wbr 3893    Or wor 4175   supcsup 6819   RRcr 7540   +oocpnf 7715   -oocmnf 7716   RR*cxr 7717    < clt 7718
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 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657  ax-arch 7658  ax-caucvg 7659
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-sup 6821  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-2 8683  df-3 8684  df-4 8685  df-n0 8876  df-z 8953  df-uz 9223  df-rp 9338  df-seqfrec 10106  df-exp 10180  df-cj 10501  df-re 10502  df-im 10503  df-rsqrt 10656  df-abs 10657
This theorem is referenced by:  xrmaxiflemval  10905  xrmaxleastlt  10911
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