MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supmul1 Unicode version

Theorem supmul1 9735
Description: The supremum function distributes over multiplication, in the sense that  A  x.  ( sup B )  =  sup ( A  x.  B
), where  A  x.  B is shorthand for  { A  x.  b  |  b  e.  B } and is defined as  C below. This is the simple version, with only one set argument; see supmul 9738 for the more general case with two set arguments. (Contributed by Mario Carneiro, 5-Jul-2013.)
Hypotheses
Ref Expression
supmul1.1  |-  C  =  { z  |  E. v  e.  B  z  =  ( A  x.  v ) }
supmul1.2  |-  ( ph  <->  ( ( 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
) ) )
Assertion
Ref Expression
supmul1  |-  ( ph  ->  ( A  x.  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
Distinct variable groups:    v, A, x, z    v, B, x, y, z    x, C
Allowed substitution hints:    ph( x, y, z, v)    A( y)    C( y, z, v)

Proof of Theorem supmul1
Dummy variables  b  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . . . . . 8  |-  w  e. 
_V
2 oveq2 5882 . . . . . . . . . . 11  |-  ( v  =  b  ->  ( A  x.  v )  =  ( A  x.  b ) )
32eqeq2d 2307 . . . . . . . . . 10  |-  ( v  =  b  ->  (
z  =  ( A  x.  v )  <->  z  =  ( A  x.  b
) ) )
43cbvrexv 2778 . . . . . . . . 9  |-  ( E. v  e.  B  z  =  ( A  x.  v )  <->  E. b  e.  B  z  =  ( A  x.  b
) )
5 eqeq1 2302 . . . . . . . . . 10  |-  ( z  =  w  ->  (
z  =  ( A  x.  b )  <->  w  =  ( A  x.  b
) ) )
65rexbidv 2577 . . . . . . . . 9  |-  ( z  =  w  ->  ( E. b  e.  B  z  =  ( A  x.  b )  <->  E. b  e.  B  w  =  ( A  x.  b
) ) )
74, 6syl5bb 248 . . . . . . . 8  |-  ( z  =  w  ->  ( E. v  e.  B  z  =  ( A  x.  v )  <->  E. b  e.  B  w  =  ( A  x.  b
) ) )
8 supmul1.1 . . . . . . . 8  |-  C  =  { z  |  E. v  e.  B  z  =  ( A  x.  v ) }
91, 7, 8elab2 2930 . . . . . . 7  |-  ( w  e.  C  <->  E. b  e.  B  w  =  ( A  x.  b
) )
10 supmul1.2 . . . . . . . . . . . . 13  |-  ( ph  <->  ( ( 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
) ) )
11 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( 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
) )  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) )
1210, 11sylbi 187 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) )
1312simp1d 967 . . . . . . . . . . 11  |-  ( ph  ->  B  C_  RR )
1413sselda 3193 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  b  e.  RR )
15 suprcl 9730 . . . . . . . . . . . 12  |-  ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
)  ->  sup ( B ,  RR ,  <  )  e.  RR )
1612, 15syl 15 . . . . . . . . . . 11  |-  ( ph  ->  sup ( B ,  RR ,  <  )  e.  RR )
1716adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  sup ( B ,  RR ,  <  )  e.  RR )
18 simpl1 958 . . . . . . . . . . . . 13  |-  ( ( ( 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 )
1910, 18sylbi 187 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR )
20 simpl2 959 . . . . . . . . . . . . 13  |-  ( ( ( 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
) )  ->  0  <_  A )
2110, 20sylbi 187 . . . . . . . . . . . 12  |-  ( ph  ->  0  <_  A )
2219, 21jca 518 . . . . . . . . . . 11  |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A )
)
2322adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  ( A  e.  RR  /\  0  <_  A ) )
24 suprub 9731 . . . . . . . . . . 11  |-  ( ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x )  /\  b  e.  B )  ->  b  <_  sup ( B ,  RR ,  <  ) )
2512, 24sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  b  <_  sup ( B ,  RR ,  <  ) )
26 lemul2a 9627 . . . . . . . . . 10  |-  ( ( ( b  e.  RR  /\ 
sup ( B ,  RR ,  <  )  e.  RR  /\  ( A  e.  RR  /\  0  <_  A ) )  /\  b  <_  sup ( B ,  RR ,  <  ) )  ->  ( A  x.  b )  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) )
2714, 17, 23, 25, 26syl31anc 1185 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B )  ->  ( A  x.  b )  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) )
28 breq1 4042 . . . . . . . . 9  |-  ( w  =  ( A  x.  b )  ->  (
w  <_  ( A  x.  sup ( B ,  RR ,  <  ) )  <-> 
( A  x.  b
)  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
2927, 28syl5ibrcom 213 . . . . . . . 8  |-  ( (
ph  /\  b  e.  B )  ->  (
w  =  ( A  x.  b )  ->  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
3029rexlimdva 2680 . . . . . . 7  |-  ( ph  ->  ( E. b  e.  B  w  =  ( A  x.  b )  ->  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
319, 30syl5bi 208 . . . . . 6  |-  ( ph  ->  ( w  e.  C  ->  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
3231ralrimiv 2638 . . . . 5  |-  ( ph  ->  A. w  e.  C  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) )
3319adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  B )  ->  A  e.  RR )
3433, 14remulcld 8879 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  B )  ->  ( A  x.  b )  e.  RR )
35 eleq1a 2365 . . . . . . . . . . 11  |-  ( ( A  x.  b )  e.  RR  ->  (
w  =  ( A  x.  b )  ->  w  e.  RR )
)
3634, 35syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  (
w  =  ( A  x.  b )  ->  w  e.  RR )
)
3736rexlimdva 2680 . . . . . . . . 9  |-  ( ph  ->  ( E. b  e.  B  w  =  ( A  x.  b )  ->  w  e.  RR ) )
389, 37syl5bi 208 . . . . . . . 8  |-  ( ph  ->  ( w  e.  C  ->  w  e.  RR ) )
3938ssrdv 3198 . . . . . . 7  |-  ( ph  ->  C  C_  RR )
40 simpr2 962 . . . . . . . . . 10  |-  ( ( ( 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
) )  ->  B  =/=  (/) )
4110, 40sylbi 187 . . . . . . . . 9  |-  ( ph  ->  B  =/=  (/) )
42 ovex 5899 . . . . . . . . . . 11  |-  ( A  x.  b )  e. 
_V
4342isseti 2807 . . . . . . . . . 10  |-  E. w  w  =  ( A  x.  b )
4443rgenw 2623 . . . . . . . . 9  |-  A. b  e.  B  E. w  w  =  ( A  x.  b )
45 r19.2z 3556 . . . . . . . . 9  |-  ( ( B  =/=  (/)  /\  A. b  e.  B  E. w  w  =  ( A  x.  b )
)  ->  E. b  e.  B  E. w  w  =  ( A  x.  b ) )
4641, 44, 45sylancl 643 . . . . . . . 8  |-  ( ph  ->  E. b  e.  B  E. w  w  =  ( A  x.  b
) )
479exbii 1572 . . . . . . . . 9  |-  ( E. w  w  e.  C  <->  E. w E. b  e.  B  w  =  ( A  x.  b ) )
48 n0 3477 . . . . . . . . 9  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
49 rexcom4 2820 . . . . . . . . 9  |-  ( E. b  e.  B  E. w  w  =  ( A  x.  b )  <->  E. w E. b  e.  B  w  =  ( A  x.  b ) )
5047, 48, 493bitr4i 268 . . . . . . . 8  |-  ( C  =/=  (/)  <->  E. b  e.  B  E. w  w  =  ( A  x.  b
) )
5146, 50sylibr 203 . . . . . . 7  |-  ( ph  ->  C  =/=  (/) )
5219, 16remulcld 8879 . . . . . . . 8  |-  ( ph  ->  ( A  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
53 breq2 4043 . . . . . . . . . 10  |-  ( x  =  ( A  x.  sup ( B ,  RR ,  <  ) )  -> 
( w  <_  x  <->  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
5453ralbidv 2576 . . . . . . . . 9  |-  ( x  =  ( A  x.  sup ( B ,  RR ,  <  ) )  -> 
( A. w  e.  C  w  <_  x  <->  A. w  e.  C  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
5554rspcev 2897 . . . . . . . 8  |-  ( ( ( A  x.  sup ( B ,  RR ,  <  ) )  e.  RR  /\ 
A. w  e.  C  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
5652, 32, 55syl2anc 642 . . . . . . 7  |-  ( ph  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
5739, 51, 563jca 1132 . . . . . 6  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
58 suprleub 9734 . . . . . 6  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  ( A  x.  sup ( B ,  RR ,  <  ) )  e.  RR )  ->  ( sup ( C ,  RR ,  <  )  <_  ( A  x.  sup ( B ,  RR ,  <  ) )  <->  A. w  e.  C  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
5957, 52, 58syl2anc 642 . . . . 5  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  <_  ( A  x.  sup ( B ,  RR ,  <  ) )  <->  A. w  e.  C  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
6032, 59mpbird 223 . . . 4  |-  ( ph  ->  sup ( C ,  RR ,  <  )  <_ 
( A  x.  sup ( B ,  RR ,  <  ) ) )
61 simpr 447 . . . . . . 7  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )
62 suprcl 9730 . . . . . . . . . 10  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  sup ( C ,  RR ,  <  )  e.  RR )
6357, 62syl 15 . . . . . . . . 9  |-  ( ph  ->  sup ( C ,  RR ,  <  )  e.  RR )
6463adantr 451 . . . . . . . 8  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  sup ( C ,  RR ,  <  )  e.  RR )
6516adantr 451 . . . . . . . 8  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  sup ( B ,  RR ,  <  )  e.  RR )
6619adantr 451 . . . . . . . 8  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  A  e.  RR )
67 n0 3477 . . . . . . . . . . . 12  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
68 0re 8854 . . . . . . . . . . . . . . . 16  |-  0  e.  RR
6968a1i 10 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  b  e.  B )  ->  0  e.  RR )
70 simpl3 960 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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  e.  B  0  <_  x )
7110, 70sylbi 187 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  B 
0  <_  x )
72 breq2 4043 . . . . . . . . . . . . . . . . 17  |-  ( x  =  b  ->  (
0  <_  x  <->  0  <_  b ) )
7372rspccva 2896 . . . . . . . . . . . . . . . 16  |-  ( ( A. x  e.  B 
0  <_  x  /\  b  e.  B )  ->  0  <_  b )
7471, 73sylan 457 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  b )
7569, 14, 17, 74, 25letrd 8989 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  sup ( B ,  RR ,  <  ) )
7675ex 423 . . . . . . . . . . . . 13  |-  ( ph  ->  ( b  e.  B  ->  0  <_  sup ( B ,  RR ,  <  ) ) )
7776exlimdv 1626 . . . . . . . . . . . 12  |-  ( ph  ->  ( E. b  b  e.  B  ->  0  <_  sup ( B ,  RR ,  <  ) ) )
7867, 77syl5bi 208 . . . . . . . . . . 11  |-  ( ph  ->  ( B  =/=  (/)  ->  0  <_  sup ( B ,  RR ,  <  ) ) )
7941, 78mpd 14 . . . . . . . . . 10  |-  ( ph  ->  0  <_  sup ( B ,  RR ,  <  ) )
8079adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  0  <_  sup ( B ,  RR ,  <  ) )
8168a1i 10 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  C )  ->  0  e.  RR )
8238imp 418 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  C )  ->  w  e.  RR )
8363adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  C )  ->  sup ( C ,  RR ,  <  )  e.  RR )
8421adantr 451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  A )
8533, 14, 84, 74mulge0d 9365 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  ( A  x.  b
) )
86 breq2 4043 . . . . . . . . . . . . . . . . . . . 20  |-  ( w  =  ( A  x.  b )  ->  (
0  <_  w  <->  0  <_  ( A  x.  b ) ) )
8785, 86syl5ibrcom 213 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  b  e.  B )  ->  (
w  =  ( A  x.  b )  -> 
0  <_  w )
)
8887rexlimdva 2680 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( E. b  e.  B  w  =  ( A  x.  b )  ->  0  <_  w
) )
899, 88syl5bi 208 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( w  e.  C  ->  0  <_  w )
)
9089imp 418 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  C )  ->  0  <_  w )
91 suprub 9731 . . . . . . . . . . . . . . . . 17  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
9257, 91sylan 457 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
9381, 82, 83, 90, 92letrd 8989 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  w  e.  C )  ->  0  <_  sup ( C ,  RR ,  <  ) )
9493ex 423 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( w  e.  C  ->  0  <_  sup ( C ,  RR ,  <  ) ) )
9594exlimdv 1626 . . . . . . . . . . . . 13  |-  ( ph  ->  ( E. w  w  e.  C  ->  0  <_  sup ( C ,  RR ,  <  ) ) )
9648, 95syl5bi 208 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  =/=  (/)  ->  0  <_  sup ( C ,  RR ,  <  ) ) )
9751, 96mpd 14 . . . . . . . . . . 11  |-  ( ph  ->  0  <_  sup ( C ,  RR ,  <  ) )
9897anim1i 551 . . . . . . . . . 10  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  (
0  <_  sup ( C ,  RR ,  <  )  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
9968a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  0  e.  RR )
100 lelttr 8928 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  sup ( C ,  RR ,  <  )  e.  RR  /\  ( A  x.  sup ( B ,  RR ,  <  ) )  e.  RR )  ->  ( ( 0  <_  sup ( C ,  RR ,  <  )  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  -> 
0  <  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
10199, 63, 52, 100syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0  <_  sup ( C ,  RR ,  <  )  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  0  <  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
102101adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  (
( 0  <_  sup ( C ,  RR ,  <  )  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  0  <  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
10398, 102mpd 14 . . . . . . . . 9  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  0  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )
104 prodgt02 9618 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\ 
sup ( B ,  RR ,  <  )  e.  RR )  /\  (
0  <_  sup ( B ,  RR ,  <  )  /\  0  < 
( A  x.  sup ( B ,  RR ,  <  ) ) ) )  ->  0  <  A
)
10566, 65, 80, 103, 104syl22anc 1183 . . . . . . . 8  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  0  <  A )
106 ltdivmul 9644 . . . . . . . 8  |-  ( ( sup ( C ,  RR ,  <  )  e.  RR  /\  sup ( B ,  RR ,  <  )  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( ( sup ( C ,  RR ,  <  )  /  A
)  <  sup ( B ,  RR ,  <  )  <->  sup ( C ,  RR ,  <  )  < 
( A  x.  sup ( B ,  RR ,  <  ) ) ) )
10764, 65, 66, 105, 106syl112anc 1186 . . . . . . 7  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  (
( sup ( C ,  RR ,  <  )  /  A )  <  sup ( B ,  RR ,  <  )  <->  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
10861, 107mpbird 223 . . . . . 6  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  ( sup ( C ,  RR ,  <  )  /  A
)  <  sup ( B ,  RR ,  <  ) )
10912adantr 451 . . . . . . 7  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) )
110105gt0ne0d 9353 . . . . . . . 8  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  A  =/=  0 )
11164, 66, 110redivcld 9604 . . . . . . 7  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  ( sup ( C ,  RR ,  <  )  /  A
)  e.  RR )
112 suprlub 9732 . . . . . . 7  |-  ( ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x )  /\  ( sup ( C ,  RR ,  <  )  /  A
)  e.  RR )  ->  ( ( sup ( C ,  RR ,  <  )  /  A
)  <  sup ( B ,  RR ,  <  )  <->  E. b  e.  B  ( sup ( C ,  RR ,  <  )  /  A )  <  b
) )
113109, 111, 112syl2anc 642 . . . . . 6  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  (
( sup ( C ,  RR ,  <  )  /  A )  <  sup ( B ,  RR ,  <  )  <->  E. b  e.  B  ( sup ( C ,  RR ,  <  )  /  A )  <  b ) )
114108, 113mpbid 201 . . . . 5  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  E. b  e.  B  ( sup ( C ,  RR ,  <  )  /  A )  <  b )
115 rspe 2617 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  B  /\  w  =  ( A  x.  b ) )  ->  E. b  e.  B  w  =  ( A  x.  b ) )
116115, 9sylibr 203 . . . . . . . . . . . . . 14  |-  ( ( b  e.  B  /\  w  =  ( A  x.  b ) )  ->  w  e.  C )
117116adantl 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  B  /\  w  =  ( A  x.  b ) ) )  ->  w  e.  C
)
118 simplrr 737 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
b  e.  B  /\  w  =  ( A  x.  b ) ) )  /\  w  e.  C
)  ->  w  =  ( A  x.  b
) )
11992adantlr 695 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
b  e.  B  /\  w  =  ( A  x.  b ) ) )  /\  w  e.  C
)  ->  w  <_  sup ( C ,  RR ,  <  ) )
120118, 119eqbrtrrd 4061 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
b  e.  B  /\  w  =  ( A  x.  b ) ) )  /\  w  e.  C
)  ->  ( A  x.  b )  <_  sup ( C ,  RR ,  <  ) )
121117, 120mpdan 649 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  B  /\  w  =  ( A  x.  b ) ) )  ->  ( A  x.  b )  <_  sup ( C ,  RR ,  <  ) )
122121expr 598 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  B )  ->  (
w  =  ( A  x.  b )  -> 
( A  x.  b
)  <_  sup ( C ,  RR ,  <  ) ) )
123122exlimdv 1626 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  ( E. w  w  =  ( A  x.  b
)  ->  ( A  x.  b )  <_  sup ( C ,  RR ,  <  ) ) )
12443, 123mpi 16 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B )  ->  ( A  x.  b )  <_  sup ( C ,  RR ,  <  ) )
125124adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  ( A  x.  b )  <_  sup ( C ,  RR ,  <  ) )
12634adantlr 695 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  ( A  x.  b )  e.  RR )
12763ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  sup ( C ,  RR ,  <  )  e.  RR )
128126, 127lenltd 8981 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  (
( A  x.  b
)  <_  sup ( C ,  RR ,  <  )  <->  -.  sup ( C ,  RR ,  <  )  <  ( A  x.  b ) ) )
129125, 128mpbid 201 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  -.  sup ( C ,  RR ,  <  )  <  ( A  x.  b )
)
13014adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  b  e.  RR )
13119ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  A  e.  RR )
132105adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  0  <  A )
133 ltdivmul 9644 . . . . . . . 8  |-  ( ( sup ( C ,  RR ,  <  )  e.  RR  /\  b  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( sup ( C ,  RR ,  <  )  /  A )  <  b  <->  sup ( C ,  RR ,  <  )  <  ( A  x.  b ) ) )
134127, 130, 131, 132, 133syl112anc 1186 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  (
( sup ( C ,  RR ,  <  )  /  A )  < 
b  <->  sup ( C ,  RR ,  <  )  < 
( A  x.  b
) ) )
135129, 134mtbird 292 . . . . . 6  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  -.  ( sup ( C ,  RR ,  <  )  /  A )  <  b
)
136135nrexdv 2659 . . . . 5  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  -.  E. b  e.  B  ( sup ( C ,  RR ,  <  )  /  A )  <  b
)
137114, 136pm2.65da 559 . . . 4  |-  ( ph  ->  -.  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )
13860, 137jca 518 . . 3  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  <_  ( A  x.  sup ( B ,  RR ,  <  ) )  /\  -.  sup ( C ,  RR ,  <  )  < 
( A  x.  sup ( B ,  RR ,  <  ) ) ) )
13963, 52eqleltd 8979 . . 3  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  =  ( A  x.  sup ( B ,  RR ,  <  ) )  <->  ( sup ( C ,  RR ,  <  )  <_  ( A  x.  sup ( B ,  RR ,  <  ) )  /\  -.  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) ) ) )
140138, 139mpbird 223 . 2  |-  ( ph  ->  sup ( C ,  RR ,  <  )  =  ( A  x.  sup ( B ,  RR ,  <  ) ) )
141140eqcomd 2301 1  |-  ( ph  ->  ( A  x.  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   class class class wbr 4039  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439
This theorem is referenced by:  supmul  9738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440
  Copyright terms: Public domain W3C validator