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Theorem supmul 9812
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 8698). (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 971 . . . . . 6  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
3 suprcl 9804 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
42, 3syl 15 . . . . 5  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
51simp3bi 972 . . . . . 6  |-  ( ph  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) )
6 suprcl 9804 . . . . . 6  |-  ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
)  ->  sup ( B ,  RR ,  <  )  e.  RR )
75, 6syl 15 . . . . 5  |-  ( ph  ->  sup ( B ,  RR ,  <  )  e.  RR )
8 recn 8917 . . . . . 6  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  sup ( A ,  RR ,  <  )  e.  CC )
9 recn 8917 . . . . . 6  |-  ( sup ( B ,  RR ,  <  )  e.  RR  ->  sup ( B ,  RR ,  <  )  e.  CC )
10 mulcom 8913 . . . . . 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 463 . . . . 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 642 . . . 4  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  ( sup ( B ,  RR ,  <  )  x.  sup ( A ,  RR ,  <  ) ) )
135simp2d 968 . . . . . . 7  |-  ( ph  ->  B  =/=  (/) )
14 n0 3540 . . . . . . 7  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
1513, 14sylib 188 . . . . . 6  |-  ( ph  ->  E. b  b  e.  B )
16 0re 8928 . . . . . . . . . 10  |-  0  e.  RR
1716a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B )  ->  0  e.  RR )
185simp1d 967 . . . . . . . . . 10  |-  ( ph  ->  B  C_  RR )
1918sselda 3256 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B )  ->  b  e.  RR )
207adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B )  ->  sup ( B ,  RR ,  <  )  e.  RR )
21 simp1r 980 . . . . . . . . . . . 12  |-  ( ( ( 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 187 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  B 
0  <_  x )
23 breq2 4108 . . . . . . . . . . . 12  |-  ( x  =  b  ->  (
0  <_  x  <->  0  <_  b ) )
2423rspccv 2957 . . . . . . . . . . 11  |-  ( A. x  e.  B  0  <_  x  ->  ( b  e.  B  ->  0  <_ 
b ) )
2522, 24syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( b  e.  B  ->  0  <_  b )
)
2625imp 418 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  b )
27 suprub 9805 . . . . . . . . . 10  |-  ( ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x )  /\  b  e.  B )  ->  b  <_  sup ( B ,  RR ,  <  ) )
285, 27sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B )  ->  b  <_  sup ( B ,  RR ,  <  ) )
2917, 19, 20, 26, 28letrd 9063 . . . . . . . 8  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  sup ( B ,  RR ,  <  ) )
3029ex 423 . . . . . . 7  |-  ( ph  ->  ( b  e.  B  ->  0  <_  sup ( B ,  RR ,  <  ) ) )
3130exlimdv 1636 . . . . . 6  |-  ( ph  ->  ( E. b  b  e.  B  ->  0  <_  sup ( B ,  RR ,  <  ) ) )
3215, 31mpd 14 . . . . 5  |-  ( ph  ->  0  <_  sup ( B ,  RR ,  <  ) )
33 simp1l 979 . . . . . 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 )
341, 33sylbi 187 . . . . 5  |-  ( ph  ->  A. x  e.  A 
0  <_  x )
35 eqid 2358 . . . . . 6  |-  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  =  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }
36 biid 227 . . . . . 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
) ) )
3735, 36supmul1 9809 . . . . 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 ,  <  ) )
387, 32, 34, 2, 37syl31anc 1185 . . . 4  |-  ( ph  ->  ( sup ( B ,  RR ,  <  )  x.  sup ( A ,  RR ,  <  ) )  =  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  ) )
3912, 38eqtrd 2390 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  ) )
40 vex 2867 . . . . . . 7  |-  w  e. 
_V
41 eqeq1 2364 . . . . . . . 8  |-  ( z  =  w  ->  (
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)  <->  w  =  ( sup ( B ,  RR ,  <  )  x.  a
) ) )
4241rexbidv 2640 . . . . . . 7  |-  ( z  =  w  ->  ( E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a )  <->  E. a  e.  A  w  =  ( sup ( B ,  RR ,  <  )  x.  a ) ) )
4340, 42elab 2990 . . . . . 6  |-  ( w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  <->  E. a  e.  A  w  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
447adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  sup ( B ,  RR ,  <  )  e.  RR )
452simp1d 967 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  RR )
4645sselda 3256 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  a  e.  RR )
47 recn 8917 . . . . . . . . . . 11  |-  ( a  e.  RR  ->  a  e.  CC )
48 mulcom 8913 . . . . . . . . . . 11  |-  ( ( sup ( B ,  RR ,  <  )  e.  CC  /\  a  e.  CC )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  =  ( a  x.  sup ( B ,  RR ,  <  ) ) )
499, 47, 48syl2an 463 . . . . . . . . . 10  |-  ( ( sup ( B ,  RR ,  <  )  e.  RR  /\  a  e.  RR )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  =  ( a  x.  sup ( B ,  RR ,  <  ) ) )
5044, 46, 49syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  =  ( a  x.  sup ( B ,  RR ,  <  ) ) )
51 breq2 4108 . . . . . . . . . . . . . 14  |-  ( x  =  a  ->  (
0  <_  x  <->  0  <_  a ) )
5251rspccv 2957 . . . . . . . . . . . . 13  |-  ( A. x  e.  A  0  <_  x  ->  ( a  e.  A  ->  0  <_ 
a ) )
5334, 52syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( a  e.  A  ->  0  <_  a )
)
5453imp 418 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  0  <_  a )
5522adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  A. x  e.  B  0  <_  x )
565adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) )
57 eqid 2358 . . . . . . . . . . . 12  |-  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  =  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }
58 biid 227 . . . . . . . . . . . 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
) ) )
5957, 58supmul1 9809 . . . . . . . . . . 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 ,  <  ) )
6046, 54, 55, 56, 59syl31anc 1185 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  (
a  x.  sup ( B ,  RR ,  <  ) )  =  sup ( { z  |  E. b  e.  B  z  =  ( a  x.  b ) } ,  RR ,  <  ) )
61 eqeq1 2364 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
z  =  ( a  x.  b )  <->  w  =  ( a  x.  b
) ) )
6261rexbidv 2640 . . . . . . . . . . . . . 14  |-  ( z  =  w  ->  ( E. b  e.  B  z  =  ( a  x.  b )  <->  E. b  e.  B  w  =  ( a  x.  b
) ) )
6340, 62elab 2990 . . . . . . . . . . . . 13  |-  ( w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  <->  E. b  e.  B  w  =  ( a  x.  b
) )
64 rspe 2680 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  A  /\  E. b  e.  B  w  =  ( a  x.  b ) )  ->  E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b ) )
65 oveq1 5952 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  =  a  ->  (
v  x.  b )  =  ( a  x.  b ) )
6665eqeq2d 2369 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  a  ->  (
z  =  ( v  x.  b )  <->  z  =  ( a  x.  b
) ) )
6766rexbidv 2640 . . . . . . . . . . . . . . . . . . 19  |-  ( v  =  a  ->  ( E. b  e.  B  z  =  ( v  x.  b )  <->  E. b  e.  B  z  =  ( a  x.  b
) ) )
6867cbvrexv 2841 . . . . . . . . . . . . . . . . . 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
) )
69612rexbidv 2662 . . . . . . . . . . . . . . . . . 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
) ) )
7068, 69syl5bb 248 . . . . . . . . . . . . . . . . 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
) ) )
71 supmul.1 . . . . . . . . . . . . . . . . 17  |-  C  =  { z  |  E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b ) }
7240, 70, 71elab2 2993 . . . . . . . . . . . . . . . 16  |-  ( w  e.  C  <->  E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) )
7364, 72sylibr 203 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  A  /\  E. b  e.  B  w  =  ( a  x.  b ) )  ->  w  e.  C )
7473ex 423 . . . . . . . . . . . . . 14  |-  ( a  e.  A  ->  ( E. b  e.  B  w  =  ( a  x.  b )  ->  w  e.  C ) )
7571, 1supmullem2 9811 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
76 suprub 9805 . . . . . . . . . . . . . . . 16  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
7776ex 423 . . . . . . . . . . . . . . 15  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  ( w  e.  C  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
7875, 77syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( w  e.  C  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
7974, 78sylan9r 639 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  A )  ->  ( E. b  e.  B  w  =  ( a  x.  b )  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
8063, 79syl5bi 208 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  (
w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
8180ralrimiv 2701 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  A )  ->  A. w  e.  { z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  sup ( C ,  RR ,  <  ) )
8246adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  a  e.  RR )
8319adantlr 695 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  b  e.  RR )
8482, 83remulcld 8953 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  (
a  x.  b )  e.  RR )
85 eleq1a 2427 . . . . . . . . . . . . . . 15  |-  ( ( a  x.  b )  e.  RR  ->  (
z  =  ( a  x.  b )  -> 
z  e.  RR ) )
8684, 85syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  e.  A )  /\  b  e.  B )  ->  (
z  =  ( a  x.  b )  -> 
z  e.  RR ) )
8786rexlimdva 2743 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  A )  ->  ( E. b  e.  B  z  =  ( a  x.  b )  ->  z  e.  RR ) )
8887abssdv 3323 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  C_  RR )
89 ovex 5970 . . . . . . . . . . . . . . . . . . 19  |-  ( a  x.  b )  e. 
_V
9089isseti 2870 . . . . . . . . . . . . . . . . . 18  |-  E. w  w  =  ( a  x.  b )
9190rgenw 2686 . . . . . . . . . . . . . . . . 17  |-  A. b  e.  B  E. w  w  =  ( a  x.  b )
92 r19.2z 3619 . . . . . . . . . . . . . . . . 17  |-  ( ( B  =/=  (/)  /\  A. b  e.  B  E. w  w  =  (
a  x.  b ) )  ->  E. b  e.  B  E. w  w  =  ( a  x.  b ) )
9313, 91, 92sylancl 643 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. b  e.  B  E. w  w  =  ( a  x.  b
) )
94 rexcom4 2883 . . . . . . . . . . . . . . . 16  |-  ( E. b  e.  B  E. w  w  =  (
a  x.  b )  <->  E. w E. b  e.  B  w  =  ( a  x.  b ) )
9593, 94sylib 188 . . . . . . . . . . . . . . 15  |-  ( ph  ->  E. w E. b  e.  B  w  =  ( a  x.  b
) )
9662cbvexv 2008 . . . . . . . . . . . . . . 15  |-  ( E. z E. b  e.  B  z  =  ( a  x.  b )  <->  E. w E. b  e.  B  w  =  ( a  x.  b ) )
9795, 96sylibr 203 . . . . . . . . . . . . . 14  |-  ( ph  ->  E. z E. b  e.  B  z  =  ( a  x.  b
) )
98 abn0 3549 . . . . . . . . . . . . . 14  |-  ( { z  |  E. b  e.  B  z  =  ( a  x.  b
) }  =/=  (/)  <->  E. z E. b  e.  B  z  =  ( a  x.  b ) )
9997, 98sylibr 203 . . . . . . . . . . . . 13  |-  ( ph  ->  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  =/=  (/) )
10099adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  { z  |  E. b  e.  B  z  =  ( a  x.  b ) }  =/=  (/) )
101 suprcl 9804 . . . . . . . . . . . . . . 15  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  sup ( C ,  RR ,  <  )  e.  RR )
10275, 101syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  sup ( C ,  RR ,  <  )  e.  RR )
103102adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  A )  ->  sup ( C ,  RR ,  <  )  e.  RR )
104 breq2 4108 . . . . . . . . . . . . . . 15  |-  ( x  =  sup ( C ,  RR ,  <  )  ->  ( w  <_  x 
<->  w  <_  sup ( C ,  RR ,  <  ) ) )
105104ralbidv 2639 . . . . . . . . . . . . . 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 ,  <  ) ) )
106105rspcev 2960 . . . . . . . . . . . . 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
)
107103, 81, 106syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  A )  ->  E. x  e.  RR  A. w  e. 
{ z  |  E. b  e.  B  z  =  ( a  x.  b ) } w  <_  x )
108 suprleub 9808 . . . . . . . . . . . 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 ,  <  ) ) )
10988, 100, 107, 103, 108syl31anc 1185 . . . . . . . . . . 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 ,  <  ) ) )
11081, 109mpbird 223 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  A )  ->  sup ( { z  |  E. b  e.  B  z  =  ( a  x.  b ) } ,  RR ,  <  )  <_  sup ( C ,  RR ,  <  ) )
11160, 110eqbrtrd 4124 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  A )  ->  (
a  x.  sup ( B ,  RR ,  <  ) )  <_  sup ( C ,  RR ,  <  ) )
11250, 111eqbrtrd 4124 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  <_  sup ( C ,  RR ,  <  ) )
113 breq1 4107 . . . . . . . 8  |-  ( w  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  ( w  <_  sup ( C ,  RR ,  <  )  <->  ( sup ( B ,  RR ,  <  )  x.  a )  <_  sup ( C ,  RR ,  <  ) ) )
114112, 113syl5ibrcom 213 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  (
w  =  ( sup ( B ,  RR ,  <  )  x.  a
)  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
115114rexlimdva 2743 . . . . . 6  |-  ( ph  ->  ( E. a  e.  A  w  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
11643, 115syl5bi 208 . . . . 5  |-  ( ph  ->  ( w  e.  {
z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  ->  w  <_  sup ( C ,  RR ,  <  ) ) )
117116ralrimiv 2701 . . . 4  |-  ( ph  ->  A. w  e.  {
z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  sup ( C ,  RR ,  <  ) )
11844, 46remulcld 8953 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( sup ( B ,  RR ,  <  )  x.  a
)  e.  RR )
119 eleq1a 2427 . . . . . . . 8  |-  ( ( sup ( B ,  RR ,  <  )  x.  a )  e.  RR  ->  ( z  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  z  e.  RR ) )
120118, 119syl 15 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  (
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)  ->  z  e.  RR ) )
121120rexlimdva 2743 . . . . . 6  |-  ( ph  ->  ( E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a )  ->  z  e.  RR ) )
122121abssdv 3323 . . . . 5  |-  ( ph  ->  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  C_  RR )
1232simp2d 968 . . . . . . . 8  |-  ( ph  ->  A  =/=  (/) )
124 ovex 5970 . . . . . . . . . 10  |-  ( sup ( B ,  RR ,  <  )  x.  a
)  e.  _V
125124isseti 2870 . . . . . . . . 9  |-  E. z 
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)
126125rgenw 2686 . . . . . . . 8  |-  A. a  e.  A  E. z 
z  =  ( sup ( B ,  RR ,  <  )  x.  a
)
127 r19.2z 3619 . . . . . . . 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
) )
128123, 126, 127sylancl 643 . . . . . . 7  |-  ( ph  ->  E. a  e.  A  E. z  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
129 rexcom4 2883 . . . . . . 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 ) )
130128, 129sylib 188 . . . . . 6  |-  ( ph  ->  E. z E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
131 abn0 3549 . . . . . 6  |-  ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  =/=  (/)  <->  E. z E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) )
132130, 131sylibr 203 . . . . 5  |-  ( ph  ->  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) }  =/=  (/) )
133104ralbidv 2639 . . . . . . 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 ,  <  ) ) )
134133rspcev 2960 . . . . . 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
)
135102, 117, 134syl2anc 642 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. w  e.  { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } w  <_  x )
136 suprleub 9808 . . . . 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 ,  <  ) ) )
137122, 132, 135, 102, 136syl31anc 1185 . . . 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 ,  <  ) ) )
138117, 137mpbird 223 . . 3  |-  ( ph  ->  sup ( { z  |  E. a  e.  A  z  =  ( sup ( B ,  RR ,  <  )  x.  a ) } ,  RR ,  <  )  <_  sup ( C ,  RR ,  <  ) )
13939, 138eqbrtrd 4124 . 2  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  <_  sup ( C ,  RR ,  <  ) )
14071, 1supmullem1 9810 . . 3  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
1414, 7remulcld 8953 . . . 4  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
142 suprleub 9808 . . . 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 ,  <  ) ) ) )
14375, 141, 142syl2anc 642 . . 3  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x. 
sup ( B ,  RR ,  <  ) ) ) )
144140, 143mpbird 223 . 2  |-  ( ph  ->  sup ( C ,  RR ,  <  )  <_ 
( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
145141, 102letri3d 9051 . 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 ,  <  ) ) ) ) )
146139, 144, 145mpbir2and 888 1  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1541    = wceq 1642    e. wcel 1710   {cab 2344    =/= wne 2521   A.wral 2619   E.wrex 2620    C_ wss 3228   (/)c0 3531   class class class wbr 4104  (class class class)co 5945   supcsup 7283   CCcc 8825   RRcr 8826   0cc0 8827    x. cmul 8832    < clt 8957    <_ cle 8958
This theorem is referenced by:  sqrlem5  11828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514
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