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Theorem xrmaxadd 11062
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 986 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  e.  RR )  ->  B  e.  RR* )
3 simpl3 987 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  e.  RR )  ->  C  e.  RR* )
4 xrmaxaddlem 11061 . . 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 1217 . 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 524 . . . . . 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 5800 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e C )  =  ( +oo +e -oo ) )
9 simp1 982 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  e.  RR* )
109ad3antrrr 484 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  A  e.  RR* )
11 simp2 983 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  B  e.  RR* )
1211ad3antrrr 484 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  B  e.  RR* )
1310, 12xaddcld 9697 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e B )  e.  RR* )
14 simp3 984 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  C  e.  RR* )
1514ad3antrrr 484 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  C  e.  RR* )
1610, 15xaddcld 9697 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e C )  e.  RR* )
1713, 16jca 304 . . . . . 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 520 . . . . . . . . . . 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 5800 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  ->  ( A +e B )  =  ( +oo +e -oo ) )
21 pnfaddmnf 9663 . . . . . . . . . 10  |-  ( +oo +e -oo )  =  0
2220, 21eqtrdi 2189 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  ->  ( A +e B )  =  0 )
2322adantr 274 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e B )  =  0 )
248, 21eqtrdi 2189 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e C )  =  0 )
2523, 24eqtr4d 2176 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e B )  =  ( A +e C ) )
2616xrleidd 9617 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e C )  <_  ( A +e C ) )
2725, 26eqbrtrd 3958 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( A +e B )  <_  ( A +e C ) )
28 xrmaxleim 11045 . . . . . 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 304 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  ( B  e.  RR*  /\  C  e.  RR* ) )
31 simplr 520 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  B  = -oo )
3231, 7eqtr4d 2176 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  B  =  C )
3315xrleidd 9617 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  C  <_  C )
3432, 33eqbrtrd 3958 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  B  <_  C )
35 xrmaxleim 11045 . . . . . . . 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 2173 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  = -oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  = -oo )
386, 37oveq12d 5800 . . . . 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 2183 . . . 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 524 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  A  = +oo )
4140oveq1d 5797 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e C )  =  ( +oo +e C ) )
4214ad3antrrr 484 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  C  e.  RR* )
43 xaddpnf2 9660 . . . . . . 7  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( +oo +e C )  = +oo )
4442, 43sylancom 417 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( +oo +e C )  = +oo )
4541, 44eqtrd 2173 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e C )  = +oo )
469, 11xaddcld 9697 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A +e B )  e.  RR* )
4746ad3antrrr 484 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e B )  e.  RR* )
48 pnfge 9605 . . . . . . . 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 3964 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e B )  <_  ( A +e C ) )
519, 14xaddcld 9697 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A +e C )  e.  RR* )
5251ad3antrrr 484 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( A +e C )  e.  RR* )
5347, 52, 28syl2anc 409 . . . . . 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 5797 . . . . . 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 484 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  B  e.  RR* )
57 xrmaxcl 11053 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  sup ( { B ,  C } ,  RR* ,  <  )  e.  RR* )
5856, 42, 57syl2anc 409 . . . . . . 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 9631 . . . . . . . . . . . 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 724 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  /\  C  =/= -oo )  ->  ( -oo  <  B  \/ -oo  <  C ) )
64 mnfxr 7846 . . . . . . . . . . 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 11058 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\ -oo  e.  RR* )  ->  ( -oo  <  sup ( { B ,  C } ,  RR* ,  <  )  <->  ( -oo  <  B  \/ -oo  <  C ) ) )
6756, 42, 65, 66syl3anc 1217 . . . . . . . . 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 9631 . . . . . . . . 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 9660 . . . . . . 7  |-  ( ( sup ( { B ,  C } ,  RR* ,  <  )  e.  RR*  /\ 
sup ( { B ,  C } ,  RR* ,  <  )  =/= -oo )  ->  ( +oo +e sup ( { B ,  C } ,  RR* ,  <  ) )  = +oo )
7358, 71, 72syl2anc 409 . . . . . 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 2173 . . . . 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 2183 . . . 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 9656 . . . . . . 7  |-  ( C  e.  RR*  -> DECID  C  = -oo )
77763ad2ant3 1005 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> DECID  C  = -oo )
7877ad2antrr 480 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  = -oo )  -> DECID 
C  = -oo )
79 dcne 2320 . . . . 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 788 . . 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 480 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  B  e.  RR* )
8314ad2antrr 480 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  C  e.  RR* )
8482, 83, 57syl2anc 409 . . . . 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 9631 . . . . . . . . . 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 723 . . . . . . 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 1217 . . . . . . 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 409 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( +oo +e sup ( { B ,  C } ,  RR* ,  <  ) )  = +oo )
96 simplr 520 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  A  = +oo )
9796oveq1d 5797 . . . 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 3607 . . . . . 6  |-  { ( A +e B ) ,  ( A +e C ) }  =  { ( A +e C ) ,  ( A +e B ) }
9998supeq1i 6883 . . . . 5  |-  sup ( { ( A +e B ) ,  ( A +e
C ) } ,  RR* ,  <  )  =  sup ( { ( A +e C ) ,  ( A +e B ) } ,  RR* ,  <  )
10051ad2antrr 480 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e C )  e. 
RR* )
10146ad2antrr 480 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e B )  e. 
RR* )
102100, 101jca 304 . . . . . . 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 9605 . . . . . . . . 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 5797 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e B )  =  ( +oo +e
B ) )
106 xaddpnf2 9660 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
10782, 106sylancom 417 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
108105, 107eqtrd 2173 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e B )  = +oo )
109104, 108breqtrrd 3964 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = +oo )  /\  B  =/= -oo )  ->  ( A +e C )  <_ 
( A +e
B ) )
110 xrmaxleim 11045 . . . . . . 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 2173 . . . . 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 2185 . . . 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 2184 . . 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 9656 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = -oo )
116 dcne 2320 . . . . . 6  |-  (DECID  B  = -oo  <->  ( B  = -oo  \/  B  =/= -oo ) )
117115, 116sylib 121 . . . . 5  |-  ( B  e.  RR*  ->  ( B  = -oo  \/  B  =/= -oo ) )
1181173ad2ant2 1004 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  = -oo  \/  B  =/= -oo ) )
119118adantr 274 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  = +oo )  ->  ( B  = -oo  \/  B  =/= -oo ) )
12081, 114, 119mpjaodan 788 . 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 524 . . . . . . 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 5800 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e C )  =  ( -oo +e +oo ) )
124 mnfaddpnf 9664 . . . . . 6  |-  ( -oo +e +oo )  =  0
125123, 124eqtrdi 2189 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e C )  =  0 )
12646ad3antrrr 484 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e B )  e.  RR* )
12751ad3antrrr 484 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e C )  e.  RR* )
128126, 127jca 304 . . . . . 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 8833 . . . . . . . 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 520 . . . . . . . . . 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 5800 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e B )  =  ( -oo +e +oo ) )
134133, 124eqtrdi 2189 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e B )  =  0 )
135134adantr 274 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  = +oo )  ->  ( A +e B )  =  0 )
136130, 135, 1253brtr4d 3968 . . . . . 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 3607 . . . . . . . . . . 11  |-  { C ,  B }  =  { B ,  C }
139138supeq1i 6883 . . . . . . . . . 10  |-  sup ( { C ,  B } ,  RR* ,  <  )  =  sup ( { B ,  C } ,  RR* ,  <  )
14014ad2antrr 480 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  C  e.  RR* )
14111ad2antrr 480 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  B  e.  RR* )
142140, 141jca 304 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( C  e. 
RR*  /\  B  e.  RR* ) )
143 pnfge 9605 . . . . . . . . . . . . . 14  |-  ( C  e.  RR*  ->  C  <_ +oo )
1441433ad2ant3 1005 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  C  <_ +oo )
145144ad2antrr 480 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  C  <_ +oo )
146145, 132breqtrrd 3964 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  C  <_  B
)
147 xrmaxleim 11045 . . . . . . . . . . 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 2187 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  =  B )
150149, 132eqtrd 2173 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  = +oo )
151150oveq2d 5798 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) )  =  ( A +e +oo ) )
152131oveq1d 5797 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e +oo )  =  ( -oo +e +oo ) )
153152, 124eqtrdi 2189 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e +oo )  =  0 )
154151, 153eqtrd 2173 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) )  =  0 )
155154adantr 274 . . . . 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 2183 . . . 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 484 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e C )  e.  RR* )
15846ad3antrrr 484 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e B )  e.  RR* )
159157, 158jca 304 . . . . . . . 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 7836 . . . . . . . . . 10  |-  0  e.  RR*
161 mnfle 9608 . . . . . . . . . 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 524 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  A  = -oo )
164163oveq1d 5797 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e C )  =  ( -oo +e C ) )
165 xaddmnf2 9662 . . . . . . . . . . 11  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  ->  ( -oo +e C )  = -oo )
166140, 165sylan 281 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( -oo +e C )  = -oo )
167164, 166eqtrd 2173 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e C )  = -oo )
168134adantr 274 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  /\  C  =/= +oo )  ->  ( A +e B )  =  0 )
169162, 167, 1683brtr4d 3968 . . . . . . . 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 2173 . . . . . 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 2185 . . . . 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 274 . . . . 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 2176 . . . 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 9655 . . . . . . 7  |-  ( C  e.  RR*  -> DECID  C  = +oo )
176 dcne 2320 . . . . . . 7  |-  (DECID  C  = +oo  <->  ( C  = +oo  \/  C  =/= +oo ) )
177175, 176sylib 121 . . . . . 6  |-  ( C  e.  RR*  ->  ( C  = +oo  \/  C  =/= +oo ) )
1781773ad2ant3 1005 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  = +oo  \/  C  =/= +oo ) )
179178ad2antrr 480 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  = +oo )  ->  ( C  = +oo  \/  C  =/= +oo ) )
180156, 174, 179mpjaodan 788 . . 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 524 . . . . . 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 5800 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  ( A +e C )  =  ( -oo +e +oo ) )
18446ad2antrr 480 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( A +e B )  e. 
RR* )
18551ad2antrr 480 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( A +e C )  e. 
RR* )
186184, 185jca 304 . . . . . . 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 520 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  A  = -oo )
188187oveq1d 5797 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( A +e B )  =  ( -oo +e
B ) )
18911ad2antrr 480 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  B  e.  RR* )
190 xaddmnf2 9662 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
191189, 190sylancom 417 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
192188, 191eqtrd 2173 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( A +e B )  = -oo )
193 mnfle 9608 . . . . . . . . 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 3958 . . . . . . 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 274 . . . . 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 274 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  e.  RR* )
19914ad3antrrr 484 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  C  e.  RR* )
200198, 199jca 304 . . . . . . . 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 9628 . . . . . . . . . . . . 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 274 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  < +oo )
206205, 182breqtrrd 3964 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  B  <  C )
207198, 199, 206xrltled 9615 . . . . . . . 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 2173 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  = +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  = +oo )
210181, 209oveq12d 5800 . . . . 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 2183 . . . 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 274 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  B  e.  RR* )
21314ad3antrrr 484 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  C  e.  RR* )
214212, 213, 57syl2anc 409 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  sup ( { B ,  C } ,  RR* ,  <  )  e.  RR* )
215204adantr 274 . . . . . . . . 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 9628 . . . . . . . . . . 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 304 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( B  < +oo  /\  C  < +oo ) )
221 pnfxr 7842 . . . . . . . . . 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 11059 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\ +oo  e.  RR* )  ->  ( sup ( { B ,  C } ,  RR* ,  <  )  < +oo  <->  ( B  < +oo  /\  C  < +oo ) ) )
224212, 213, 222, 223syl3anc 1217 . . . . . . . 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 9628 . . . . . . . 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 9662 . . . . . 6  |-  ( ( sup ( { B ,  C } ,  RR* ,  <  )  e.  RR*  /\ 
sup ( { B ,  C } ,  RR* ,  <  )  =/= +oo )  ->  ( -oo +e sup ( { B ,  C } ,  RR* ,  <  ) )  = -oo )
230214, 228, 229syl2anc 409 . . . . 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 524 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  A  = -oo )
232231oveq1d 5797 . . . . 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 274 . . . . . . 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 5797 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( A +e C )  =  ( -oo +e C ) )
235233, 234eqtrd 2173 . . . . . 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 417 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  /\  C  =/= +oo )  ->  ( -oo +e C )  = -oo )
237235, 236eqtrd 2173 . . . . 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 2184 . . . 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 480 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  A  = -oo )  /\  B  =/= +oo )  ->  ( C  = +oo  \/  C  =/= +oo ) )
240211, 238, 239mpjaodan 788 . . 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 9655 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = +oo )
2422413ad2ant2 1004 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> DECID  B  = +oo )
243 dcne 2320 . . . . 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 274 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  = -oo )  ->  ( B  = +oo  \/  B  =/= +oo ) )
246180, 240, 245mpjaodan 788 . 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 9593 . . . 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 1003 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2505, 120, 246, 249mpjao3dan 1286 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 698  DECID wdc 820    \/ w3o 962    /\ w3a 963    = wceq 1332    e. wcel 1481    =/= wne 2309   {cpr 3533   class class class wbr 3937  (class class class)co 5782   supcsup 6877   RRcr 7643   0cc0 7644   +oocpnf 7821   -oocmnf 7822   RR*cxr 7823    < clt 7824    <_ cle 7825   +ecxad 9587
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-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764
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-eu 2003  df-mo 2004  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-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-sup 6879  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-rp 9471  df-xneg 9589  df-xadd 9590  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803
This theorem is referenced by:  xrminadd  11076
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