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Theorem supmul 9932
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 8817). (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 973 . . . . . 6  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
3 suprcl 9924 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
42, 3syl 16 . . . . 5  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
51simp3bi 974 . . . . . 6  |-  ( ph  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) )
6 suprcl 9924 . . . . . 6  |-  ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
)  ->  sup ( B ,  RR ,  <  )  e.  RR )
75, 6syl 16 . . . . 5  |-  ( ph  ->  sup ( B ,  RR ,  <  )  e.  RR )
8 recn 9036 . . . . . 6  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  sup ( A ,  RR ,  <  )  e.  CC )
9 recn 9036 . . . . . 6  |-  ( sup ( B ,  RR ,  <  )  e.  RR  ->  sup ( B ,  RR ,  <  )  e.  CC )
10 mulcom 9032 . . . . . 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 464 . . . . 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 643 . . . 4  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  ( sup ( B ,  RR ,  <  )  x.  sup ( A ,  RR ,  <  ) ) )
135simp2d 970 . . . . . . 7  |-  ( ph  ->  B  =/=  (/) )
14 n0 3597 . . . . . . 7  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
1513, 14sylib 189 . . . . . 6  |-  ( ph  ->  E. b  b  e.  B )
16 0re 9047 . . . . . . . 8  |-  0  e.  RR
1716a1i 11 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  0  e.  RR )
185simp1d 969 . . . . . . . 8  |-  ( ph  ->  B  C_  RR )
1918sselda 3308 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  b  e.  RR )
207adantr 452 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  sup ( B ,  RR ,  <  )  e.  RR )
21 simp1r 982 . . . . . . . . . 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 )
221, 21sylbi 188 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  B 
0  <_  x )
23 breq2 4176 . . . . . . . . . 10  |-  ( x  =  b  ->  (
0  <_  x  <->  0  <_  b ) )
2423rspccv 3009 . . . . . . . . 9  |-  ( A. x  e.  B  0  <_  x  ->  ( b  e.  B  ->  0  <_ 
b ) )
2522, 24syl 16 . . . . . . . 8  |-  ( ph  ->  ( b  e.  B  ->  0  <_  b )
)
2625imp 419 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  b )
27 suprub 9925 . . . . . . . 8  |-  ( ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x )  /\  b  e.  B )  ->  b  <_  sup ( B ,  RR ,  <  ) )
285, 27sylan 458 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  b  <_  sup ( B ,  RR ,  <  ) )
2917, 19, 20, 26, 28letrd 9183 . . . . . 6  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  sup ( B ,  RR ,  <  ) )
3015, 29exlimddv 1645 . . . . 5  |-  ( ph  ->  0  <_  sup ( B ,  RR ,  <  ) )
31 simp1l 981 . . . . . 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 )
321, 31sylbi 188 . . . . 5  |-  ( ph  ->  A. x  e.  A 
0  <_  x )
33 eqid 2404 . . . . . 6  |-  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  =  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }
34 biid 228 . . . . . 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
) ) )
3533, 34supmul1 9929 . . . . 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 ,  <  ) )
367, 30, 32, 2, 35syl31anc 1187 . . . 4  |-  ( ph  ->  ( sup ( B ,  RR ,  <  )  x.  sup ( A ,  RR ,  <  ) )  =  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  ) )
3712, 36eqtrd 2436 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  ) )
38 vex 2919 . . . . . . 7  |-  w  e. 
_V
39 eqeq1 2410 . . . . . . . 8  |-  ( z  =  w  ->  (
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)  <->  w  =  ( sup ( B ,  RR ,  <  )  x.  a
) ) )
4039rexbidv 2687 . . . . . . 7  |-  ( z  =  w  ->  ( E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a )  <->  E. a  e.  A  w  =  ( sup ( B ,  RR ,  <  )  x.  a ) ) )
4138, 40elab 3042 . . . . . 6  |-  ( w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  <->  E. a  e.  A  w  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
427adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  sup ( B ,  RR ,  <  )  e.  RR )
432simp1d 969 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  RR )
4443sselda 3308 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  a  e.  RR )
45 recn 9036 . . . . . . . . . . 11  |-  ( a  e.  RR  ->  a  e.  CC )
46 mulcom 9032 . . . . . . . . . . 11  |-  ( ( sup ( B ,  RR ,  <  )  e.  CC  /\  a  e.  CC )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  =  ( a  x.  sup ( B ,  RR ,  <  ) ) )
479, 45, 46syl2an 464 . . . . . . . . . 10  |-  ( ( sup ( B ,  RR ,  <  )  e.  RR  /\  a  e.  RR )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  =  ( a  x.  sup ( B ,  RR ,  <  ) ) )
4842, 44, 47syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  =  ( a  x.  sup ( B ,  RR ,  <  ) ) )
49 breq2 4176 . . . . . . . . . . . . . 14  |-  ( x  =  a  ->  (
0  <_  x  <->  0  <_  a ) )
5049rspccv 3009 . . . . . . . . . . . . 13  |-  ( A. x  e.  A  0  <_  x  ->  ( a  e.  A  ->  0  <_ 
a ) )
5132, 50syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( a  e.  A  ->  0  <_  a )
)
5251imp 419 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  0  <_  a )
5322adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  A. x  e.  B  0  <_  x )
545adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) )
55 eqid 2404 . . . . . . . . . . . 12  |-  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  =  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }
56 biid 228 . . . . . . . . . . . 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
) ) )
5755, 56supmul1 9929 . . . . . . . . . . 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 ,  <  ) )
5844, 52, 53, 54, 57syl31anc 1187 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  (
a  x.  sup ( B ,  RR ,  <  ) )  =  sup ( { z  |  E. b  e.  B  z  =  ( a  x.  b ) } ,  RR ,  <  ) )
59 eqeq1 2410 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
z  =  ( a  x.  b )  <->  w  =  ( a  x.  b
) ) )
6059rexbidv 2687 . . . . . . . . . . . . . 14  |-  ( z  =  w  ->  ( E. b  e.  B  z  =  ( a  x.  b )  <->  E. b  e.  B  w  =  ( a  x.  b
) ) )
6138, 60elab 3042 . . . . . . . . . . . . 13  |-  ( w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  <->  E. b  e.  B  w  =  ( a  x.  b
) )
62 rspe 2727 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  A  /\  E. b  e.  B  w  =  ( a  x.  b ) )  ->  E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b ) )
63 oveq1 6047 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  =  a  ->  (
v  x.  b )  =  ( a  x.  b ) )
6463eqeq2d 2415 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  a  ->  (
z  =  ( v  x.  b )  <->  z  =  ( a  x.  b
) ) )
6564rexbidv 2687 . . . . . . . . . . . . . . . . . . 19  |-  ( v  =  a  ->  ( E. b  e.  B  z  =  ( v  x.  b )  <->  E. b  e.  B  z  =  ( a  x.  b
) ) )
6665cbvrexv 2893 . . . . . . . . . . . . . . . . . 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
) )
67592rexbidv 2709 . . . . . . . . . . . . . . . . . 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
) ) )
6866, 67syl5bb 249 . . . . . . . . . . . . . . . . 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
) ) )
69 supmul.1 . . . . . . . . . . . . . . . . 17  |-  C  =  { z  |  E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b ) }
7038, 68, 69elab2 3045 . . . . . . . . . . . . . . . 16  |-  ( w  e.  C  <->  E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) )
7162, 70sylibr 204 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  A  /\  E. b  e.  B  w  =  ( a  x.  b ) )  ->  w  e.  C )
7271ex 424 . . . . . . . . . . . . . 14  |-  ( a  e.  A  ->  ( E. b  e.  B  w  =  ( a  x.  b )  ->  w  e.  C ) )
7369, 1supmullem2 9931 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
74 suprub 9925 . . . . . . . . . . . . . . . 16  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
7574ex 424 . . . . . . . . . . . . . . 15  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  ( w  e.  C  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
7673, 75syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( w  e.  C  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
7772, 76sylan9r 640 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  A )  ->  ( E. b  e.  B  w  =  ( a  x.  b )  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
7861, 77syl5bi 209 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  (
w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
7978ralrimiv 2748 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  A. w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  sup ( C ,  RR ,  <  ) )
8044adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  a  e.  RR )
8119adantlr 696 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  b  e.  RR )
8280, 81remulcld 9072 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  (
a  x.  b )  e.  RR )
83 eleq1a 2473 . . . . . . . . . . . . . . 15  |-  ( ( a  x.  b )  e.  RR  ->  (
z  =  ( a  x.  b )  -> 
z  e.  RR ) )
8482, 83syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  (
z  =  ( a  x.  b )  -> 
z  e.  RR ) )
8584rexlimdva 2790 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  A )  ->  ( E. b  e.  B  z  =  ( a  x.  b )  ->  z  e.  RR ) )
8685abssdv 3377 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  C_  RR )
87 ovex 6065 . . . . . . . . . . . . . . . . . . 19  |-  ( a  x.  b )  e. 
_V
8887isseti 2922 . . . . . . . . . . . . . . . . . 18  |-  E. w  w  =  ( a  x.  b )
8988rgenw 2733 . . . . . . . . . . . . . . . . 17  |-  A. b  e.  B  E. w  w  =  ( a  x.  b )
90 r19.2z 3677 . . . . . . . . . . . . . . . . 17  |-  ( ( B  =/=  (/)  /\  A. b  e.  B  E. w  w  =  (
a  x.  b ) )  ->  E. b  e.  B  E. w  w  =  ( a  x.  b ) )
9113, 89, 90sylancl 644 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. b  e.  B  E. w  w  =  ( a  x.  b
) )
92 rexcom4 2935 . . . . . . . . . . . . . . . 16  |-  ( E. b  e.  B  E. w  w  =  (
a  x.  b )  <->  E. w E. b  e.  B  w  =  ( a  x.  b ) )
9391, 92sylib 189 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E. w E. b  e.  B  w  =  ( a  x.  b
) )
9460cbvexv 2053 . . . . . . . . . . . . . . 15  |-  ( E. z E. b  e.  B  z  =  ( a  x.  b )  <->  E. w E. b  e.  B  w  =  ( a  x.  b ) )
9593, 94sylibr 204 . . . . . . . . . . . . . 14  |-  ( ph  ->  E. z E. b  e.  B  z  =  ( a  x.  b
) )
96 abn0 3606 . . . . . . . . . . . . . 14  |-  ( { z  |  E. b  e.  B  z  =  ( a  x.  b
) }  =/=  (/)  <->  E. z E. b  e.  B  z  =  ( a  x.  b ) )
9795, 96sylibr 204 . . . . . . . . . . . . 13  |-  ( ph  ->  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  =/=  (/) )
9897adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  =/=  (/) )
99 suprcl 9924 . . . . . . . . . . . . . . 15  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  sup ( C ,  RR ,  <  )  e.  RR )
10073, 99syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  sup ( C ,  RR ,  <  )  e.  RR )
101100adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  A )  ->  sup ( C ,  RR ,  <  )  e.  RR )
102 breq2 4176 . . . . . . . . . . . . . . 15  |-  ( x  =  sup ( C ,  RR ,  <  )  ->  ( w  <_  x 
<->  w  <_  sup ( C ,  RR ,  <  ) ) )
103102ralbidv 2686 . . . . . . . . . . . . . 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 ,  <  ) ) )
104103rspcev 3012 . . . . . . . . . . . . 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
)
105101, 79, 104syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  E. x  e.  RR  A. w  e. 
{ z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  x )
106 suprleub 9928 . . . . . . . . . . . 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 ,  <  ) ) )
10786, 98, 105, 101, 106syl31anc 1187 . . . . . . . . . . 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 ,  <  ) ) )
10879, 107mpbird 224 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  sup ( { z  |  E. b  e.  B  z  =  ( a  x.  b ) } ,  RR ,  <  )  <_  sup ( C ,  RR ,  <  ) )
10958, 108eqbrtrd 4192 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  (
a  x.  sup ( B ,  RR ,  <  ) )  <_  sup ( C ,  RR ,  <  ) )
11048, 109eqbrtrd 4192 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  <_  sup ( C ,  RR ,  <  ) )
111 breq1 4175 . . . . . . . 8  |-  ( w  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  ( w  <_  sup ( C ,  RR ,  <  )  <->  ( sup ( B ,  RR ,  <  )  x.  a )  <_  sup ( C ,  RR ,  <  ) ) )
112110, 111syl5ibrcom 214 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( sup ( B ,  RR ,  <  )  x.  a
)  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
113112rexlimdva 2790 . . . . . 6  |-  ( ph  ->  ( E. a  e.  A  w  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
11441, 113syl5bi 209 . . . . 5  |-  ( ph  ->  ( w  e.  {
z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
115114ralrimiv 2748 . . . 4  |-  ( ph  ->  A. w  e.  {
z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  sup ( C ,  RR ,  <  ) )
11642, 44remulcld 9072 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  e.  RR )
117 eleq1a 2473 . . . . . . . 8  |-  ( ( sup ( B ,  RR ,  <  )  x.  a )  e.  RR  ->  ( z  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  z  e.  RR ) )
118116, 117syl 16 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  (
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)  ->  z  e.  RR ) )
119118rexlimdva 2790 . . . . . 6  |-  ( ph  ->  ( E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  z  e.  RR ) )
120119abssdv 3377 . . . . 5  |-  ( ph  ->  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  C_  RR )
1212simp2d 970 . . . . . . . 8  |-  ( ph  ->  A  =/=  (/) )
122 ovex 6065 . . . . . . . . . 10  |-  ( sup ( B ,  RR ,  <  )  x.  a
)  e.  _V
123122isseti 2922 . . . . . . . . 9  |-  E. z 
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)
124123rgenw 2733 . . . . . . . 8  |-  A. a  e.  A  E. z 
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)
125 r19.2z 3677 . . . . . . . 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
) )
126121, 124, 125sylancl 644 . . . . . . 7  |-  ( ph  ->  E. a  e.  A  E. z  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
127 rexcom4 2935 . . . . . . 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 ) )
128126, 127sylib 189 . . . . . 6  |-  ( ph  ->  E. z E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
129 abn0 3606 . . . . . 6  |-  ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  =/=  (/)  <->  E. z E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
130128, 129sylibr 204 . . . . 5  |-  ( ph  ->  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  =/=  (/) )
131102ralbidv 2686 . . . . . . 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 ,  <  ) ) )
132131rspcev 3012 . . . . . 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
)
133100, 115, 132syl2anc 643 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  x )
134 suprleub 9928 . . . . 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 ,  <  ) ) )
135120, 130, 133, 100, 134syl31anc 1187 . . . 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 ,  <  ) ) )
136115, 135mpbird 224 . . 3  |-  ( ph  ->  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  )  <_  sup ( C ,  RR ,  <  ) )
13737, 136eqbrtrd 4192 . 2  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  <_  sup ( C ,  RR ,  <  ) )
13869, 1supmullem1 9930 . . 3  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
1394, 7remulcld 9072 . . . 4  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
140 suprleub 9928 . . . 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 ,  <  ) ) ) )
14173, 139, 140syl2anc 643 . . 3  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x. 
sup ( B ,  RR ,  <  ) ) ) )
142138, 141mpbird 224 . 2  |-  ( ph  ->  sup ( C ,  RR ,  <  )  <_ 
( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
143139, 100letri3d 9171 . 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 ,  <  ) ) ) ) )
144137, 142, 143mpbir2and 889 1  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667    C_ wss 3280   (/)c0 3588   class class class wbr 4172  (class class class)co 6040   supcsup 7403   CCcc 8944   RRcr 8945   0cc0 8946    x. cmul 8951    < clt 9076    <_ cle 9077
This theorem is referenced by:  sqrlem5  12007
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-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-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-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
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