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

Theorem xrmaxltsup 10866
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 952 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  A  e.  RR* )
2 simpl2 953 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  B  e.  RR* )
3 xrmaxcl 10860 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  sup ( { A ,  B } ,  RR* ,  <  )  e.  RR* )
41, 2, 3syl2anc 406 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  sup ( { A ,  B } ,  RR* ,  <  )  e.  RR* )
5 simpl3 954 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  C  e.  RR* )
6 xrmax1sup 10861 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  sup ( { A ,  B } ,  RR* ,  <  ) )
763adant3 969 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  <_  sup ( { A ,  B } ,  RR* ,  <  ) )
87adantr 272 . . . 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 9434 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  A  <  C )
11 xrmax2sup 10862 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  <_  sup ( { A ,  B } ,  RR* ,  <  ) )
121, 2, 11syl2anc 406 . . . 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 9434 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  ->  B  <  C )
1410, 13jca 302 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  sup ( { A ,  B } ,  RR* ,  <  )  <  C )  -> 
( A  <  C  /\  B  <  C ) )
15 simplr 500 . . . . . . 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 504 . . . . . . 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 10863 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  sup ( { A ,  B } ,  RR ,  <  ) )
1815, 16, 17syl2anc 406 . . . . . 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 512 . . . . . . 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 10830 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( sup ( { A ,  B } ,  RR ,  <  )  <  C  <->  ( A  <  C  /\  B  < 
C ) ) )
2215, 16, 20, 21syl3anc 1184 . . . . . . 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 3895 . . . . 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 500 . . . . . . . . 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 504 . . . . . . . . 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 10822 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
2825, 26, 27syl2anc 406 . . . . . . . 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 2168 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( sup ( { A ,  B } ,  RR* ,  <  )  e.  RR  <->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR ) )
3025, 26, 29syl2anc 406 . . . . . . . 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 9408 . . . . . . 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 3901 . . . . 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 501 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  <  C )
3736ad3antrrr 479 . . . . . 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 9415 . . . . . . . . 9  |-  ( A  e.  RR*  ->  -.  A  < -oo )
39383ad2ant1 970 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -.  A  < -oo )
4039ad4antr 481 . . . . . . 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 3887 . . . . . . 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 639 . . . . . 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 590 . . . . 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 9404 . . . . . . . 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 972 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  RR  \/  C  = +oo  \/  C  = -oo ) )
4847ad3antrrr 479 . . . . 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 1253 . . . 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 475 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = +oo )  ->  A  <  C
)
51 pnfnlt 9414 . . . . . . . 8  |-  ( C  e.  RR*  ->  -. +oo  <  C )
52513ad2ant3 972 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -. +oo 
<  C )
5352ad3antrrr 479 . . . . . 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 3885 . . . . . 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 639 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = +oo )  ->  -.  A  <  C )
5750, 56pm2.21dd 590 . . . 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 9419 . . . . . . . . 9  |-  ( B  e.  RR*  -> -oo  <_  B )
60593ad2ant2 971 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> -oo  <_  B )
6160ad3antrrr 479 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  -> -oo  <_  B )
6258, 61eqbrtrd 3895 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  A  <_  B
)
63 simp1 949 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  e.  RR* )
6463ad3antrrr 479 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  A  e.  RR* )
65 simp2 950 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  B  e.  RR* )
6665ad3antrrr 479 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  B  e.  RR* )
67 xrmaxleim 10852 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B ) )
6864, 66, 67syl2anc 406 . . . . . 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 502 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  <  C )
7170ad2antrr 475 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  /\  B  e.  RR )  /\  A  = -oo )  ->  B  <  C
)
7269, 71eqbrtrd 3895 . . . 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 9404 . . . . . . 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 970 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
7675ad2antrr 475 . . . 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 1253 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C )
78 simplrr 506 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = +oo )  ->  B  <  C )
79 breq1 3878 . . . . . 6  |-  ( B  = +oo  ->  ( B  <  C  <-> +oo  <  C
) )
8079adantl 273 . . . . 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 475 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = +oo )  ->  -. +oo 
<  C )
8381, 82pm2.21dd 590 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = +oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C )
84 prcom 3546 . . . . . 6  |-  { B ,  A }  =  { A ,  B }
8584supeq1i 6790 . . . . 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 9419 . . . . . . . . 9  |-  ( A  e.  RR*  -> -oo  <_  A )
88873ad2ant1 970 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> -oo  <_  A )
8988ad2antrr 475 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  -> -oo  <_  A )
9086, 89eqbrtrd 3895 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  B  <_  A )
91 simpll2 989 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  B  e.  RR* )
92 simpll1 988 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  A  e.  RR* )
93 xrmaxleim 10852 . . . . . . 7  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  ->  sup ( { B ,  A } ,  RR* ,  <  )  =  A ) )
9491, 92, 93syl2anc 406 . . . . . 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 2146 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  A )
97 simplrl 505 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  A  <  C )
9896, 97eqbrtrd 3895 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR* )  /\  ( A  < 
C  /\  B  <  C ) )  /\  B  = -oo )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C )
99 elxr 9404 . . . . . 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 971 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
102101adantr 272 . . 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 1253 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  C  /\  B  <  C ) )  ->  sup ( { A ,  B } ,  RR* ,  <  )  <  C )
10414, 103impbida 566 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 929    /\ w3a 930    = wceq 1299    e. wcel 1448   {cpr 3475   class class class wbr 3875   supcsup 6784   RRcr 7499   +oocpnf 7669   -oocmnf 7670   RR*cxr 7671    < clt 7672    <_ cle 7673
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-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:  xrmaxadd  10869  xrltmininf  10878  iooinsup  10885
  Copyright terms: Public domain W3C validator