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Theorem xrofsup 24079
Description: The supremum is preserved by extended addition set operation. (provided minus infinity is not involved as it does not behave well with addition) (Contributed by Thierry Arnoux, 20-Mar-2017.)
Hypotheses
Ref Expression
xrofsup.1  |-  ( ph  ->  X  C_  RR* )
xrofsup.2  |-  ( ph  ->  Y  C_  RR* )
xrofsup.3  |-  ( ph  ->  sup ( X ,  RR* ,  <  )  =/= 
-oo )
xrofsup.4  |-  ( ph  ->  sup ( Y ,  RR* ,  <  )  =/= 
-oo )
xrofsup.5  |-  ( ph  ->  Z  =  ( + e " ( X  X.  Y ) ) )
Assertion
Ref Expression
xrofsup  |-  ( ph  ->  sup ( Z ,  RR* ,  <  )  =  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )

Proof of Theorem xrofsup
Dummy variables  a 
b  k  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrofsup.5 . . 3  |-  ( ph  ->  Z  =  ( + e " ( X  X.  Y ) ) )
2 xrofsup.1 . . . . . . . . . 10  |-  ( ph  ->  X  C_  RR* )
32sseld 3307 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  X  ->  x  e.  RR* )
)
4 xrofsup.2 . . . . . . . . . 10  |-  ( ph  ->  Y  C_  RR* )
54sseld 3307 . . . . . . . . 9  |-  ( ph  ->  ( y  e.  Y  ->  y  e.  RR* )
)
63, 5anim12d 547 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  X  /\  y  e.  Y )  ->  (
x  e.  RR*  /\  y  e.  RR* ) ) )
76imp 419 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  Y ) )  -> 
( x  e.  RR*  /\  y  e.  RR* )
)
8 xaddcl 10779 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x + e y )  e.  RR* )
97, 8syl 16 . . . . . 6  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  Y ) )  -> 
( x + e
y )  e.  RR* )
109ralrimivva 2758 . . . . 5  |-  ( ph  ->  A. x  e.  X  A. y  e.  Y  ( x + e
y )  e.  RR* )
11 fveq2 5687 . . . . . . . 8  |-  ( u  =  <. x ,  y
>.  ->  ( + e `  u )  =  ( + e `  <. x ,  y >. )
)
12 df-ov 6043 . . . . . . . 8  |-  ( x + e y )  =  ( + e `  <. x ,  y
>. )
1311, 12syl6eqr 2454 . . . . . . 7  |-  ( u  =  <. x ,  y
>.  ->  ( + e `  u )  =  ( x + e y ) )
1413eleq1d 2470 . . . . . 6  |-  ( u  =  <. x ,  y
>.  ->  ( ( + e `  u )  e.  RR*  <->  ( x + e y )  e. 
RR* ) )
1514ralxp 4975 . . . . 5  |-  ( A. u  e.  ( X  X.  Y ) ( + e `  u )  e.  RR*  <->  A. x  e.  X  A. y  e.  Y  ( x + e
y )  e.  RR* )
1610, 15sylibr 204 . . . 4  |-  ( ph  ->  A. u  e.  ( X  X.  Y ) ( + e `  u )  e.  RR* )
17 xaddf 10766 . . . . . 6  |-  + e : ( RR*  X.  RR* )
--> RR*
18 ffun 5552 . . . . . 6  |-  ( + e : ( RR*  X. 
RR* ) --> RR*  ->  Fun 
+ e )
1917, 18ax-mp 8 . . . . 5  |-  Fun  + e
20 xpss12 4940 . . . . . . 7  |-  ( ( X  C_  RR*  /\  Y  C_ 
RR* )  ->  ( X  X.  Y )  C_  ( RR*  X.  RR* )
)
212, 4, 20syl2anc 643 . . . . . 6  |-  ( ph  ->  ( X  X.  Y
)  C_  ( RR*  X. 
RR* ) )
2217fdmi 5555 . . . . . 6  |-  dom  + e  =  ( RR*  X. 
RR* )
2321, 22syl6sseqr 3355 . . . . 5  |-  ( ph  ->  ( X  X.  Y
)  C_  dom  + e
)
24 funimass4 5736 . . . . 5  |-  ( ( Fun  + e  /\  ( X  X.  Y
)  C_  dom  + e
)  ->  ( ( + e " ( X  X.  Y ) ) 
C_  RR*  <->  A. u  e.  ( X  X.  Y ) ( + e `  u )  e.  RR* ) )
2519, 23, 24sylancr 645 . . . 4  |-  ( ph  ->  ( ( + e " ( X  X.  Y ) )  C_  RR*  <->  A. u  e.  ( X  X.  Y ) ( + e `  u
)  e.  RR* )
)
2616, 25mpbird 224 . . 3  |-  ( ph  ->  ( + e "
( X  X.  Y
) )  C_  RR* )
271, 26eqsstrd 3342 . 2  |-  ( ph  ->  Z  C_  RR* )
28 supxrcl 10849 . . . 4  |-  ( X 
C_  RR*  ->  sup ( X ,  RR* ,  <  )  e.  RR* )
292, 28syl 16 . . 3  |-  ( ph  ->  sup ( X ,  RR* ,  <  )  e. 
RR* )
30 supxrcl 10849 . . . 4  |-  ( Y 
C_  RR*  ->  sup ( Y ,  RR* ,  <  )  e.  RR* )
314, 30syl 16 . . 3  |-  ( ph  ->  sup ( Y ,  RR* ,  <  )  e. 
RR* )
3229, 31xaddcld 10836 . 2  |-  ( ph  ->  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
)  e.  RR* )
331eleq2d 2471 . . . . 5  |-  ( ph  ->  ( z  e.  Z  <->  z  e.  ( + e " ( X  X.  Y ) ) ) )
3433pm5.32i 619 . . . 4  |-  ( (
ph  /\  z  e.  Z )  <->  ( ph  /\  z  e.  ( + e " ( X  X.  Y ) ) ) )
35 nfvd 1627 . . . . 5  |-  ( (
ph  /\  z  e.  ( + e " ( X  X.  Y ) ) )  ->  F/ x  z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) ) )
36 nfvd 1627 . . . . 5  |-  ( (
ph  /\  z  e.  ( + e " ( X  X.  Y ) ) )  ->  F/ y 
z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) ) )
372ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  X  C_ 
RR* )
38 simprl 733 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  x  e.  X )
39 supxrub 10859 . . . . . . . . . . 11  |-  ( ( X  C_  RR*  /\  x  e.  X )  ->  x  <_  sup ( X ,  RR* ,  <  ) )
4037, 38, 39syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  x  <_  sup ( X ,  RR* ,  <  ) )
414ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  Y  C_ 
RR* )
42 simprr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  y  e.  Y )
43 supxrub 10859 . . . . . . . . . . 11  |-  ( ( Y  C_  RR*  /\  y  e.  Y )  ->  y  <_  sup ( Y ,  RR* ,  <  ) )
4441, 42, 43syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  y  <_  sup ( Y ,  RR* ,  <  ) )
4537, 38sseldd 3309 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  x  e.  RR* )
4641, 42sseldd 3309 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  y  e.  RR* )
4737, 28syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  sup ( X ,  RR* ,  <  )  e.  RR* )
4841, 30syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  sup ( Y ,  RR* ,  <  )  e.  RR* )
49 xle2add 10794 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  ( sup ( X ,  RR* ,  <  )  e.  RR*  /\  sup ( Y ,  RR* ,  <  )  e.  RR* ) )  -> 
( ( x  <_  sup ( X ,  RR* ,  <  )  /\  y  <_  sup ( Y ,  RR* ,  <  ) )  ->  ( x + e y )  <_ 
( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) ) )
5045, 46, 47, 48, 49syl22anc 1185 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  (
( x  <_  sup ( X ,  RR* ,  <  )  /\  y  <_  sup ( Y ,  RR* ,  <  ) )  ->  ( x + e y )  <_ 
( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) ) )
5140, 44, 50mp2and 661 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( + e "
( X  X.  Y
) ) )  /\  ( x  e.  X  /\  y  e.  Y
) )  ->  (
x + e y )  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) ) )
5251ralrimivva 2758 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( + e " ( X  X.  Y ) ) )  ->  A. x  e.  X  A. y  e.  Y  ( x + e y )  <_ 
( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )
53 fvelima 5737 . . . . . . . . . . 11  |-  ( ( Fun  + e  /\  z  e.  ( + e " ( X  X.  Y ) ) )  ->  E. u  e.  ( X  X.  Y ) ( + e `  u )  =  z )
5419, 53mpan 652 . . . . . . . . . 10  |-  ( z  e.  ( + e " ( X  X.  Y ) )  ->  E. u  e.  ( X  X.  Y ) ( + e `  u
)  =  z )
5554adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( + e " ( X  X.  Y ) ) )  ->  E. u  e.  ( X  X.  Y
) ( + e `  u )  =  z )
5613eqeq1d 2412 . . . . . . . . . 10  |-  ( u  =  <. x ,  y
>.  ->  ( ( + e `  u )  =  z  <->  ( x + e y )  =  z ) )
5756rexxp 4976 . . . . . . . . 9  |-  ( E. u  e.  ( X  X.  Y ) ( + e `  u
)  =  z  <->  E. x  e.  X  E. y  e.  Y  ( x + e y )  =  z )
5855, 57sylib 189 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( + e " ( X  X.  Y ) ) )  ->  E. x  e.  X  E. y  e.  Y  ( x + e y )  =  z )
5952, 58r19.29d2r 2811 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( + e " ( X  X.  Y ) ) )  ->  E. x  e.  X  E. y  e.  Y  ( (
x + e y )  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) )  /\  ( x + e y )  =  z ) )
60 ancom 438 . . . . . . . 8  |-  ( ( ( x + e
y )  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) )  /\  ( x + e
y )  =  z )  <->  ( ( x + e y )  =  z  /\  (
x + e y )  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) ) ) )
61602rexbii 2693 . . . . . . 7  |-  ( E. x  e.  X  E. y  e.  Y  (
( x + e
y )  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) )  /\  ( x + e
y )  =  z )  <->  E. x  e.  X  E. y  e.  Y  ( ( x + e y )  =  z  /\  ( x + e y )  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) ) ) )
6259, 61sylib 189 . . . . . 6  |-  ( (
ph  /\  z  e.  ( + e " ( X  X.  Y ) ) )  ->  E. x  e.  X  E. y  e.  Y  ( (
x + e y )  =  z  /\  ( x + e
y )  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) ) ) )
63 breq1 4175 . . . . . . . . 9  |-  ( ( x + e y )  =  z  -> 
( ( x + e y )  <_ 
( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
)  <->  z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) ) ) )
6463biimpa 471 . . . . . . . 8  |-  ( ( ( x + e
y )  =  z  /\  ( x + e y )  <_ 
( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )
6564reximi 2773 . . . . . . 7  |-  ( E. y  e.  Y  ( ( x + e
y )  =  z  /\  ( x + e y )  <_ 
( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  E. y  e.  Y  z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )
6665reximi 2773 . . . . . 6  |-  ( E. x  e.  X  E. y  e.  Y  (
( x + e
y )  =  z  /\  ( x + e y )  <_ 
( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  E. x  e.  X  E. y  e.  Y  z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )
6762, 66syl 16 . . . . 5  |-  ( (
ph  /\  z  e.  ( + e " ( X  X.  Y ) ) )  ->  E. x  e.  X  E. y  e.  Y  z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )
6835, 36, 6719.9d2r 23922 . . . 4  |-  ( (
ph  /\  z  e.  ( + e " ( X  X.  Y ) ) )  ->  z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )
6934, 68sylbi 188 . . 3  |-  ( (
ph  /\  z  e.  Z )  ->  z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )
7069ralrimiva 2749 . 2  |-  ( ph  ->  A. z  e.  Z  z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) ) )
712ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  X  C_ 
RR* )
724ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  Y  C_ 
RR* )
73 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  z  e.  RR )
7429ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  sup ( X ,  RR* ,  <  )  e.  RR* )
7531ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  sup ( Y ,  RR* ,  <  )  e.  RR* )
76 xrofsup.3 . . . . . . . 8  |-  ( ph  ->  sup ( X ,  RR* ,  <  )  =/= 
-oo )
7776ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  sup ( X ,  RR* ,  <  )  =/=  -oo )
78 xrofsup.4 . . . . . . . 8  |-  ( ph  ->  sup ( Y ,  RR* ,  <  )  =/= 
-oo )
7978ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  sup ( Y ,  RR* ,  <  )  =/=  -oo )
80 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )
8173, 74, 75, 77, 79, 80xlt2addrd 24077 . . . . . 6  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  E. a  e.  RR*  E. b  e. 
RR*  ( z  =  ( a + e
b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )
82 nfv 1626 . . . . . . . 8  |-  F/ b ( X  C_  RR*  /\  Y  C_ 
RR* )
83 nfcv 2540 . . . . . . . . 9  |-  F/_ b RR*
84 nfre1 2722 . . . . . . . . 9  |-  F/ b E. b  e.  RR*  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) )
8583, 84nfrex 2721 . . . . . . . 8  |-  F/ b E. a  e.  RR*  E. b  e.  RR*  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) )
8682, 85nfan 1842 . . . . . . 7  |-  F/ b ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  E. a  e.  RR*  E. b  e.  RR*  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )
87 nfvd 1627 . . . . . . 7  |-  ( ( ( X  C_  RR*  /\  Y  C_ 
RR* )  /\  E. a  e.  RR*  E. b  e.  RR*  ( z  =  ( a + e
b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  ->  F/ a E. v  e.  X  E. w  e.  Y  z  <  ( v + e w ) )
88 nfvd 1627 . . . . . . 7  |-  ( ( ( X  C_  RR*  /\  Y  C_ 
RR* )  /\  E. a  e.  RR*  E. b  e.  RR*  ( z  =  ( a + e
b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  ->  F/ b E. v  e.  X  E. w  e.  Y  z  <  ( v + e w ) )
89 id 20 . . . . . . . . . . . 12  |-  ( ( X  C_  RR*  /\  Y  C_ 
RR* )  ->  ( X  C_  RR*  /\  Y  C_  RR* ) )
9089ralrimivw 2750 . . . . . . . . . . 11  |-  ( ( X  C_  RR*  /\  Y  C_ 
RR* )  ->  A. b  e.  RR*  ( X  C_  RR* 
/\  Y  C_  RR* )
)
9190ralrimivw 2750 . . . . . . . . . 10  |-  ( ( X  C_  RR*  /\  Y  C_ 
RR* )  ->  A. a  e.  RR*  A. b  e. 
RR*  ( X  C_  RR* 
/\  Y  C_  RR* )
)
9291adantr 452 . . . . . . . . 9  |-  ( ( ( X  C_  RR*  /\  Y  C_ 
RR* )  /\  E. a  e.  RR*  E. b  e.  RR*  ( z  =  ( a + e
b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  ->  A. a  e.  RR*  A. b  e. 
RR*  ( X  C_  RR* 
/\  Y  C_  RR* )
)
93 simpr 448 . . . . . . . . 9  |-  ( ( ( X  C_  RR*  /\  Y  C_ 
RR* )  /\  E. a  e.  RR*  E. b  e.  RR*  ( z  =  ( a + e
b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  ->  E. a  e.  RR*  E. b  e. 
RR*  ( z  =  ( a + e
b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )
9492, 93r19.29d2r 2811 . . . . . . . 8  |-  ( ( ( X  C_  RR*  /\  Y  C_ 
RR* )  /\  E. a  e.  RR*  E. b  e.  RR*  ( z  =  ( a + e
b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  ->  E. a  e.  RR*  E. b  e. 
RR*  ( ( X 
C_  RR*  /\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )
95 simplll 735 . . . . . . . . . . . . . . 15  |-  ( ( ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  /\  ( a  e.  RR*  /\  b  e.  RR* ) )  ->  X  C_  RR* )
96 simprl 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  /\  ( a  e.  RR*  /\  b  e.  RR* ) )  -> 
a  e.  RR* )
97 simplr2 1000 . . . . . . . . . . . . . . 15  |-  ( ( ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  /\  ( a  e.  RR*  /\  b  e.  RR* ) )  -> 
a  <  sup ( X ,  RR* ,  <  ) )
98 supxrlub 10860 . . . . . . . . . . . . . . . 16  |-  ( ( X  C_  RR*  /\  a  e.  RR* )  ->  (
a  <  sup ( X ,  RR* ,  <  )  <->  E. v  e.  X  a  <  v ) )
9998biimpa 471 . . . . . . . . . . . . . . 15  |-  ( ( ( X  C_  RR*  /\  a  e.  RR* )  /\  a  <  sup ( X ,  RR* ,  <  ) )  ->  E. v  e.  X  a  <  v )
10095, 96, 97, 99syl21anc 1183 . . . . . . . . . . . . . 14  |-  ( ( ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  /\  ( a  e.  RR*  /\  b  e.  RR* ) )  ->  E. v  e.  X  a  <  v )
101 simpllr 736 . . . . . . . . . . . . . . 15  |-  ( ( ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  /\  ( a  e.  RR*  /\  b  e.  RR* ) )  ->  Y  C_  RR* )
102 simprr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  /\  ( a  e.  RR*  /\  b  e.  RR* ) )  -> 
b  e.  RR* )
103 simplr3 1001 . . . . . . . . . . . . . . 15  |-  ( ( ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  /\  ( a  e.  RR*  /\  b  e.  RR* ) )  -> 
b  <  sup ( Y ,  RR* ,  <  ) )
104 supxrlub 10860 . . . . . . . . . . . . . . . 16  |-  ( ( Y  C_  RR*  /\  b  e.  RR* )  ->  (
b  <  sup ( Y ,  RR* ,  <  )  <->  E. w  e.  Y  b  <  w ) )
105104biimpa 471 . . . . . . . . . . . . . . 15  |-  ( ( ( Y  C_  RR*  /\  b  e.  RR* )  /\  b  <  sup ( Y ,  RR* ,  <  ) )  ->  E. w  e.  Y  b  <  w )
106101, 102, 103, 105syl21anc 1183 . . . . . . . . . . . . . 14  |-  ( ( ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  /\  ( a  e.  RR*  /\  b  e.  RR* ) )  ->  E. w  e.  Y  b  <  w )
107 reeanv 2835 . . . . . . . . . . . . . 14  |-  ( E. v  e.  X  E. w  e.  Y  (
a  <  v  /\  b  <  w )  <->  ( E. v  e.  X  a  <  v  /\  E. w  e.  Y  b  <  w ) )
108100, 106, 107sylanbrc 646 . . . . . . . . . . . . 13  |-  ( ( ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  /\  ( a  e.  RR*  /\  b  e.  RR* ) )  ->  E. v  e.  X  E. w  e.  Y  ( a  <  v  /\  b  <  w ) )
109108ancoms 440 . . . . . . . . . . . 12  |-  ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  ->  E. v  e.  X  E. w  e.  Y  ( a  <  v  /\  b  < 
w ) )
110 simplrr 738 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  ( ( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  ( v  e.  X  /\  w  e.  Y  /\  ( a  <  v  /\  b  <  w ) ) )  ->  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )
1111103anassrs 1175 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  -> 
( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )
112111simp1d 969 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  -> 
z  =  ( a + e b ) )
113 simp-4l 743 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  -> 
( a  e.  RR*  /\  b  e.  RR* )
)
114 simplrl 737 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  ( ( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  ( v  e.  X  /\  w  e.  Y  /\  ( a  <  v  /\  b  <  w ) ) )  ->  ( X  C_  RR*  /\  Y  C_  RR* ) )
1151143anassrs 1175 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  -> 
( X  C_  RR*  /\  Y  C_ 
RR* ) )
116115simpld 446 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  ->  X  C_  RR* )
117 simpllr 736 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  -> 
v  e.  X )
118116, 117sseldd 3309 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  -> 
v  e.  RR* )
119115simprd 450 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  ->  Y  C_  RR* )
120 simplr 732 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  ->  w  e.  Y )
121119, 120sseldd 3309 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  ->  w  e.  RR* )
122113, 118, 121jca32 522 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  -> 
( ( a  e. 
RR*  /\  b  e.  RR* )  /\  ( v  e.  RR*  /\  w  e.  RR* ) ) )
123 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  -> 
( a  <  v  /\  b  <  w ) )
124122, 123jca 519 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  -> 
( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
v  e.  RR*  /\  w  e.  RR* ) )  /\  ( a  <  v  /\  b  <  w ) ) )
125 xlt2add 10795 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  ( v  e.  RR*  /\  w  e.  RR* )
)  ->  ( (
a  <  v  /\  b  <  w )  -> 
( a + e
b )  <  (
v + e w ) ) )
126125imp 419 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  ( v  e.  RR*  /\  w  e.  RR* ) )  /\  ( a  <  v  /\  b  <  w ) )  ->  ( a + e b )  < 
( v + e
w ) )
127 breq1 4175 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  ( a + e b )  -> 
( z  <  (
v + e w )  <->  ( a + e b )  < 
( v + e
w ) ) )
128127biimpar 472 . . . . . . . . . . . . . . . . 17  |-  ( ( z  =  ( a + e b )  /\  ( a + e b )  < 
( v + e
w ) )  -> 
z  <  ( v + e w ) )
129126, 128sylan2 461 . . . . . . . . . . . . . . . 16  |-  ( ( z  =  ( a + e b )  /\  ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
v  e.  RR*  /\  w  e.  RR* ) )  /\  ( a  <  v  /\  b  <  w ) ) )  ->  z  <  ( v + e
w ) )
130112, 124, 129syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  /\  ( a  <  v  /\  b  < 
w ) )  -> 
z  <  ( v + e w ) )
131130ex 424 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  /\  w  e.  Y
)  ->  ( (
a  <  v  /\  b  <  w )  -> 
z  <  ( v + e w ) ) )
132131reximdva 2778 . . . . . . . . . . . . 13  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  ( ( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  /\  v  e.  X )  ->  ( E. w  e.  Y  ( a  < 
v  /\  b  <  w )  ->  E. w  e.  Y  z  <  ( v + e w ) ) )
133132reximdva 2778 . . . . . . . . . . . 12  |-  ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  ->  ( E. v  e.  X  E. w  e.  Y  ( a  <  v  /\  b  <  w )  ->  E. v  e.  X  E. w  e.  Y  z  <  ( v + e w ) ) )
134109, 133mpd 15 . . . . . . . . . . 11  |-  ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  ( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) ) )  ->  E. v  e.  X  E. w  e.  Y  z  <  ( v + e w ) )
135134ex 424 . . . . . . . . . 10  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
( ( X  C_  RR* 
/\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  ->  E. v  e.  X  E. w  e.  Y  z  <  ( v + e w ) ) )
136135reximdva 2778 . . . . . . . . 9  |-  ( a  e.  RR*  ->  ( E. b  e.  RR*  (
( X  C_  RR*  /\  Y  C_ 
RR* )  /\  (
z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  ->  E. b  e.  RR*  E. v  e.  X  E. w  e.  Y  z  <  (
v + e w ) ) )
137136reximia 2771 . . . . . . . 8  |-  ( E. a  e.  RR*  E. b  e.  RR*  ( ( X 
C_  RR*  /\  Y  C_  RR* )  /\  ( z  =  ( a + e b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  ->  E. a  e.  RR*  E. b  e. 
RR*  E. v  e.  X  E. w  e.  Y  z  <  ( v + e w ) )
13894, 137syl 16 . . . . . . 7  |-  ( ( ( X  C_  RR*  /\  Y  C_ 
RR* )  /\  E. a  e.  RR*  E. b  e.  RR*  ( z  =  ( a + e
b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  ->  E. a  e.  RR*  E. b  e. 
RR*  E. v  e.  X  E. w  e.  Y  z  <  ( v + e w ) )
13986, 87, 88, 13819.9d2rf 23921 . . . . . 6  |-  ( ( ( X  C_  RR*  /\  Y  C_ 
RR* )  /\  E. a  e.  RR*  E. b  e.  RR*  ( z  =  ( a + e
b )  /\  a  <  sup ( X ,  RR* ,  <  )  /\  b  <  sup ( Y ,  RR* ,  <  ) ) )  ->  E. v  e.  X  E. w  e.  Y  z  <  ( v + e w ) )
14071, 72, 81, 139syl21anc 1183 . . . . 5  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  E. v  e.  X  E. w  e.  Y  z  <  ( v + e w ) )
141 simprl 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( v  e.  X  /\  w  e.  Y ) )  -> 
v  e.  X )
142 simprr 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( v  e.  X  /\  w  e.  Y ) )  ->  w  e.  Y )
14323adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( v  e.  X  /\  w  e.  Y ) )  -> 
( X  X.  Y
)  C_  dom  + e
)
144141, 142, 19, 143elovimad 24004 . . . . . . . . 9  |-  ( (
ph  /\  ( v  e.  X  /\  w  e.  Y ) )  -> 
( v + e
w )  e.  ( + e " ( X  X.  Y ) ) )
1451eleq2d 2471 . . . . . . . . . 10  |-  ( ph  ->  ( ( v + e w )  e.  Z  <->  ( v + e w )  e.  ( + e "
( X  X.  Y
) ) ) )
146145adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( v  e.  X  /\  w  e.  Y ) )  -> 
( ( v + e w )  e.  Z  <->  ( v + e w )  e.  ( + e "
( X  X.  Y
) ) ) )
147144, 146mpbird 224 . . . . . . . 8  |-  ( (
ph  /\  ( v  e.  X  /\  w  e.  Y ) )  -> 
( v + e
w )  e.  Z
)
148 simpr 448 . . . . . . . . 9  |-  ( ( ( ph  /\  (
v  e.  X  /\  w  e.  Y )
)  /\  k  =  ( v + e
w ) )  -> 
k  =  ( v + e w ) )
149148breq2d 4184 . . . . . . . 8  |-  ( ( ( ph  /\  (
v  e.  X  /\  w  e.  Y )
)  /\  k  =  ( v + e
w ) )  -> 
( z  <  k  <->  z  <  ( v + e w ) ) )
150147, 149rspcedv 3016 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  X  /\  w  e.  Y ) )  -> 
( z  <  (
v + e w )  ->  E. k  e.  Z  z  <  k ) )
151150rexlimdvva 2797 . . . . . 6  |-  ( ph  ->  ( E. v  e.  X  E. w  e.  Y  z  <  (
v + e w )  ->  E. k  e.  Z  z  <  k ) )
152151ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  ( E. v  e.  X  E. w  e.  Y  z  <  ( v + e w )  ->  E. k  e.  Z  z  <  k ) )
153140, 152mpd 15 . . . 4  |-  ( ( ( ph  /\  z  e.  RR )  /\  z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )  ->  E. k  e.  Z  z  <  k )
154153ex 424 . . 3  |-  ( (
ph  /\  z  e.  RR )  ->  ( z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) )  ->  E. k  e.  Z  z  <  k ) )
155154ralrimiva 2749 . 2  |-  ( ph  ->  A. z  e.  RR  ( z  <  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) )  ->  E. k  e.  Z  z  <  k ) )
156 supxr2 10848 . 2  |-  ( ( ( Z  C_  RR*  /\  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) )  e. 
RR* )  /\  ( A. z  e.  Z  z  <_  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  ) )  /\  A. z  e.  RR  ( z  < 
( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
)  ->  E. k  e.  Z  z  <  k ) ) )  ->  sup ( Z ,  RR* ,  <  )  =  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )
15727, 32, 70, 155, 156syl22anc 1185 1  |-  ( ph  ->  sup ( Z ,  RR* ,  <  )  =  ( sup ( X ,  RR* ,  <  ) + e sup ( Y ,  RR* ,  <  )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    C_ wss 3280   <.cop 3777   class class class wbr 4172    X. cxp 4835   dom cdm 4837   "cima 4840   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040   supcsup 7403   RRcr 8945    -oocmnf 9074   RR*cxr 9075    < clt 9076    <_ cle 9077   + ecxad 10664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-rp 10569  df-xneg 10666  df-xadd 10667
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