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Theorem xrmaxiflemcom 11292
Description: Lemma for xrmaxif 11294. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
Assertion
Ref Expression
xrmaxiflemcom  |-  ( ( 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 ,  <  )
) ) ) ) )

Proof of Theorem xrmaxiflemcom
StepHypRef Expression
1 simpr 110 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  B  = +oo )
21iftrued 3556 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )  = +oo )
3 xrpnfdc 9874 . . . . . . 7  |-  ( A  e.  RR*  -> DECID  A  = +oo )
43ad2antrr 488 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  -> DECID 
A  = +oo )
5 exmiddc 837 . . . . . 6  |-  (DECID  A  = +oo  ->  ( A  = +oo  \/  -.  A  = +oo ) )
64, 5syl 14 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( A  = +oo  \/  -.  A  = +oo ) )
7 eqid 2189 . . . . . . . 8  |- +oo  = +oo
87biantru 302 . . . . . . 7  |-  ( A  = +oo  <->  ( A  = +oo  /\ +oo  = +oo ) )
98a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( A  = +oo  <->  ( A  = +oo  /\ +oo  = +oo ) ) )
10 xrmnfdc 9875 . . . . . . . . . . 11  |-  ( A  e.  RR*  -> DECID  A  = -oo )
1110ad2antrr 488 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  -> DECID 
A  = -oo )
12 exmiddc 837 . . . . . . . . . 10  |-  (DECID  A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo ) )
1311, 12syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( A  = -oo  \/  -.  A  = -oo ) )
14 iba 300 . . . . . . . . . . . 12  |-  ( +oo  =  B  ->  ( A  = -oo  <->  ( A  = -oo  /\ +oo  =  B ) ) )
1514eqcoms 2192 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( A  = -oo  <->  ( A  = -oo  /\ +oo  =  B ) ) )
1615adantl 277 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( A  = -oo  <->  ( A  = -oo  /\ +oo  =  B ) ) )
171iftrued 3556 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) )  = +oo )
1817eqcomd 2195 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  -> +oo  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) )
1918biantrud 304 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( -.  A  = -oo  <->  ( -.  A  = -oo  /\ +oo  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) ) )
2016, 19orbi12d 794 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( ( A  = -oo  \/  -.  A  = -oo )  <->  ( ( A  = -oo  /\ +oo  =  B )  \/  ( -.  A  = -oo  /\ +oo  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) ) ) )
2113, 20mpbid 147 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( ( A  = -oo  /\ +oo  =  B )  \/  ( -.  A  = -oo  /\ +oo  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) )
22 eqifdc 3584 . . . . . . . . 9  |-  (DECID  A  = -oo  ->  ( +oo  =  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) )  <->  ( ( A  = -oo  /\ +oo  =  B )  \/  ( -.  A  = -oo  /\ +oo  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) ) )
2311, 22syl 14 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( +oo  =  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) )  <->  ( ( A  = -oo  /\ +oo  =  B )  \/  ( -.  A  = -oo  /\ +oo  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) ) )
2421, 23mpbird 167 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  -> +oo  =  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) )
2524biantrud 304 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( -.  A  = +oo  <->  ( -.  A  = +oo  /\ +oo  =  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) ) )
269, 25orbi12d 794 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( ( A  = +oo  \/  -.  A  = +oo )  <->  ( ( A  = +oo  /\ +oo  = +oo )  \/  ( -.  A  = +oo  /\ +oo  =  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) ) ) )
276, 26mpbid 147 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( ( A  = +oo  /\ +oo  = +oo )  \/  ( -.  A  = +oo  /\ +oo  =  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) ) ) )
28 eqifdc 3584 . . . . 5  |-  (DECID  A  = +oo  ->  ( +oo  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) )  <->  ( ( A  = +oo  /\ +oo  = +oo )  \/  ( -.  A  = +oo  /\ +oo  =  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) ) ) ) )
294, 28syl 14 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( +oo  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) )  <->  ( ( A  = +oo  /\ +oo  = +oo )  \/  ( -.  A  = +oo  /\ +oo  =  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) ) ) ) )
3027, 29mpbird 167 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  -> +oo  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) )
312, 30eqtrd 2222 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  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 ,  <  )
) ) ) ) )
323, 5syl 14 . . . . . . . 8  |-  ( A  e.  RR*  ->  ( A  = +oo  \/  -.  A  = +oo )
)
3332ad3antrrr 492 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( A  = +oo  \/  -.  A  = +oo )
)
34 pm4.24 395 . . . . . . . . 9  |-  ( A  = +oo  <->  ( A  = +oo  /\  A  = +oo ) )
3534a1i 9 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( A  = +oo  <->  ( A  = +oo  /\  A  = +oo ) ) )
3610ad3antrrr 492 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  -> DECID  A  = -oo )
3736, 12syl 14 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( A  = -oo  \/  -.  A  = -oo )
)
38 simpr 110 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  B  = -oo )
3938eqeq2d 2201 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( A  =  B  <->  A  = -oo ) )
4039anbi2d 464 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  (
( A  = -oo  /\  A  =  B )  <-> 
( A  = -oo  /\  A  = -oo )
) )
41 anidm 396 . . . . . . . . . . . . 13  |-  ( ( A  = -oo  /\  A  = -oo )  <->  A  = -oo )
4240, 41bitr2di 197 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( A  = -oo  <->  ( A  = -oo  /\  A  =  B ) ) )
43 eqid 2189 . . . . . . . . . . . . . 14  |-  A  =  A
4443biantru 302 . . . . . . . . . . . . 13  |-  ( -.  A  = -oo  <->  ( -.  A  = -oo  /\  A  =  A ) )
4544a1i 9 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( -.  A  = -oo  <->  ( -.  A  = -oo  /\  A  =  A ) ) )
4642, 45orbi12d 794 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  (
( A  = -oo  \/  -.  A  = -oo ) 
<->  ( ( A  = -oo  /\  A  =  B )  \/  ( -.  A  = -oo  /\  A  =  A ) ) ) )
4737, 46mpbid 147 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  (
( A  = -oo  /\  A  =  B )  \/  ( -.  A  = -oo  /\  A  =  A ) ) )
48 eqifdc 3584 . . . . . . . . . . 11  |-  (DECID  A  = -oo  ->  ( A  =  if ( A  = -oo ,  B ,  A )  <->  ( ( A  = -oo  /\  A  =  B )  \/  ( -.  A  = -oo  /\  A  =  A ) ) ) )
4936, 48syl 14 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( A  =  if ( A  = -oo ,  B ,  A )  <->  ( ( A  = -oo  /\  A  =  B )  \/  ( -.  A  = -oo  /\  A  =  A ) ) ) )
5047, 49mpbird 167 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  =  if ( A  = -oo ,  B ,  A ) )
5150biantrud 304 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( -.  A  = +oo  <->  ( -.  A  = +oo  /\  A  =  if ( A  = -oo ,  B ,  A )
) ) )
5235, 51orbi12d 794 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  (
( A  = +oo  \/  -.  A  = +oo ) 
<->  ( ( A  = +oo  /\  A  = +oo )  \/  ( -.  A  = +oo  /\  A  =  if ( A  = -oo ,  B ,  A )
) ) ) )
5333, 52mpbid 147 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  (
( A  = +oo  /\  A  = +oo )  \/  ( -.  A  = +oo  /\  A  =  if ( A  = -oo ,  B ,  A ) ) ) )
543ad3antrrr 492 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  -> DECID  A  = +oo )
55 eqifdc 3584 . . . . . . 7  |-  (DECID  A  = +oo  ->  ( A  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  A ) )  <->  ( ( A  = +oo  /\  A  = +oo )  \/  ( -.  A  = +oo  /\  A  =  if ( A  = -oo ,  B ,  A )
) ) ) )
5654, 55syl 14 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( A  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  A ) )  <->  ( ( A  = +oo  /\  A  = +oo )  \/  ( -.  A  = +oo  /\  A  =  if ( A  = -oo ,  B ,  A )
) ) ) )
5753, 56mpbird 167 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  A ) ) )
5838iftrued 3556 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) )  =  A )
5938iftrued 3556 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
)  =  A )
6059ifeq2d 3567 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  if ( A  = -oo ,  B ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) )  =  if ( A  = -oo ,  B ,  A ) )
6160ifeq2d 3567 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) )  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  A ) ) )
6257, 58, 613eqtr4d 2232 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  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 ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) )
63 simpr 110 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  B  = -oo )
6463iffalsed 3559 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  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 ,  <  )
) ) )
6563iffalsed 3559 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) )  =  sup ( { B ,  A } ,  RR ,  <  ) )
6665ifeq2d 3567 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  if ( A  = -oo ,  B ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) )  =  if ( A  = -oo ,  B ,  sup ( { B ,  A } ,  RR ,  <  )
) )
6766ifeq2d 3567 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) )  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { B ,  A } ,  RR ,  <  )
) ) )
68 maxcom 11247 . . . . . . 7  |-  sup ( { B ,  A } ,  RR ,  <  )  =  sup ( { A ,  B } ,  RR ,  <  )
69 ifeq2 3553 . . . . . . 7  |-  ( sup ( { B ,  A } ,  RR ,  <  )  =  sup ( { A ,  B } ,  RR ,  <  )  ->  if ( A  = -oo ,  B ,  sup ( { B ,  A } ,  RR ,  <  ) )  =  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) )
70 ifeq2 3553 . . . . . . 7  |-  ( if ( A  = -oo ,  B ,  sup ( { B ,  A } ,  RR ,  <  )
)  =  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
)  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { B ,  A } ,  RR ,  <  ) ) )  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )
7168, 69, 70mp2b 8 . . . . . 6  |-  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { B ,  A } ,  RR ,  <  ) ) )  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )
7267, 71eqtrdi 2238 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) )  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )
7364, 72eqtr4d 2225 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  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 ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) )
74 xrmnfdc 9875 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = -oo )
75 exmiddc 837 . . . . . 6  |-  (DECID  B  = -oo  ->  ( B  = -oo  \/  -.  B  = -oo ) )
7674, 75syl 14 . . . . 5  |-  ( B  e.  RR*  ->  ( B  = -oo  \/  -.  B  = -oo )
)
7776ad2antlr 489 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  ( B  = -oo  \/  -.  B  = -oo ) )
7862, 73, 77mpjaodan 799 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  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 ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) )
79 simpr 110 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  -.  B  = +oo )
8079iffalsed 3559 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  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 ,  <  )
) ) ) )
8179iffalsed 3559 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) )  =  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) )
8281ifeq2d 3567 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) )  =  if ( A  = -oo ,  B ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) )
8382ifeq2d 3567 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) )  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) ) )
8478, 80, 833eqtr4d 2232 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  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 ,  <  )
) ) ) ) )
85 xrpnfdc 9874 . . . 4  |-  ( B  e.  RR*  -> DECID  B  = +oo )
86 exmiddc 837 . . . 4  |-  (DECID  B  = +oo  ->  ( B  = +oo  \/  -.  B  = +oo ) )
8785, 86syl 14 . . 3  |-  ( B  e.  RR*  ->  ( B  = +oo  \/  -.  B  = +oo )
)
8887adantl 277 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  = +oo  \/  -.  B  = +oo )
)
8931, 84, 88mpjaodan 799 1  |-  ( ( 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 ,  <  )
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835    = wceq 1364    e. wcel 2160   ifcif 3549   {cpr 3608   supcsup 7012   RRcr 7841   +oocpnf 8020   -oocmnf 8021   RR*cxr 8022    < clt 8023
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-sep 4136  ax-pow 4192  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934
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-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-sup 7014  df-pnf 8025  df-mnf 8026  df-xr 8027
This theorem is referenced by:  xrmaxiflemval  11293
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