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Theorem xrmaxiflemlub 11224
Description: Lemma for xrmaxif 11227. 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 11221 . . . . 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 9767 . . . . 5  |-  <  Or  RR*
8 sowlin 4314 . . . . 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 1238 . . 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 3539 . . . . . . . . 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 4024 . . . . . . . 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 4026 . . . . . . 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 3542 . . . . . . . . . . . 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 4024 . . . . . . . . . . 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 3539 . . . . . . . . . 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 4024 . . . . . . . . 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 9750 . . . . . . . . . . 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 620 . . . . . . . 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 3542 . . . . . . . . . . . . 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 4024 . . . . . . . . . . . 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 3539 . . . . . . . . . . 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 4024 . . . . . . . . . 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 9785 . . . . . . . . . . . . 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 620 . . . . . . . . 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 3542 . . . . . . . . . . . . 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 4024 . . . . . . . . . . . 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 3539 . . . . . . . . . . 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 4024 . . . . . . . . . 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 9747 . . . . . . . . . . . . . . . . 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 1350 . . . . . . . . . . . . . 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 9747 . . . . . . . . . . . . . . . . 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 1350 . . . . . . . . . . . . . 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 3542 . . . . . . . . . . . . . 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 4024 . . . . . . . . . . . . 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 11192 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  e.  RR  /\  A  <  sup ( { A ,  B } ,  RR ,  <  ) ) )  -> 
( A  <  A  \/  A  <  B ) )
8068, 75, 68, 78, 79syl22anc 1239 . . . . . . . . . . . 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 729 . . . . . . . . . . 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 1349 . . . . . . . . . 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 9814 . . . . . . . . . . . 12  |-  ( A  e.  RR*  -> DECID  A  = -oo )
84 exmiddc 836 . . . . . . . . . . . 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 798 . . . . . . . . 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 9813 . . . . . . . . . . 11  |-  ( A  e.  RR*  -> DECID  A  = +oo )
89 exmiddc 836 . . . . . . . . . . 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 798 . . . . . . . 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 9814 . . . . . . . . . 10  |-  ( B  e.  RR*  -> DECID  B  = -oo )
94 exmiddc 836 . . . . . . . . . 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 798 . . . . . . 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 9813 . . . . . . . 8  |-  ( B  e.  RR*  -> DECID  B  = +oo )
99 exmiddc 836 . . . . . . . 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 798 . . . . . 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 11223 . . . . 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 4024 . . . 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 788 . 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 708  DECID wdc 834    \/ w3o 977    /\ w3a 978    = wceq 1353    e. wcel 2146   ifcif 3532   {cpr 3590   class class class wbr 3998    Or wor 4289   supcsup 6971   RRcr 7785   +oocpnf 7963   -oocmnf 7964   RR*cxr 7965    < clt 7966
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-n0 9150  df-z 9227  df-uz 9502  df-rp 9625  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821  df-rsqrt 10975  df-abs 10976
This theorem is referenced by:  xrmaxiflemval  11226  xrmaxleastlt  11232
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