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Theorem supmul 9968
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 8853). (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 9960 . . . . . 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 9960 . . . . . 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 9072 . . . . . 6  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  sup ( A ,  RR ,  <  )  e.  CC )
9 recn 9072 . . . . . 6  |-  ( sup ( B ,  RR ,  <  )  e.  RR  ->  sup ( B ,  RR ,  <  )  e.  CC )
10 mulcom 9068 . . . . . 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 3629 . . . . . . 7  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
1513, 14sylib 189 . . . . . 6  |-  ( ph  ->  E. b  b  e.  B )
16 0re 9083 . . . . . . . 8  |-  0  e.  RR
1716a1i 11 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  0  e.  RR )
185simp1d 969 . . . . . . . 8  |-  ( ph  ->  B  C_  RR )
1918sselda 3340 . . . . . . 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 4208 . . . . . . . . . 10  |-  ( x  =  b  ->  (
0  <_  x  <->  0  <_  b ) )
2423rspccv 3041 . . . . . . . . 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 9961 . . . . . . . 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 9219 . . . . . 6  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  sup ( B ,  RR ,  <  ) )
3015, 29exlimddv 1648 . . . . 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 2435 . . . . . 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 9965 . . . . 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 2467 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  ) )
38 vex 2951 . . . . . . 7  |-  w  e. 
_V
39 eqeq1 2441 . . . . . . . 8  |-  ( z  =  w  ->  (
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)  <->  w  =  ( sup ( B ,  RR ,  <  )  x.  a
) ) )
4039rexbidv 2718 . . . . . . 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 3074 . . . . . 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 3340 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  a  e.  RR )
45 recn 9072 . . . . . . . . . . 11  |-  ( a  e.  RR  ->  a  e.  CC )
46 mulcom 9068 . . . . . . . . . . 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 4208 . . . . . . . . . . . . . 14  |-  ( x  =  a  ->  (
0  <_  x  <->  0  <_  a ) )
5049rspccv 3041 . . . . . . . . . . . . 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 2435 . . . . . . . . . . . 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 9965 . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
z  =  ( a  x.  b )  <->  w  =  ( a  x.  b
) ) )
6059rexbidv 2718 . . . . . . . . . . . . . 14  |-  ( z  =  w  ->  ( E. b  e.  B  z  =  ( a  x.  b )  <->  E. b  e.  B  w  =  ( a  x.  b
) ) )
6138, 60elab 3074 . . . . . . . . . . . . 13  |-  ( w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  <->  E. b  e.  B  w  =  ( a  x.  b
) )
62 rspe 2759 . . . . . . . . . . . . . . . 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 6080 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  =  a  ->  (
v  x.  b )  =  ( a  x.  b ) )
6463eqeq2d 2446 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  a  ->  (
z  =  ( v  x.  b )  <->  z  =  ( a  x.  b
) ) )
6564rexbidv 2718 . . . . . . . . . . . . . . . . . . 19  |-  ( v  =  a  ->  ( E. b  e.  B  z  =  ( v  x.  b )  <->  E. b  e.  B  z  =  ( a  x.  b
) ) )
6665cbvrexv 2925 . . . . . . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . . . . . 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 3077 . . . . . . . . . . . . . . . 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 9967 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
74 suprub 9961 . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . 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 9108 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  (
a  x.  b )  e.  RR )
83 eleq1a 2504 . . . . . . . . . . . . . . 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 2822 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  A )  ->  ( E. b  e.  B  z  =  ( a  x.  b )  ->  z  e.  RR ) )
8685abssdv 3409 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  C_  RR )
87 ovex 6098 . . . . . . . . . . . . . . . . . . 19  |-  ( a  x.  b )  e. 
_V
8887isseti 2954 . . . . . . . . . . . . . . . . . 18  |-  E. w  w  =  ( a  x.  b )
8988rgenw 2765 . . . . . . . . . . . . . . . . 17  |-  A. b  e.  B  E. w  w  =  ( a  x.  b )
90 r19.2z 3709 . . . . . . . . . . . . . . . . 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 2967 . . . . . . . . . . . . . . . 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 1985 . . . . . . . . . . . . . . 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 3638 . . . . . . . . . . . . . 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 9960 . . . . . . . . . . . . . . 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 4208 . . . . . . . . . . . . . . 15  |-  ( x  =  sup ( C ,  RR ,  <  )  ->  ( w  <_  x 
<->  w  <_  sup ( C ,  RR ,  <  ) ) )
103102ralbidv 2717 . . . . . . . . . . . . . 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 3044 . . . . . . . . . . . . 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 9964 . . . . . . . . . . . 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 4224 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  (
a  x.  sup ( B ,  RR ,  <  ) )  <_  sup ( C ,  RR ,  <  ) )
11048, 109eqbrtrd 4224 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  <_  sup ( C ,  RR ,  <  ) )
111 breq1 4207 . . . . . . . 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 2822 . . . . . 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 2780 . . . 4  |-  ( ph  ->  A. w  e.  {
z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  sup ( C ,  RR ,  <  ) )
11642, 44remulcld 9108 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  e.  RR )
117 eleq1a 2504 . . . . . . . 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 2822 . . . . . 6  |-  ( ph  ->  ( E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  z  e.  RR ) )
120119abssdv 3409 . . . . 5  |-  ( ph  ->  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  C_  RR )
1212simp2d 970 . . . . . . . 8  |-  ( ph  ->  A  =/=  (/) )
122 ovex 6098 . . . . . . . . . 10  |-  ( sup ( B ,  RR ,  <  )  x.  a
)  e.  _V
123122isseti 2954 . . . . . . . . 9  |-  E. z 
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)
124123rgenw 2765 . . . . . . . 8  |-  A. a  e.  A  E. z 
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)
125 r19.2z 3709 . . . . . . . 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 2967 . . . . . . 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 3638 . . . . . 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 2717 . . . . . . 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 3044 . . . . . 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 9964 . . . . 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 4224 . 2  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  <_  sup ( C ,  RR ,  <  ) )
13869, 1supmullem1 9966 . . 3  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
1394, 7remulcld 9108 . . . 4  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
140 suprleub 9964 . . . 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 9207 . 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 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698    C_ wss 3312   (/)c0 3620   class class class wbr 4204  (class class class)co 6073   supcsup 7437   CCcc 8980   RRcr 8981   0cc0 8982    x. cmul 8987    < clt 9112    <_ cle 9113
This theorem is referenced by:  sqrlem5  12044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670
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