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Theorem xrmaxiflemlub 11288
Description: Lemma for xrmaxif 11291. 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 11285 . . . . 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 411 . . . 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 9826 . . . . 5  |-  <  Or  RR*
8 sowlin 4338 . . . . 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 424 . . . 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 1249 . . 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 276 . . . . 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 276 . . . . . 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 276 . . . . . 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 528 . . . . . . . . 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 110 . . . . . . . . . 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 3556 . . . . . . . . 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 4044 . . . . . . . 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 4046 . . . . . . 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 528 . . . . . . . . . . . 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 110 . . . . . . . . . . . . 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 3559 . . . . . . . . . . . 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 4044 . . . . . . . . . . 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 276 . . . . . . . . . 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 110 . . . . . . . . . . 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 3556 . . . . . . . . . 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 4044 . . . . . . . . 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 9809 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  -.  A  <  A )
293, 28syl 14 . . . . . . . . . 10  |-  ( ph  ->  -.  A  <  A
)
3029ad3antrrr 492 . . . . . . . . 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 621 . . . . . . . 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 276 . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . 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 3559 . . . . . . . . . . . . 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 4044 . . . . . . . . . . . 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 276 . . . . . . . . . . 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 110 . . . . . . . . . . . 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 3556 . . . . . . . . . . 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 4044 . . . . . . . . . 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 9844 . . . . . . . . . . . . 13  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
413, 40syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  = +oo  <->  -.  A  < +oo ) )
4241ad4antr 494 . . . . . . . . . . 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 147 . . . . . . . . . 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 621 . . . . . . . . 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 276 . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . 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 3559 . . . . . . . . . . . . 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 4044 . . . . . . . . . . . 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 276 . . . . . . . . . . 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 110 . . . . . . . . . . . 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 3556 . . . . . . . . . . 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 4044 . . . . . . . . . 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 496 . . . . . . . . . . 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 543 . . . . . . . . . . . . . . . . 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 542 . . . . . . . . . . . . . . . . 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 306 . . . . . . . . . . . . . . . 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 534 . . . . . . . . . . . . . . . 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 306 . . . . . . . . . . . . . . 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 528 . . . . . . . . . . . . . . 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 306 . . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . 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 528 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = +oo )
63 simpr 110 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = -oo )
64 elxr 9806 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
653, 64sylib 122 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
6665ad4antr 494 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
6762, 63, 66ecase23d 1361 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  e.  RR )
6860, 61, 67syl2anc 411 . . . . . . . . . . . . 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 542 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = +oo )
70 simpllr 534 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = -oo )
71 elxr 9806 . . . . . . . . . . . . . . . . 17  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
724, 71sylib 122 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
7372ad4antr 494 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
7469, 70, 73ecase23d 1361 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  B  e.  RR )
7560, 61, 74syl2anc 411 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 3559 . . . . . . . . . . . . . 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 4044 . . . . . . . . . . . . 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 11256 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  e.  RR  /\  A  <  sup ( { A ,  B } ,  RR ,  <  ) ) )  -> 
( A  <  A  \/  A  <  B ) )
8068, 75, 68, 78, 79syl22anc 1250 . . . . . . . . . . . 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 730 . . . . . . . . . . 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 1360 . . . . . . . . . 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 9873 . . . . . . . . . . . 12  |-  ( A  e.  RR*  -> DECID  A  = -oo )
84 exmiddc 837 . . . . . . . . . . . 12  |-  (DECID  A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo ) )
853, 83, 843syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( A  = -oo  \/  -.  A  = -oo ) )
8685ad4antr 494 . . . . . . . . . 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 799 . . . . . . . . 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 9872 . . . . . . . . . . 11  |-  ( A  e.  RR*  -> DECID  A  = +oo )
89 exmiddc 837 . . . . . . . . . . 11  |-  (DECID  A  = +oo  ->  ( A  = +oo  \/  -.  A  = +oo ) )
903, 88, 893syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( A  = +oo  \/  -.  A  = +oo ) )
9190ad3antrrr 492 . . . . . . . . 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 799 . . . . . . . 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 9873 . . . . . . . . . 10  |-  ( B  e.  RR*  -> DECID  B  = -oo )
94 exmiddc 837 . . . . . . . . . 10  |-  (DECID  B  = -oo  ->  ( B  = -oo  \/  -.  B  = -oo ) )
954, 93, 943syl 17 . . . . . . . . 9  |-  ( ph  ->  ( B  = -oo  \/  -.  B  = -oo ) )
9695ad2antrr 488 . . . . . . . 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 799 . . . . . . 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 9872 . . . . . . . 8  |-  ( B  e.  RR*  -> DECID  B  = +oo )
99 exmiddc 837 . . . . . . . 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 799 . . . . . 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 11287 . . . . 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 4044 . . . 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 115 . . 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 789 . 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 104    <-> wb 105    \/ wo 709  DECID wdc 835    \/ w3o 979    /\ w3a 980    = wceq 1364    e. wcel 2160   ifcif 3549   {cpr 3608   class class class wbr 4018    Or wor 4313   supcsup 7011   RRcr 7840   +oocpnf 8019   -oocmnf 8020   RR*cxr 8021    < clt 8022
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-sup 7013  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-rp 9684  df-seqfrec 10477  df-exp 10551  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040
This theorem is referenced by:  xrmaxiflemval  11290  xrmaxleastlt  11296
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