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Theorem supmul 10601
Description: The supremum function distributes over multiplication, in the sense that  ( sup A
)  x.  ( sup B )  =  sup ( A  x.  B
), where  A  x.  B is shorthand for  { a  x.  b  |  a  e.  A ,  b  e.  B } and is defined as  C below. We made use of this in our definition of multiplication in the Dedekind cut construction of the reals (see df-mp 9427). (Contributed by Mario Carneiro, 5-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2014.)
Hypotheses
Ref Expression
supmul.1  |-  C  =  { z  |  E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b ) }
supmul.2  |-  ( ph  <->  ( ( A. x  e.  A  0  <_  x  /\  A. x  e.  B 
0  <_  x )  /\  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  ( B 
C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) ) )
Assertion
Ref Expression
supmul  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
Distinct variable groups:    A, b,
v, x, y, z    B, b, v, x, y, z    x, C    ph, b,
z
Allowed substitution hints:    ph( x, y, v)    C( y, z, v, b)

Proof of Theorem supmul
Dummy variables  a  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supmul.2 . . . . . . 7  |-  ( ph  <->  ( ( A. x  e.  A  0  <_  x  /\  A. x  e.  B 
0  <_  x )  /\  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  ( B 
C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) ) )
21simp2bi 1046 . . . . . 6  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
3 suprcl 10591 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
42, 3syl 17 . . . . 5  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
51simp3bi 1047 . . . . . 6  |-  ( ph  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) )
6 suprcl 10591 . . . . . 6  |-  ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
)  ->  sup ( B ,  RR ,  <  )  e.  RR )
75, 6syl 17 . . . . 5  |-  ( ph  ->  sup ( B ,  RR ,  <  )  e.  RR )
8 recn 9647 . . . . . 6  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  sup ( A ,  RR ,  <  )  e.  CC )
9 recn 9647 . . . . . 6  |-  ( sup ( B ,  RR ,  <  )  e.  RR  ->  sup ( B ,  RR ,  <  )  e.  CC )
10 mulcom 9643 . . . . . 6  |-  ( ( sup ( A ,  RR ,  <  )  e.  CC  /\  sup ( B ,  RR ,  <  )  e.  CC )  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  ( sup ( B ,  RR ,  <  )  x. 
sup ( A ,  RR ,  <  ) ) )
118, 9, 10syl2an 485 . . . . 5  |-  ( ( sup ( A ,  RR ,  <  )  e.  RR  /\  sup ( B ,  RR ,  <  )  e.  RR )  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  ( sup ( B ,  RR ,  <  )  x. 
sup ( A ,  RR ,  <  ) ) )
124, 7, 11syl2anc 673 . . . 4  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  ( sup ( B ,  RR ,  <  )  x.  sup ( A ,  RR ,  <  ) ) )
135simp2d 1043 . . . . . . 7  |-  ( ph  ->  B  =/=  (/) )
14 n0 3732 . . . . . . 7  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
1513, 14sylib 201 . . . . . 6  |-  ( ph  ->  E. b  b  e.  B )
16 0red 9662 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  0  e.  RR )
175simp1d 1042 . . . . . . . 8  |-  ( ph  ->  B  C_  RR )
1817sselda 3418 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  b  e.  RR )
197adantr 472 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  sup ( B ,  RR ,  <  )  e.  RR )
20 simp1r 1055 . . . . . . . . . 10  |-  ( ( ( A. x  e.  A  0  <_  x  /\  A. x  e.  B 
0  <_  x )  /\  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  ( B 
C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) )  ->  A. x  e.  B  0  <_  x )
211, 20sylbi 200 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  B 
0  <_  x )
22 breq2 4399 . . . . . . . . . 10  |-  ( x  =  b  ->  (
0  <_  x  <->  0  <_  b ) )
2322rspccv 3133 . . . . . . . . 9  |-  ( A. x  e.  B  0  <_  x  ->  ( b  e.  B  ->  0  <_ 
b ) )
2421, 23syl 17 . . . . . . . 8  |-  ( ph  ->  ( b  e.  B  ->  0  <_  b )
)
2524imp 436 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  b )
26 suprub 10592 . . . . . . . 8  |-  ( ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x )  /\  b  e.  B )  ->  b  <_  sup ( B ,  RR ,  <  ) )
275, 26sylan 479 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  b  <_  sup ( B ,  RR ,  <  ) )
2816, 18, 19, 25, 27letrd 9809 . . . . . 6  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  sup ( B ,  RR ,  <  ) )
2915, 28exlimddv 1789 . . . . 5  |-  ( ph  ->  0  <_  sup ( B ,  RR ,  <  ) )
30 simp1l 1054 . . . . . 6  |-  ( ( ( A. x  e.  A  0  <_  x  /\  A. x  e.  B 
0  <_  x )  /\  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  ( B 
C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) )  ->  A. x  e.  A  0  <_  x )
311, 30sylbi 200 . . . . 5  |-  ( ph  ->  A. x  e.  A 
0  <_  x )
32 eqid 2471 . . . . . 6  |-  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  =  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }
33 biid 244 . . . . . 6  |-  ( ( ( sup ( B ,  RR ,  <  )  e.  RR  /\  0  <_  sup ( B ,  RR ,  <  )  /\  A. x  e.  A  0  <_  x )  /\  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
) )  <->  ( ( sup ( B ,  RR ,  <  )  e.  RR  /\  0  <_  sup ( B ,  RR ,  <  )  /\  A. x  e.  A  0  <_  x )  /\  ( A 
C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
) ) )
3432, 33supmul1 10598 . . . . 5  |-  ( ( ( sup ( B ,  RR ,  <  )  e.  RR  /\  0  <_  sup ( B ,  RR ,  <  )  /\  A. x  e.  A  0  <_  x )  /\  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
) )  ->  ( sup ( B ,  RR ,  <  )  x.  sup ( A ,  RR ,  <  ) )  =  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  ) )
357, 29, 31, 2, 34syl31anc 1295 . . . 4  |-  ( ph  ->  ( sup ( B ,  RR ,  <  )  x.  sup ( A ,  RR ,  <  ) )  =  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  ) )
3612, 35eqtrd 2505 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  ) )
37 vex 3034 . . . . . . 7  |-  w  e. 
_V
38 eqeq1 2475 . . . . . . . 8  |-  ( z  =  w  ->  (
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)  <->  w  =  ( sup ( B ,  RR ,  <  )  x.  a
) ) )
3938rexbidv 2892 . . . . . . 7  |-  ( z  =  w  ->  ( E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a )  <->  E. a  e.  A  w  =  ( sup ( B ,  RR ,  <  )  x.  a ) ) )
4037, 39elab 3173 . . . . . 6  |-  ( w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  <->  E. a  e.  A  w  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
417adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  sup ( B ,  RR ,  <  )  e.  RR )
422simp1d 1042 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  RR )
4342sselda 3418 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  a  e.  RR )
44 recn 9647 . . . . . . . . . . 11  |-  ( a  e.  RR  ->  a  e.  CC )
45 mulcom 9643 . . . . . . . . . . 11  |-  ( ( sup ( B ,  RR ,  <  )  e.  CC  /\  a  e.  CC )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  =  ( a  x.  sup ( B ,  RR ,  <  ) ) )
469, 44, 45syl2an 485 . . . . . . . . . 10  |-  ( ( sup ( B ,  RR ,  <  )  e.  RR  /\  a  e.  RR )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  =  ( a  x.  sup ( B ,  RR ,  <  ) ) )
4741, 43, 46syl2anc 673 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  =  ( a  x.  sup ( B ,  RR ,  <  ) ) )
48 breq2 4399 . . . . . . . . . . . . . 14  |-  ( x  =  a  ->  (
0  <_  x  <->  0  <_  a ) )
4948rspccv 3133 . . . . . . . . . . . . 13  |-  ( A. x  e.  A  0  <_  x  ->  ( a  e.  A  ->  0  <_ 
a ) )
5031, 49syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( a  e.  A  ->  0  <_  a )
)
5150imp 436 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  0  <_  a )
5221adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  A. x  e.  B  0  <_  x )
535adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) )
54 eqid 2471 . . . . . . . . . . . 12  |-  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  =  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }
55 biid 244 . . . . . . . . . . . 12  |-  ( ( ( a  e.  RR  /\  0  <_  a  /\  A. x  e.  B  0  <_  x )  /\  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) )  <->  ( (
a  e.  RR  /\  0  <_  a  /\  A. x  e.  B  0  <_  x )  /\  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) ) )
5654, 55supmul1 10598 . . . . . . . . . . 11  |-  ( ( ( a  e.  RR  /\  0  <_  a  /\  A. x  e.  B  0  <_  x )  /\  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) )  ->  (
a  x.  sup ( B ,  RR ,  <  ) )  =  sup ( { z  |  E. b  e.  B  z  =  ( a  x.  b ) } ,  RR ,  <  ) )
5743, 51, 52, 53, 56syl31anc 1295 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  (
a  x.  sup ( B ,  RR ,  <  ) )  =  sup ( { z  |  E. b  e.  B  z  =  ( a  x.  b ) } ,  RR ,  <  ) )
58 eqeq1 2475 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
z  =  ( a  x.  b )  <->  w  =  ( a  x.  b
) ) )
5958rexbidv 2892 . . . . . . . . . . . . . 14  |-  ( z  =  w  ->  ( E. b  e.  B  z  =  ( a  x.  b )  <->  E. b  e.  B  w  =  ( a  x.  b
) ) )
6037, 59elab 3173 . . . . . . . . . . . . 13  |-  ( w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  <->  E. b  e.  B  w  =  ( a  x.  b
) )
61 rspe 2844 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  A  /\  E. b  e.  B  w  =  ( a  x.  b ) )  ->  E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b ) )
62 oveq1 6315 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  =  a  ->  (
v  x.  b )  =  ( a  x.  b ) )
6362eqeq2d 2481 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  a  ->  (
z  =  ( v  x.  b )  <->  z  =  ( a  x.  b
) ) )
6463rexbidv 2892 . . . . . . . . . . . . . . . . . . 19  |-  ( v  =  a  ->  ( E. b  e.  B  z  =  ( v  x.  b )  <->  E. b  e.  B  z  =  ( a  x.  b
) ) )
6564cbvrexv 3006 . . . . . . . . . . . . . . . . . 18  |-  ( E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b )  <->  E. a  e.  A  E. b  e.  B  z  =  ( a  x.  b
) )
66582rexbidv 2897 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  w  ->  ( E. a  e.  A  E. b  e.  B  z  =  ( a  x.  b )  <->  E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) ) )
6765, 66syl5bb 265 . . . . . . . . . . . . . . . . 17  |-  ( z  =  w  ->  ( E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b )  <->  E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) ) )
68 supmul.1 . . . . . . . . . . . . . . . . 17  |-  C  =  { z  |  E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b ) }
6937, 67, 68elab2 3176 . . . . . . . . . . . . . . . 16  |-  ( w  e.  C  <->  E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) )
7061, 69sylibr 217 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  A  /\  E. b  e.  B  w  =  ( a  x.  b ) )  ->  w  e.  C )
7170ex 441 . . . . . . . . . . . . . 14  |-  ( a  e.  A  ->  ( E. b  e.  B  w  =  ( a  x.  b )  ->  w  e.  C ) )
7268, 1supmullem2 10600 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
73 suprub 10592 . . . . . . . . . . . . . . . 16  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
7473ex 441 . . . . . . . . . . . . . . 15  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  ( w  e.  C  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
7572, 74syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( w  e.  C  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
7671, 75sylan9r 670 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  A )  ->  ( E. b  e.  B  w  =  ( a  x.  b )  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
7760, 76syl5bi 225 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  (
w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
7877ralrimiv 2808 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  A. w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  sup ( C ,  RR ,  <  ) )
7943adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  a  e.  RR )
8018adantlr 729 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  b  e.  RR )
8179, 80remulcld 9689 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  (
a  x.  b )  e.  RR )
82 eleq1a 2544 . . . . . . . . . . . . . . 15  |-  ( ( a  x.  b )  e.  RR  ->  (
z  =  ( a  x.  b )  -> 
z  e.  RR ) )
8381, 82syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  (
z  =  ( a  x.  b )  -> 
z  e.  RR ) )
8483rexlimdva 2871 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  A )  ->  ( E. b  e.  B  z  =  ( a  x.  b )  ->  z  e.  RR ) )
8584abssdv 3489 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  C_  RR )
86 ovex 6336 . . . . . . . . . . . . . . . . . . 19  |-  ( a  x.  b )  e. 
_V
8786isseti 3037 . . . . . . . . . . . . . . . . . 18  |-  E. w  w  =  ( a  x.  b )
8887rgenw 2768 . . . . . . . . . . . . . . . . 17  |-  A. b  e.  B  E. w  w  =  ( a  x.  b )
89 r19.2z 3849 . . . . . . . . . . . . . . . . 17  |-  ( ( B  =/=  (/)  /\  A. b  e.  B  E. w  w  =  (
a  x.  b ) )  ->  E. b  e.  B  E. w  w  =  ( a  x.  b ) )
9013, 88, 89sylancl 675 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. b  e.  B  E. w  w  =  ( a  x.  b
) )
91 rexcom4 3053 . . . . . . . . . . . . . . . 16  |-  ( E. b  e.  B  E. w  w  =  (
a  x.  b )  <->  E. w E. b  e.  B  w  =  ( a  x.  b ) )
9290, 91sylib 201 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E. w E. b  e.  B  w  =  ( a  x.  b
) )
9359cbvexv 2130 . . . . . . . . . . . . . . 15  |-  ( E. z E. b  e.  B  z  =  ( a  x.  b )  <->  E. w E. b  e.  B  w  =  ( a  x.  b ) )
9492, 93sylibr 217 . . . . . . . . . . . . . 14  |-  ( ph  ->  E. z E. b  e.  B  z  =  ( a  x.  b
) )
95 abn0 3754 . . . . . . . . . . . . . 14  |-  ( { z  |  E. b  e.  B  z  =  ( a  x.  b
) }  =/=  (/)  <->  E. z E. b  e.  B  z  =  ( a  x.  b ) )
9694, 95sylibr 217 . . . . . . . . . . . . 13  |-  ( ph  ->  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  =/=  (/) )
9796adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  =/=  (/) )
98 suprcl 10591 . . . . . . . . . . . . . . 15  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  sup ( C ,  RR ,  <  )  e.  RR )
9972, 98syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  sup ( C ,  RR ,  <  )  e.  RR )
10099adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  A )  ->  sup ( C ,  RR ,  <  )  e.  RR )
101 breq2 4399 . . . . . . . . . . . . . . 15  |-  ( x  =  sup ( C ,  RR ,  <  )  ->  ( w  <_  x 
<->  w  <_  sup ( C ,  RR ,  <  ) ) )
102101ralbidv 2829 . . . . . . . . . . . . . 14  |-  ( x  =  sup ( C ,  RR ,  <  )  ->  ( A. w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  x  <->  A. w  e.  {
z  |  E. b  e.  B  z  =  ( a  x.  b
) } w  <_  sup ( C ,  RR ,  <  ) ) )
103102rspcev 3136 . . . . . . . . . . . . 13  |-  ( ( sup ( C ,  RR ,  <  )  e.  RR  /\  A. w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  sup ( C ,  RR ,  <  ) )  ->  E. x  e.  RR  A. w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  x
)
104100, 78, 103syl2anc 673 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  E. x  e.  RR  A. w  e. 
{ z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  x )
105 suprleub 10595 . . . . . . . . . . . 12  |-  ( ( ( { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  C_  RR  /\  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  =/=  (/) 
/\  E. x  e.  RR  A. w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  x
)  /\  sup ( C ,  RR ,  <  )  e.  RR )  ->  ( sup ( { z  |  E. b  e.  B  z  =  ( a  x.  b ) } ,  RR ,  <  )  <_  sup ( C ,  RR ,  <  )  <->  A. w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  sup ( C ,  RR ,  <  ) ) )
10685, 97, 104, 100, 105syl31anc 1295 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( { z  |  E. b  e.  B  z  =  ( a  x.  b ) } ,  RR ,  <  )  <_  sup ( C ,  RR ,  <  )  <->  A. w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  sup ( C ,  RR ,  <  ) ) )
10778, 106mpbird 240 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  sup ( { z  |  E. b  e.  B  z  =  ( a  x.  b ) } ,  RR ,  <  )  <_  sup ( C ,  RR ,  <  ) )
10857, 107eqbrtrd 4416 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  (
a  x.  sup ( B ,  RR ,  <  ) )  <_  sup ( C ,  RR ,  <  ) )
10947, 108eqbrtrd 4416 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  <_  sup ( C ,  RR ,  <  ) )
110 breq1 4398 . . . . . . . 8  |-  ( w  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  ( w  <_  sup ( C ,  RR ,  <  )  <->  ( sup ( B ,  RR ,  <  )  x.  a )  <_  sup ( C ,  RR ,  <  ) ) )
111109, 110syl5ibrcom 230 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( sup ( B ,  RR ,  <  )  x.  a
)  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
112111rexlimdva 2871 . . . . . 6  |-  ( ph  ->  ( E. a  e.  A  w  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
11340, 112syl5bi 225 . . . . 5  |-  ( ph  ->  ( w  e.  {
z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
114113ralrimiv 2808 . . . 4  |-  ( ph  ->  A. w  e.  {
z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  sup ( C ,  RR ,  <  ) )
11541, 43remulcld 9689 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  e.  RR )
116 eleq1a 2544 . . . . . . . 8  |-  ( ( sup ( B ,  RR ,  <  )  x.  a )  e.  RR  ->  ( z  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  z  e.  RR ) )
117115, 116syl 17 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  (
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)  ->  z  e.  RR ) )
118117rexlimdva 2871 . . . . . 6  |-  ( ph  ->  ( E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  z  e.  RR ) )
119118abssdv 3489 . . . . 5  |-  ( ph  ->  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  C_  RR )
1202simp2d 1043 . . . . . . . 8  |-  ( ph  ->  A  =/=  (/) )
121 ovex 6336 . . . . . . . . . 10  |-  ( sup ( B ,  RR ,  <  )  x.  a
)  e.  _V
122121isseti 3037 . . . . . . . . 9  |-  E. z 
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)
123122rgenw 2768 . . . . . . . 8  |-  A. a  e.  A  E. z 
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)
124 r19.2z 3849 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  A. a  e.  A  E. z  z  =  ( sup ( B ,  RR ,  <  )  x.  a
) )  ->  E. a  e.  A  E. z 
z  =  ( sup ( B ,  RR ,  <  )  x.  a
) )
125120, 123, 124sylancl 675 . . . . . . 7  |-  ( ph  ->  E. a  e.  A  E. z  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
126 rexcom4 3053 . . . . . . 7  |-  ( E. a  e.  A  E. z  z  =  ( sup ( B ,  RR ,  <  )  x.  a
)  <->  E. z E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
127125, 126sylib 201 . . . . . 6  |-  ( ph  ->  E. z E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
128 abn0 3754 . . . . . 6  |-  ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  =/=  (/)  <->  E. z E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
129127, 128sylibr 217 . . . . 5  |-  ( ph  ->  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  =/=  (/) )
130101ralbidv 2829 . . . . . . 7  |-  ( x  =  sup ( C ,  RR ,  <  )  ->  ( A. w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  x  <->  A. w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  sup ( C ,  RR ,  <  ) ) )
131130rspcev 3136 . . . . . 6  |-  ( ( sup ( C ,  RR ,  <  )  e.  RR  /\  A. w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  sup ( C ,  RR ,  <  ) )  ->  E. x  e.  RR  A. w  e. 
{ z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  x
)
13299, 114, 131syl2anc 673 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  x )
133 suprleub 10595 . . . . 5  |-  ( ( ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  C_  RR  /\  {
z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  =/=  (/) 
/\  E. x  e.  RR  A. w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  x )  /\  sup ( C ,  RR ,  <  )  e.  RR )  ->  ( sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  )  <_  sup ( C ,  RR ,  <  )  <->  A. w  e.  {
z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  sup ( C ,  RR ,  <  ) ) )
134119, 129, 132, 99, 133syl31anc 1295 . . . 4  |-  ( ph  ->  ( sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  )  <_  sup ( C ,  RR ,  <  )  <->  A. w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  sup ( C ,  RR ,  <  ) ) )
135114, 134mpbird 240 . . 3  |-  ( ph  ->  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  )  <_  sup ( C ,  RR ,  <  ) )
13636, 135eqbrtrd 4416 . 2  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  <_  sup ( C ,  RR ,  <  ) )
13768, 1supmullem1 10599 . . 3  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
1384, 7remulcld 9689 . . . 4  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
139 suprleub 10595 . . . 4  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR )  ->  ( sup ( C ,  RR ,  <  )  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x. 
sup ( B ,  RR ,  <  ) ) ) )
14072, 138, 139syl2anc 673 . . 3  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x. 
sup ( B ,  RR ,  <  ) ) ) )
141137, 140mpbird 240 . 2  |-  ( ph  ->  sup ( C ,  RR ,  <  )  <_ 
( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
142138, 99letri3d 9794 . 2  |-  ( ph  ->  ( ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  )  <->  ( ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  <_  sup ( C ,  RR ,  <  )  /\  sup ( C ,  RR ,  <  )  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) ) ) )
143136, 141, 142mpbir2and 936 1  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756   E.wrex 2757    C_ wss 3390   (/)c0 3722   class class class wbr 4395  (class class class)co 6308   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557    x. cmul 9562    < clt 9693    <_ cle 9694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292
This theorem is referenced by:  sqrlem5  13387
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