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Theorem xrmaxiflemcom 11049
Description: Lemma for xrmaxif 11051. 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 109 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  B  = +oo )
21iftrued 3485 . . 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 9654 . . . . . . 7  |-  ( A  e.  RR*  -> DECID  A  = +oo )
43ad2antrr 480 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  -> DECID 
A  = +oo )
5 exmiddc 822 . . . . . 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 2140 . . . . . . . 8  |- +oo  = +oo
87biantru 300 . . . . . . 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 9655 . . . . . . . . . . 11  |-  ( A  e.  RR*  -> DECID  A  = -oo )
1110ad2antrr 480 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  -> DECID 
A  = -oo )
12 exmiddc 822 . . . . . . . . . 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 298 . . . . . . . . . . . 12  |-  ( +oo  =  B  ->  ( A  = -oo  <->  ( A  = -oo  /\ +oo  =  B ) ) )
1514eqcoms 2143 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( A  = -oo  <->  ( A  = -oo  /\ +oo  =  B ) ) )
1615adantl 275 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  ( A  = -oo  <->  ( A  = -oo  /\ +oo  =  B ) ) )
171iftrued 3485 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) )  = +oo )
1817eqcomd 2146 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  -> +oo  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
) ) )
1918biantrud 302 . . . . . . . . . 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 783 . . . . . . . . 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 146 . . . . . . . 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 3510 . . . . . . . . 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 166 . . . . . . 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 302 . . . . . 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 783 . . . . 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 146 . . . 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 3510 . . . . 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 166 . . 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 2173 . 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 484 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( A  = +oo  \/  -.  A  = +oo )
)
34 pm4.24 393 . . . . . . . . 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 484 . . . . . . . . . . . 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 109 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  B  = -oo )
3938eqeq2d 2152 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( A  =  B  <->  A  = -oo ) )
4039anbi2d 460 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  (
( A  = -oo  /\  A  =  B )  <-> 
( A  = -oo  /\  A  = -oo )
) )
41 anidm 394 . . . . . . . . . . . . 13  |-  ( ( A  = -oo  /\  A  = -oo )  <->  A  = -oo )
4240, 41syl6rbb 196 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( A  = -oo  <->  ( A  = -oo  /\  A  =  B ) ) )
43 eqid 2140 . . . . . . . . . . . . . 14  |-  A  =  A
4443biantru 300 . . . . . . . . . . . . 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 783 . . . . . . . . . . 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 146 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  (
( A  = -oo  /\  A  =  B )  \/  ( -.  A  = -oo  /\  A  =  A ) ) )
48 eqifdc 3510 . . . . . . . . . . 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 166 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  =  if ( A  = -oo ,  B ,  A ) )
5150biantrud 302 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  ( -.  A  = +oo  <->  ( -.  A  = +oo  /\  A  =  if ( A  = -oo ,  B ,  A )
) ) )
5235, 51orbi12d 783 . . . . . . 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 146 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  (
( A  = +oo  /\  A  = +oo )  \/  ( -.  A  = +oo  /\  A  =  if ( A  = -oo ,  B ,  A ) ) ) )
543ad3antrrr 484 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  -> DECID  A  = +oo )
55 eqifdc 3510 . . . . . . 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 166 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  A ) ) )
5838iftrued 3485 . . . . 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 3485 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  )
)  =  A )
6059ifeq2d 3494 . . . . . 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 3494 . . . . 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 2183 . . . 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 109 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  B  = -oo )
6463iffalsed 3488 . . . . 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 3488 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) )  =  sup ( { B ,  A } ,  RR ,  <  ) )
6665ifeq2d 3494 . . . . . . 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 3494 . . . . . 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 11006 . . . . . . 7  |-  sup ( { B ,  A } ,  RR ,  <  )  =  sup ( { A ,  B } ,  RR ,  <  )
69 ifeq2 3482 . . . . . . 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 3482 . . . . . . 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 2189 . . . . 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 2176 . . . 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 9655 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = -oo )
75 exmiddc 822 . . . . . 6  |-  (DECID  B  = -oo  ->  ( B  = -oo  \/  -.  B  = -oo ) )
7674, 75syl 14 . . . . 5  |-  ( B  e.  RR*  ->  ( B  = -oo  \/  -.  B  = -oo )
)
7776ad2antlr 481 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  ( B  = -oo  \/  -.  B  = -oo ) )
7862, 73, 77mpjaodan 788 . . 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 109 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  -.  B  = +oo )
8079iffalsed 3488 . . 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 3488 . . . . 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 3494 . . . 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 3494 . . 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 2183 . 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 9654 . . . 4  |-  ( B  e.  RR*  -> DECID  B  = +oo )
86 exmiddc 822 . . . 4  |-  (DECID  B  = +oo  ->  ( B  = +oo  \/  -.  B  = +oo ) )
8785, 86syl 14 . . 3  |-  ( B  e.  RR*  ->  ( B  = +oo  \/  -.  B  = +oo )
)
8887adantl 275 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  = +oo  \/  -.  B  = +oo )
)
8931, 84, 88mpjaodan 788 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 103    <-> wb 104    \/ wo 698  DECID wdc 820    = wceq 1332    e. wcel 1481   ifcif 3478   {cpr 3532   supcsup 6876   RRcr 7642   +oocpnf 7820   -oocmnf 7821   RR*cxr 7822    < clt 7823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-uni 3744  df-sup 6878  df-pnf 7825  df-mnf 7826  df-xr 7827
This theorem is referenced by:  xrmaxiflemval  11050
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