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Theorem xrmaxltsup 11034
Description: Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
Assertion
Ref Expression
xrmaxltsup  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( sup ( { A ,  B } ,  RR* ,  <  )  <  C  <->  ( A  <  C  /\  B  < 
C ) ) )

Proof of Theorem xrmaxltsup
StepHypRef Expression
1 simpl1 984 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  A  e.  RR* )
2 simpl2 985 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  B  e.  RR* )
3 xrmaxcl 11028 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  sup ( { A ,  B } ,  RR* ,  <  )  e.  RR* )
41, 2, 3syl2anc 408 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  sup ( { A ,  B } ,  RR* ,  <  )  e.  RR* )
5 simpl3 986 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  C  e.  RR* )
6 xrmax1sup 11029 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  sup ( { A ,  B } ,  RR* ,  <  ) )
763adant3 1001 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  <_  sup ( { A ,  B } ,  RR* ,  <  ) )
87adantr 274 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  A  <_  sup ( { A ,  B } ,  RR* ,  <  ) )
9 simpr 109 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C )
101, 4, 5, 8, 9xrlelttrd 9600 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  A  <  C )
11 xrmax2sup 11030 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  <_  sup ( { A ,  B } ,  RR* ,  <  ) )
121, 2, 11syl2anc 408 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  B  <_  sup ( { A ,  B } ,  RR* ,  <  ) )
132, 4, 5, 12, 9xrlelttrd 9600 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  B  <  C )
1410, 13jca 304 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  -> 
( A  <  C  /\  B  <  C ) )
15 simplr 519 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  e.  RR )  ->  A  e.  RR )
16 simpllr 523 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  e.  RR )  ->  B  e.  RR )
17 xrmaxrecl 11031 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  sup ( { A ,  B } ,  RR ,  <  ) )
1815, 16, 17syl2anc 408 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  e.  RR )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  sup ( { A ,  B } ,  RR ,  <  ) )
19 simp-4r 531 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  e.  RR )  ->  ( A  < 
C  /\  B  <  C ) )
20 simpr 109 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  e.  RR )  ->  C  e.  RR )
21 maxltsup 10997 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( sup ( { A ,  B } ,  RR ,  <  )  <  C  <->  ( A  <  C  /\  B  < 
C ) ) )
2215, 16, 20, 21syl3anc 1216 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  e.  RR )  ->  ( sup ( { A ,  B } ,  RR ,  <  )  <  C  <->  ( A  < 
C  /\  B  <  C ) ) )
2319, 22mpbird 166 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  <  C
)
2418, 23eqbrtrd 3950 . . . . 5  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  e.  RR )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C
)
25 simplr 519 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = +oo )  ->  A  e.  RR )
26 simpllr 523 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = +oo )  ->  B  e.  RR )
27 maxcl 10989 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
2825, 26, 27syl2anc 408 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = +oo )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
2917eleq1d 2208 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( sup ( { A ,  B } ,  RR* ,  <  )  e.  RR  <->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR ) )
3025, 26, 29syl2anc 408 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = +oo )  ->  ( sup ( { A ,  B } ,  RR* ,  <  )  e.  RR  <->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR ) )
3128, 30mpbird 166 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = +oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  e.  RR )
32 ltpnf 9574 . . . . . . 7  |-  ( sup ( { A ,  B } ,  RR* ,  <  )  e.  RR  ->  sup ( { A ,  B } ,  RR* ,  <  )  < +oo )
3331, 32syl 14 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = +oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  < +oo )
34 simpr 109 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = +oo )  ->  C  = +oo )
3533, 34breqtrrd 3956 . . . . 5  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = +oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C
)
36 simprl 520 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  <  C )
3736ad3antrrr 483 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = -oo )  ->  A  <  C
)
38 nltmnf 9581 . . . . . . . . 9  |-  ( A  e.  RR*  ->  -.  A  < -oo )
39383ad2ant1 1002 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -.  A  < -oo )
4039ad4antr 485 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = -oo )  ->  -.  A  < -oo )
41 simpr 109 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = -oo )  ->  C  = -oo )
4241breq2d 3941 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = -oo )  ->  ( A  < 
C  <->  A  < -oo )
)
4340, 42mtbird 662 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = -oo )  ->  -.  A  <  C )
4437, 43pm2.21dd 609 . . . . 5  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  /\  C  = -oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C
)
45 elxr 9570 . . . . . . . 8  |-  ( C  e.  RR*  <->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
4645biimpi 119 . . . . . . 7  |-  ( C  e.  RR*  ->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
47463ad2ant3 1004 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
4847ad3antrrr 483 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  ->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
4924, 35, 44, 48mpjao3dan 1285 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  e.  RR )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C
)
5036ad2antrr 479 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = +oo )  ->  A  <  C
)
51 pnfnlt 9580 . . . . . . . 8  |-  ( C  e.  RR*  ->  -. +oo  <  C )
52513ad2ant3 1004 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -. +oo 
<  C )
5352ad3antrrr 483 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = +oo )  ->  -. +oo  <  C
)
54 simpr 109 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = +oo )  ->  A  = +oo )
5554breq1d 3939 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = +oo )  ->  ( A  < 
C  <-> +oo  <  C )
)
5653, 55mtbird 662 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = +oo )  ->  -.  A  <  C )
5750, 56pm2.21dd 609 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = +oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C
)
58 simpr 109 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  A  = -oo )
59 mnfle 9585 . . . . . . . . 9  |-  ( B  e.  RR*  -> -oo  <_  B )
60593ad2ant2 1003 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> -oo  <_  B )
6160ad3antrrr 483 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  -> -oo  <_  B )
6258, 61eqbrtrd 3950 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  A  <_  B
)
63 simp1 981 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  e.  RR* )
6463ad3antrrr 483 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  A  e.  RR* )
65 simp2 982 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  B  e.  RR* )
6665ad3antrrr 483 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  B  e.  RR* )
67 xrmaxleim 11020 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B ) )
6864, 66, 67syl2anc 408 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B ) )
6962, 68mpd 13 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B )
70 simprr 521 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  <  C )
7170ad2antrr 479 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  B  <  C
)
7269, 71eqbrtrd 3950 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C
)
73 elxr 9570 . . . . . . 7  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
7473biimpi 119 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
75743ad2ant1 1002 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
7675ad2antrr 479 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  e.  RR )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
7749, 57, 72, 76mpjao3dan 1285 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C )
78 simplrr 525 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = +oo )  ->  B  <  C )
79 breq1 3932 . . . . . 6  |-  ( B  = +oo  ->  ( B  <  C  <-> +oo  <  C
) )
8079adantl 275 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = +oo )  ->  ( B  <  C  <-> +oo  <  C
) )
8178, 80mpbid 146 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = +oo )  -> +oo  <  C )
8252ad2antrr 479 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = +oo )  ->  -. +oo 
<  C )
8381, 82pm2.21dd 609 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = +oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C )
84 prcom 3599 . . . . . 6  |-  { B ,  A }  =  { A ,  B }
8584supeq1i 6875 . . . . 5  |-  sup ( { B ,  A } ,  RR* ,  <  )  =  sup ( { A ,  B } ,  RR* ,  <  )
86 simpr 109 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  B  = -oo )
87 mnfle 9585 . . . . . . . . 9  |-  ( A  e.  RR*  -> -oo  <_  A )
88873ad2ant1 1002 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> -oo  <_  A )
8988ad2antrr 479 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  -> -oo  <_  A )
9086, 89eqbrtrd 3950 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  B  <_  A )
91 simpll2 1021 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  B  e.  RR* )
92 simpll1 1020 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  A  e.  RR* )
93 xrmaxleim 11020 . . . . . . 7  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  ->  sup ( { B ,  A } ,  RR* ,  <  )  =  A ) )
9491, 92, 93syl2anc 408 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  ( B  <_  A  ->  sup ( { B ,  A } ,  RR* ,  <  )  =  A ) )
9590, 94mpd 13 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  sup ( { B ,  A } ,  RR* ,  <  )  =  A )
9685, 95syl5eqr 2186 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  A )
97 simplrl 524 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  A  <  C )
9896, 97eqbrtrd 3950 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C )
99 elxr 9570 . . . . . 6  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
10099biimpi 119 . . . . 5  |-  ( B  e.  RR*  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
1011003ad2ant2 1003 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
102101adantr 274 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  -> 
( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
10377, 83, 98, 102mpjao3dan 1285 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C )
10414, 103impbida 585 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( sup ( { A ,  B } ,  RR* ,  <  )  <  C  <->  ( A  <  C  /\  B  < 
C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 961    /\ w3a 962    = wceq 1331    e. wcel 1480   {cpr 3528   class class class wbr 3929   supcsup 6869   RRcr 7626   +oocpnf 7804   -oocmnf 7805   RR*cxr 7806    < clt 7807    <_ cle 7808
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-rp 9449  df-seqfrec 10226  df-exp 10300  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778
This theorem is referenced by:  xrmaxadd  11037  xrltmininf  11046  iooinsup  11053  xmetxpbl  12686  txmetcnp  12696
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