ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrmaxadd Unicode version

Theorem xrmaxadd 10869
Description: Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
Assertion
Ref Expression
xrmaxadd  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )

Proof of Theorem xrmaxadd
StepHypRef Expression
1 simpr 109 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  e.  RR )  ->  A  e.  RR )
2 simpl2 953 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  e.  RR )  ->  B  e.  RR* )
3 simpl3 954 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  e.  RR )  ->  C  e.  RR* )
4 xrmaxaddlem 10868 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
51, 2, 3, 4syl3anc 1184 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  e.  RR )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
6 simpllr 504 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  A  = +oo )
7 simpr 109 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  C  = -oo )
86, 7oveq12d 5724 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e C )  =  ( +oo +e -oo ) )
9 simp1 949 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  e.  RR* )
109ad3antrrr 479 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  A  e.  RR* )
11 simp2 950 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  B  e.  RR* )
1211ad3antrrr 479 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  B  e.  RR* )
1310, 12xaddcld 9508 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e B )  e.  RR* )
14 simp3 951 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  C  e.  RR* )
1514ad3antrrr 479 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  C  e.  RR* )
1610, 15xaddcld 9508 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e C )  e.  RR* )
1713, 16jca 302 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  (
( A +e
B )  e.  RR*  /\  ( A +e
C )  e.  RR* ) )
18 simplr 500 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  ->  A  = +oo )
19 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  ->  B  = -oo )
2018, 19oveq12d 5724 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  ->  ( A +e B )  =  ( +oo +e -oo ) )
21 pnfaddmnf 9474 . . . . . . . . . 10  |-  ( +oo +e -oo )  =  0
2220, 21syl6eq 2148 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  ->  ( A +e B )  =  0 )
2322adantr 272 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e B )  =  0 )
248, 21syl6eq 2148 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e C )  =  0 )
2523, 24eqtr4d 2135 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e B )  =  ( A +e C ) )
2616xrleidd 9428 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e C )  <_  ( A +e C ) )
2725, 26eqbrtrd 3895 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e B )  <_  ( A +e C ) )
28 xrmaxleim 10852 . . . . . 6  |-  ( ( ( A +e
B )  e.  RR*  /\  ( A +e
C )  e.  RR* )  ->  ( ( A +e B )  <_  ( A +e C )  ->  sup ( { ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +e C ) ) )
2917, 27, 28sylc 62 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e
C ) )
3012, 15jca 302 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( B  e.  RR*  /\  C  e.  RR* ) )
31 simplr 500 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  B  = -oo )
3231, 7eqtr4d 2135 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  B  =  C )
3315xrleidd 9428 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  C  <_  C )
3432, 33eqbrtrd 3895 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  B  <_  C )
35 xrmaxleim 10852 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B  <_  C  ->  sup ( { B ,  C } ,  RR* ,  <  )  =  C ) )
3630, 34, 35sylc 62 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  =  C )
3736, 7eqtrd 2132 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  = -oo )
386, 37oveq12d 5724 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  )
)  =  ( +oo +e -oo )
)
398, 29, 383eqtr4d 2142 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
40 simpllr 504 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  A  = +oo )
4140oveq1d 5721 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e C )  =  ( +oo +e C ) )
4214ad3antrrr 479 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  C  e.  RR* )
43 xaddpnf2 9471 . . . . . . 7  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( +oo +e C )  = +oo )
4442, 43sylancom 414 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( +oo +e C )  = +oo )
4541, 44eqtrd 2132 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e C )  = +oo )
469, 11xaddcld 9508 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A +e B )  e.  RR* )
4746ad3antrrr 479 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e B )  e.  RR* )
48 pnfge 9416 . . . . . . . 8  |-  ( ( A +e B )  e.  RR*  ->  ( A +e B )  <_ +oo )
4947, 48syl 14 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e B )  <_ +oo )
5049, 45breqtrrd 3901 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e B )  <_  ( A +e C ) )
519, 14xaddcld 9508 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A +e C )  e.  RR* )
5251ad3antrrr 479 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e C )  e.  RR* )
5347, 52, 28syl2anc 406 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  (
( A +e
B )  <_  ( A +e C )  ->  sup ( { ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +e C ) ) )
5450, 53mpd 13 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e
C ) )
5540oveq1d 5721 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  )
)  =  ( +oo +e sup ( { B ,  C } ,  RR* ,  <  )
) )
5611ad3antrrr 479 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  B  e.  RR* )
57 xrmaxcl 10860 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  sup ( { B ,  C } ,  RR* ,  <  )  e.  RR* )
5856, 42, 57syl2anc 406 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  e.  RR* )
59 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  C  =/= -oo )
60 nmnfgt 9442 . . . . . . . . . . . 12  |-  ( C  e.  RR*  ->  ( -oo  <  C  <->  C  =/= -oo )
)
6142, 60syl 14 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( -oo  <  C  <->  C  =/= -oo ) )
6259, 61mpbird 166 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  -> -oo  <  C )
6362olcd 694 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( -oo  <  B  \/ -oo  <  C ) )
64 mnfxr 7694 . . . . . . . . . . 11  |- -oo  e.  RR*
6564a1i 9 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  -> -oo  e.  RR* )
66 xrltmaxsup 10865 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\ -oo  e.  RR* )  ->  ( -oo  <  sup ( { B ,  C } ,  RR* ,  <  )  <->  ( -oo  <  B  \/ -oo  <  C ) ) )
6756, 42, 65, 66syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( -oo  <  sup ( { B ,  C } ,  RR* ,  <  )  <->  ( -oo  <  B  \/ -oo  <  C ) ) )
6863, 67mpbird 166 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  -> -oo  <  sup ( { B ,  C } ,  RR* ,  <  ) )
69 nmnfgt 9442 . . . . . . . . 9  |-  ( sup ( { B ,  C } ,  RR* ,  <  )  e.  RR*  ->  ( -oo  <  sup ( { B ,  C } ,  RR* ,  <  )  <->  sup ( { B ,  C } ,  RR* ,  <  )  =/= -oo ) )
7058, 69syl 14 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( -oo  <  sup ( { B ,  C } ,  RR* ,  <  )  <->  sup ( { B ,  C } ,  RR* ,  <  )  =/= -oo ) )
7168, 70mpbid 146 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  =/= -oo )
72 xaddpnf2 9471 . . . . . . 7  |-  ( ( sup ( { B ,  C } ,  RR* ,  <  )  e.  RR*  /\ 
sup ( { B ,  C } ,  RR* ,  <  )  =/= -oo )  ->  ( +oo +e sup ( { B ,  C } ,  RR* ,  <  ) )  = +oo )
7358, 71, 72syl2anc 406 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( +oo +e sup ( { B ,  C } ,  RR* ,  <  )
)  = +oo )
7455, 73eqtrd 2132 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  )
)  = +oo )
7545, 54, 743eqtr4d 2142 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
76 xrmnfdc 9467 . . . . . . 7  |-  ( C  e.  RR*  -> DECID  C  = -oo )
77763ad2ant3 972 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> DECID  C  = -oo )
7877ad2antrr 475 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  -> DECID 
C  = -oo )
79 dcne 2278 . . . . 5  |-  (DECID  C  = -oo  <->  ( C  = -oo  \/  C  =/= -oo ) )
8078, 79sylib 121 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  ->  ( C  = -oo  \/  C  =/= -oo ) )
8139, 75, 80mpjaodan 753 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  ->  sup ( { ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
8211ad2antrr 475 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  B  e.  RR* )
8314ad2antrr 475 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  C  e.  RR* )
8482, 83, 57syl2anc 406 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  e.  RR* )
85 simpr 109 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  B  =/= -oo )
86 nmnfgt 9442 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  ( -oo  <  B  <->  B  =/= -oo )
)
8782, 86syl 14 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( -oo  <  B  <-> 
B  =/= -oo )
)
8885, 87mpbird 166 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  -> -oo  <  B )
8988orcd 693 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( -oo  <  B  \/ -oo  <  C
) )
9064a1i 9 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  -> -oo  e.  RR* )
9182, 83, 90, 66syl3anc 1184 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( -oo  <  sup ( { B ,  C } ,  RR* ,  <  )  <-> 
( -oo  <  B  \/ -oo 
<  C ) ) )
9289, 91mpbird 166 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  -> -oo  <  sup ( { B ,  C } ,  RR* ,  <  )
)
9384, 69syl 14 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( -oo  <  sup ( { B ,  C } ,  RR* ,  <  )  <->  sup ( { B ,  C } ,  RR* ,  <  )  =/= -oo ) )
9492, 93mpbid 146 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  =/= -oo )
9584, 94, 72syl2anc 406 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( +oo +e sup ( { B ,  C } ,  RR* ,  <  ) )  = +oo )
96 simplr 500 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  A  = +oo )
9796oveq1d 5721 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) )  =  ( +oo +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
98 prcom 3546 . . . . . 6  |-  { ( A +e B ) ,  ( A +e C ) }  =  { ( A +e C ) ,  ( A +e B ) }
9998supeq1i 6790 . . . . 5  |-  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  sup ( { ( A +e C ) ,  ( A +e B ) } ,  RR* ,  <  )
10051ad2antrr 475 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e C )  e. 
RR* )
10146ad2antrr 475 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e B )  e. 
RR* )
102100, 101jca 302 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( ( A +e C )  e.  RR*  /\  ( A +e B )  e.  RR* ) )
103 pnfge 9416 . . . . . . . . 9  |-  ( ( A +e C )  e.  RR*  ->  ( A +e C )  <_ +oo )
104100, 103syl 14 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e C )  <_ +oo )
10596oveq1d 5721 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e B )  =  ( +oo +e
B ) )
106 xaddpnf2 9471 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
10782, 106sylancom 414 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
108105, 107eqtrd 2132 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e B )  = +oo )
109104, 108breqtrrd 3901 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e C )  <_ 
( A +e
B ) )
110 xrmaxleim 10852 . . . . . . 7  |-  ( ( ( A +e
C )  e.  RR*  /\  ( A +e
B )  e.  RR* )  ->  ( ( A +e C )  <_  ( A +e B )  ->  sup ( { ( A +e C ) ,  ( A +e B ) } ,  RR* ,  <  )  =  ( A +e B ) ) )
111102, 109, 110sylc 62 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  sup ( { ( A +e C ) ,  ( A +e B ) } ,  RR* ,  <  )  =  ( A +e B ) )
112111, 108eqtrd 2132 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  sup ( { ( A +e C ) ,  ( A +e B ) } ,  RR* ,  <  )  = +oo )
11399, 112syl5eq 2144 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  sup ( { ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  = +oo )
11495, 97, 1133eqtr4rd 2143 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  sup ( { ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
115 xrmnfdc 9467 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = -oo )
116 dcne 2278 . . . . . 6  |-  (DECID  B  = -oo  <->  ( B  = -oo  \/  B  =/= -oo ) )
117115, 116sylib 121 . . . . 5  |-  ( B  e.  RR*  ->  ( B  = -oo  \/  B  =/= -oo ) )
1181173ad2ant2 971 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  = -oo  \/  B  =/= -oo ) )
119118adantr 272 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  = +oo )  ->  ( B  = -oo  \/  B  =/= -oo ) )
12081, 114, 119mpjaodan 753 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  = +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
121 simpllr 504 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  A  = -oo )
122 simpr 109 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  C  = +oo )
123121, 122oveq12d 5724 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e C )  =  ( -oo +e +oo ) )
124 mnfaddpnf 9475 . . . . . 6  |-  ( -oo +e +oo )  =  0
125123, 124syl6eq 2148 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e C )  =  0 )
12646ad3antrrr 479 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e B )  e.  RR* )
12751ad3antrrr 479 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e C )  e.  RR* )
128126, 127jca 302 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  (
( A +e
B )  e.  RR*  /\  ( A +e
C )  e.  RR* ) )
129 0le0 8667 . . . . . . . 8  |-  0  <_  0
130129a1i 9 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  0  <_  0 )
131 simplr 500 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  A  = -oo )
132 simpr 109 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  B  = +oo )
133131, 132oveq12d 5724 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e B )  =  ( -oo +e +oo ) )
134133, 124syl6eq 2148 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e B )  =  0 )
135134adantr 272 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e B )  =  0 )
136130, 135, 1253brtr4d 3905 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e B )  <_  ( A +e C ) )
137128, 136, 28sylc 62 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e
C ) )
138 prcom 3546 . . . . . . . . . . 11  |-  { C ,  B }  =  { B ,  C }
139138supeq1i 6790 . . . . . . . . . 10  |-  sup ( { C ,  B } ,  RR* ,  <  )  =  sup ( { B ,  C } ,  RR* ,  <  )
14014ad2antrr 475 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  C  e.  RR* )
14111ad2antrr 475 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  B  e.  RR* )
142140, 141jca 302 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( C  e. 
RR*  /\  B  e.  RR* ) )
143 pnfge 9416 . . . . . . . . . . . . . 14  |-  ( C  e.  RR*  ->  C  <_ +oo )
1441433ad2ant3 972 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  C  <_ +oo )
145144ad2antrr 475 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  C  <_ +oo )
146145, 132breqtrrd 3901 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  C  <_  B
)
147 xrmaxleim 10852 . . . . . . . . . . 11  |-  ( ( C  e.  RR*  /\  B  e.  RR* )  ->  ( C  <_  B  ->  sup ( { C ,  B } ,  RR* ,  <  )  =  B ) )
148142, 146, 147sylc 62 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  sup ( { C ,  B } ,  RR* ,  <  )  =  B )
149139, 148syl5eqr 2146 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  =  B )
150149, 132eqtrd 2132 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  = +oo )
151150oveq2d 5722 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) )  =  ( A +e +oo ) )
152131oveq1d 5721 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e +oo )  =  ( -oo +e +oo ) )
153152, 124syl6eq 2148 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e +oo )  =  0 )
154151, 153eqtrd 2132 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) )  =  0 )
155154adantr 272 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  )
)  =  0 )
156125, 137, 1553eqtr4d 2142 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
15751ad3antrrr 479 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e C )  e.  RR* )
15846ad3antrrr 479 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e B )  e.  RR* )
159157, 158jca 302 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  (
( A +e
C )  e.  RR*  /\  ( A +e
B )  e.  RR* ) )
160 0xr 7684 . . . . . . . . . 10  |-  0  e.  RR*
161 mnfle 9419 . . . . . . . . . 10  |-  ( 0  e.  RR*  -> -oo  <_  0 )
162160, 161mp1i 10 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  -> -oo  <_  0 )
163 simpllr 504 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  = -oo )
164163oveq1d 5721 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e C )  =  ( -oo +e C ) )
165 xaddmnf2 9473 . . . . . . . . . . 11  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  ->  ( -oo +e C )  = -oo )
166140, 165sylan 279 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( -oo +e C )  = -oo )
167164, 166eqtrd 2132 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e C )  = -oo )
168134adantr 272 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e B )  =  0 )
169162, 167, 1683brtr4d 3905 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e C )  <_  ( A +e B ) )
170159, 169, 110sylc 62 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  sup ( { ( A +e C ) ,  ( A +e
B ) } ,  RR* ,  <  )  =  ( A +e
B ) )
171170, 168eqtrd 2132 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  sup ( { ( A +e C ) ,  ( A +e
B ) } ,  RR* ,  <  )  =  0 )
17299, 171syl5eq 2144 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  0 )
173154adantr 272 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  )
)  =  0 )
174172, 173eqtr4d 2135 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
175 xrpnfdc 9466 . . . . . . 7  |-  ( C  e.  RR*  -> DECID  C  = +oo )
176 dcne 2278 . . . . . . 7  |-  (DECID  C  = +oo  <->  ( C  = +oo  \/  C  =/= +oo ) )
177175, 176sylib 121 . . . . . 6  |-  ( C  e.  RR*  ->  ( C  = +oo  \/  C  =/= +oo ) )
1781773ad2ant3 972 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  = +oo  \/  C  =/= +oo ) )
179178ad2antrr 475 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( C  = +oo  \/  C  =/= +oo ) )
180156, 174, 179mpjaodan 753 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  sup ( { ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
181 simpllr 504 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  A  = -oo )
182 simpr 109 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  C  = +oo )
183181, 182oveq12d 5724 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e C )  =  ( -oo +e +oo ) )
18446ad2antrr 475 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( A +e B )  e. 
RR* )
18551ad2antrr 475 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( A +e C )  e. 
RR* )
186184, 185jca 302 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( ( A +e B )  e.  RR*  /\  ( A +e C )  e.  RR* ) )
187 simplr 500 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  A  = -oo )
188187oveq1d 5721 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( A +e B )  =  ( -oo +e
B ) )
18911ad2antrr 475 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  B  e.  RR* )
190 xaddmnf2 9473 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
191189, 190sylancom 414 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
192188, 191eqtrd 2132 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( A +e B )  = -oo )
193 mnfle 9419 . . . . . . . . 9  |-  ( ( A +e C )  e.  RR*  -> -oo 
<_  ( A +e
C ) )
194185, 193syl 14 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  -> -oo  <_  ( A +e C ) )
195192, 194eqbrtrd 3895 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( A +e B )  <_ 
( A +e
C ) )
196186, 195, 28sylc 62 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  sup ( { ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +e C ) )
197196adantr 272 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e
C ) )
198189adantr 272 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  e.  RR* )
19914ad3antrrr 479 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  C  e.  RR* )
200198, 199jca 302 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( B  e.  RR*  /\  C  e.  RR* ) )
201 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  B  =/= +oo )
202 npnflt 9439 . . . . . . . . . . . . 13  |-  ( B  e.  RR*  ->  ( B  < +oo  <->  B  =/= +oo )
)
203189, 202syl 14 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( B  < +oo 
<->  B  =/= +oo )
)
204201, 203mpbird 166 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  B  < +oo )
205204adantr 272 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  < +oo )
206205, 182breqtrrd 3901 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  <  C )
207198, 199, 206xrltled 9426 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  <_  C )
208200, 207, 35sylc 62 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  =  C )
209208, 182eqtrd 2132 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  = +oo )
210181, 209oveq12d 5724 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  )
)  =  ( -oo +e +oo )
)
211183, 197, 2103eqtr4d 2142 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
212189adantr 272 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  e.  RR* )
21314ad3antrrr 479 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  e.  RR* )
214212, 213, 57syl2anc 406 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  e.  RR* )
215204adantr 272 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  < +oo )
216 simpr 109 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  =/= +oo )
217 npnflt 9439 . . . . . . . . . . 11  |-  ( C  e.  RR*  ->  ( C  < +oo  <->  C  =/= +oo )
)
218213, 217syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( C  < +oo  <->  C  =/= +oo )
)
219216, 218mpbird 166 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  < +oo )
220215, 219jca 302 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B  < +oo  /\  C  < +oo ) )
221 pnfxr 7690 . . . . . . . . . 10  |- +oo  e.  RR*
222221a1i 9 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  -> +oo  e.  RR* )
223 xrmaxltsup 10866 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\ +oo  e.  RR* )  ->  ( sup ( { B ,  C } ,  RR* ,  <  )  < +oo  <->  ( B  < +oo  /\  C  < +oo ) ) )
224212, 213, 222, 223syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( sup ( { B ,  C } ,  RR* ,  <  )  < +oo  <->  ( B  < +oo  /\  C  < +oo ) ) )
225220, 224mpbird 166 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  < +oo )
226 npnflt 9439 . . . . . . . 8  |-  ( sup ( { B ,  C } ,  RR* ,  <  )  e.  RR*  ->  ( sup ( { B ,  C } ,  RR* ,  <  )  < +oo  <->  sup ( { B ,  C } ,  RR* ,  <  )  =/= +oo ) )
227214, 226syl 14 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( sup ( { B ,  C } ,  RR* ,  <  )  < +oo  <->  sup ( { B ,  C } ,  RR* ,  <  )  =/= +oo ) )
228225, 227mpbid 146 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  =/= +oo )
229 xaddmnf2 9473 . . . . . 6  |-  ( ( sup ( { B ,  C } ,  RR* ,  <  )  e.  RR*  /\ 
sup ( { B ,  C } ,  RR* ,  <  )  =/= +oo )  ->  ( -oo +e sup ( { B ,  C } ,  RR* ,  <  ) )  = -oo )
230214, 228, 229syl2anc 406 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( -oo +e sup ( { B ,  C } ,  RR* ,  <  )
)  = -oo )
231 simpllr 504 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  = -oo )
232231oveq1d 5721 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  )
)  =  ( -oo +e sup ( { B ,  C } ,  RR* ,  <  )
) )
233196adantr 272 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e
C ) )
234231oveq1d 5721 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( A +e C )  =  ( -oo +e C ) )
235233, 234eqtrd 2132 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( -oo +e
C ) )
236213, 165sylancom 414 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( -oo +e C )  = -oo )
237235, 236eqtrd 2132 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  = -oo )
238230, 232, 2373eqtr4rd 2143 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
239178ad2antrr 475 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( C  = +oo  \/  C  =/= +oo ) )
240211, 238, 239mpjaodan 753 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  sup ( { ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
241 xrpnfdc 9466 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = +oo )
2422413ad2ant2 971 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> DECID  B  = +oo )
243 dcne 2278 . . . . 5  |-  (DECID  B  = +oo  <->  ( B  = +oo  \/  B  =/= +oo ) )
244242, 243sylib 121 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  = +oo  \/  B  =/= +oo ) )
245244adantr 272 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  = -oo )  ->  ( B  = +oo  \/  B  =/= +oo ) )
246180, 240, 245mpjaodan 753 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  = -oo )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
247 elxr 9404 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
248247biimpi 119 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2492483ad2ant1 970 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2505, 120, 246, 249mpjao3dan 1253 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 670  DECID wdc 786    \/ w3o 929    /\ w3a 930    = wceq 1299    e. wcel 1448    =/= wne 2267   {cpr 3475   class class class wbr 3875  (class class class)co 5706   supcsup 6784   RRcr 7499   0cc0 7500   +oocpnf 7669   -oocmnf 7670   RR*cxr 7671    < clt 7672    <_ cle 7673   +ecxad 9398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-sup 6786  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-xneg 9400  df-xadd 9401  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611
This theorem is referenced by:  xrminadd  10883
  Copyright terms: Public domain W3C validator