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Theorem supmul1 10528
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 10531 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 3112 . . . . . . . 8  |-  w  e. 
_V
2 oveq2 6304 . . . . . . . . . . 11  |-  ( v  =  b  ->  ( A  x.  v )  =  ( A  x.  b ) )
32eqeq2d 2471 . . . . . . . . . 10  |-  ( v  =  b  ->  (
z  =  ( A  x.  v )  <->  z  =  ( A  x.  b
) ) )
43cbvrexv 3085 . . . . . . . . 9  |-  ( E. v  e.  B  z  =  ( A  x.  v )  <->  E. b  e.  B  z  =  ( A  x.  b
) )
5 eqeq1 2461 . . . . . . . . . 10  |-  ( z  =  w  ->  (
z  =  ( A  x.  b )  <->  w  =  ( A  x.  b
) ) )
65rexbidv 2968 . . . . . . . . 9  |-  ( z  =  w  ->  ( E. b  e.  B  z  =  ( A  x.  b )  <->  E. b  e.  B  w  =  ( A  x.  b
) ) )
74, 6syl5bb 257 . . . . . . . 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 3249 . . . . . . 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 461 . . . . . . . . . . . . 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 195 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) )
1312simp1d 1008 . . . . . . . . . . 11  |-  ( ph  ->  B  C_  RR )
1413sselda 3499 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  b  e.  RR )
15 suprcl 10523 . . . . . . . . . . . 12  |-  ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
)  ->  sup ( B ,  RR ,  <  )  e.  RR )
1612, 15syl 16 . . . . . . . . . . 11  |-  ( ph  ->  sup ( B ,  RR ,  <  )  e.  RR )
1716adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  sup ( B ,  RR ,  <  )  e.  RR )
18 simpl1 999 . . . . . . . . . . . . 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 195 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR )
20 simpl2 1000 . . . . . . . . . . . . 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 195 . . . . . . . . . . . 12  |-  ( ph  ->  0  <_  A )
2219, 21jca 532 . . . . . . . . . . 11  |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A )
)
2322adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  ( A  e.  RR  /\  0  <_  A ) )
24 suprub 10524 . . . . . . . . . . 11  |-  ( ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x )  /\  b  e.  B )  ->  b  <_  sup ( B ,  RR ,  <  ) )
2512, 24sylan 471 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  b  <_  sup ( B ,  RR ,  <  ) )
26 lemul2a 10418 . . . . . . . . . 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 1231 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B )  ->  ( A  x.  b )  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) )
28 breq1 4459 . . . . . . . . 9  |-  ( w  =  ( A  x.  b )  ->  (
w  <_  ( A  x.  sup ( B ,  RR ,  <  ) )  <-> 
( A  x.  b
)  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
2927, 28syl5ibrcom 222 . . . . . . . 8  |-  ( (
ph  /\  b  e.  B )  ->  (
w  =  ( A  x.  b )  ->  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
3029rexlimdva 2949 . . . . . . 7  |-  ( ph  ->  ( E. b  e.  B  w  =  ( A  x.  b )  ->  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
319, 30syl5bi 217 . . . . . 6  |-  ( ph  ->  ( w  e.  C  ->  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
3231ralrimiv 2869 . . . . 5  |-  ( ph  ->  A. w  e.  C  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) )
3319adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  B )  ->  A  e.  RR )
3433, 14remulcld 9641 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  B )  ->  ( A  x.  b )  e.  RR )
35 eleq1a 2540 . . . . . . . . . . 11  |-  ( ( A  x.  b )  e.  RR  ->  (
w  =  ( A  x.  b )  ->  w  e.  RR )
)
3634, 35syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  (
w  =  ( A  x.  b )  ->  w  e.  RR )
)
3736rexlimdva 2949 . . . . . . . . 9  |-  ( ph  ->  ( E. b  e.  B  w  =  ( A  x.  b )  ->  w  e.  RR ) )
389, 37syl5bi 217 . . . . . . . 8  |-  ( ph  ->  ( w  e.  C  ->  w  e.  RR ) )
3938ssrdv 3505 . . . . . . 7  |-  ( ph  ->  C  C_  RR )
40 simpr2 1003 . . . . . . . . . 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 195 . . . . . . . . 9  |-  ( ph  ->  B  =/=  (/) )
42 ovex 6324 . . . . . . . . . . 11  |-  ( A  x.  b )  e. 
_V
4342isseti 3115 . . . . . . . . . 10  |-  E. w  w  =  ( A  x.  b )
4443rgenw 2818 . . . . . . . . 9  |-  A. b  e.  B  E. w  w  =  ( A  x.  b )
45 r19.2z 3921 . . . . . . . . 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 662 . . . . . . . 8  |-  ( ph  ->  E. b  e.  B  E. w  w  =  ( A  x.  b
) )
479exbii 1668 . . . . . . . . 9  |-  ( E. w  w  e.  C  <->  E. w E. b  e.  B  w  =  ( A  x.  b ) )
48 n0 3803 . . . . . . . . 9  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
49 rexcom4 3129 . . . . . . . . 9  |-  ( E. b  e.  B  E. w  w  =  ( A  x.  b )  <->  E. w E. b  e.  B  w  =  ( A  x.  b ) )
5047, 48, 493bitr4i 277 . . . . . . . 8  |-  ( C  =/=  (/)  <->  E. b  e.  B  E. w  w  =  ( A  x.  b
) )
5146, 50sylibr 212 . . . . . . 7  |-  ( ph  ->  C  =/=  (/) )
5219, 16remulcld 9641 . . . . . . . 8  |-  ( ph  ->  ( A  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
53 breq2 4460 . . . . . . . . . 10  |-  ( x  =  ( A  x.  sup ( B ,  RR ,  <  ) )  -> 
( w  <_  x  <->  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
5453ralbidv 2896 . . . . . . . . 9  |-  ( x  =  ( A  x.  sup ( B ,  RR ,  <  ) )  -> 
( A. w  e.  C  w  <_  x  <->  A. w  e.  C  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
5554rspcev 3210 . . . . . . . 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 661 . . . . . . 7  |-  ( ph  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
5739, 51, 563jca 1176 . . . . . 6  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
58 suprleub 10527 . . . . . 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 661 . . . . 5  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  <_  ( A  x.  sup ( B ,  RR ,  <  ) )  <->  A. w  e.  C  w  <_  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
6032, 59mpbird 232 . . . 4  |-  ( ph  ->  sup ( C ,  RR ,  <  )  <_ 
( A  x.  sup ( B ,  RR ,  <  ) ) )
61 simpr 461 . . . . . . 7  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )
62 suprcl 10523 . . . . . . . . . 10  |-  ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x
)  ->  sup ( C ,  RR ,  <  )  e.  RR )
6357, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  sup ( C ,  RR ,  <  )  e.  RR )
6463adantr 465 . . . . . . . 8  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  sup ( C ,  RR ,  <  )  e.  RR )
6516adantr 465 . . . . . . . 8  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  sup ( B ,  RR ,  <  )  e.  RR )
6619adantr 465 . . . . . . . 8  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  A  e.  RR )
67 n0 3803 . . . . . . . . . . . 12  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
68 0red 9614 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  b  e.  B )  ->  0  e.  RR )
69 simpl3 1001 . . . . . . . . . . . . . . . . 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 )
7010, 69sylbi 195 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  B 
0  <_  x )
71 breq2 4460 . . . . . . . . . . . . . . . . 17  |-  ( x  =  b  ->  (
0  <_  x  <->  0  <_  b ) )
7271rspccva 3209 . . . . . . . . . . . . . . . 16  |-  ( ( A. x  e.  B 
0  <_  x  /\  b  e.  B )  ->  0  <_  b )
7370, 72sylan 471 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  b )
7468, 14, 17, 73, 25letrd 9756 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  sup ( B ,  RR ,  <  ) )
7574ex 434 . . . . . . . . . . . . 13  |-  ( ph  ->  ( b  e.  B  ->  0  <_  sup ( B ,  RR ,  <  ) ) )
7675exlimdv 1725 . . . . . . . . . . . 12  |-  ( ph  ->  ( E. b  b  e.  B  ->  0  <_  sup ( B ,  RR ,  <  ) ) )
7767, 76syl5bi 217 . . . . . . . . . . 11  |-  ( ph  ->  ( B  =/=  (/)  ->  0  <_  sup ( B ,  RR ,  <  ) ) )
7841, 77mpd 15 . . . . . . . . . 10  |-  ( ph  ->  0  <_  sup ( B ,  RR ,  <  ) )
7978adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  0  <_  sup ( B ,  RR ,  <  ) )
80 0red 9614 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  C )  ->  0  e.  RR )
8138imp 429 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  C )  ->  w  e.  RR )
8263adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  C )  ->  sup ( C ,  RR ,  <  )  e.  RR )
8321adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  A )
8433, 14, 83, 73mulge0d 10150 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  b  e.  B )  ->  0  <_  ( A  x.  b
) )
85 breq2 4460 . . . . . . . . . . . . . . . . . . . 20  |-  ( w  =  ( A  x.  b )  ->  (
0  <_  w  <->  0  <_  ( A  x.  b ) ) )
8684, 85syl5ibrcom 222 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  b  e.  B )  ->  (
w  =  ( A  x.  b )  -> 
0  <_  w )
)
8786rexlimdva 2949 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( E. b  e.  B  w  =  ( A  x.  b )  ->  0  <_  w
) )
889, 87syl5bi 217 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( w  e.  C  ->  0  <_  w )
)
8988imp 429 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  C )  ->  0  <_  w )
90 suprub 10524 . . . . . . . . . . . . . . . . 17  |-  ( ( ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x )  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
9157, 90sylan 471 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  C )  ->  w  <_  sup ( C ,  RR ,  <  ) )
9280, 81, 82, 89, 91letrd 9756 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  w  e.  C )  ->  0  <_  sup ( C ,  RR ,  <  ) )
9392ex 434 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( w  e.  C  ->  0  <_  sup ( C ,  RR ,  <  ) ) )
9493exlimdv 1725 . . . . . . . . . . . . 13  |-  ( ph  ->  ( E. w  w  e.  C  ->  0  <_  sup ( C ,  RR ,  <  ) ) )
9548, 94syl5bi 217 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  =/=  (/)  ->  0  <_  sup ( C ,  RR ,  <  ) ) )
9651, 95mpd 15 . . . . . . . . . . 11  |-  ( ph  ->  0  <_  sup ( C ,  RR ,  <  ) )
9796anim1i 568 . . . . . . . . . 10  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  (
0  <_  sup ( C ,  RR ,  <  )  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
98 0red 9614 . . . . . . . . . . . 12  |-  ( ph  ->  0  e.  RR )
99 lelttr 9692 . . . . . . . . . . . 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 ,  <  ) ) ) )
10098, 63, 52, 99syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 0  <_  sup ( C ,  RR ,  <  )  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  0  <  ( A  x.  sup ( B ,  RR ,  <  ) ) ) )
101100adantr 465 . . . . . . . . . 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 ,  <  ) ) ) )
10297, 101mpd 15 . . . . . . . . 9  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  0  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )
103 prodgt02 10409 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\ 
sup ( B ,  RR ,  <  )  e.  RR )  /\  (
0  <_  sup ( B ,  RR ,  <  )  /\  0  < 
( A  x.  sup ( B ,  RR ,  <  ) ) ) )  ->  0  <  A
)
10466, 65, 79, 102, 103syl22anc 1229 . . . . . . . 8  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  0  <  A )
105 ltdivmul 10438 . . . . . . . 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 ,  <  ) ) ) )
10664, 65, 66, 104, 105syl112anc 1232 . . . . . . 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 ,  <  ) ) ) )
10761, 106mpbird 232 . . . . . 6  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  ( sup ( C ,  RR ,  <  )  /  A
)  <  sup ( B ,  RR ,  <  ) )
10812adantr 465 . . . . . . 7  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
) )
109104gt0ne0d 10138 . . . . . . . 8  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  A  =/=  0 )
11064, 66, 109redivcld 10393 . . . . . . 7  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  ( sup ( C ,  RR ,  <  )  /  A
)  e.  RR )
111 suprlub 10525 . . . . . . 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
) )
112108, 110, 111syl2anc 661 . . . . . 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 ) )
113107, 112mpbid 210 . . . . 5  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  E. b  e.  B  ( sup ( C ,  RR ,  <  )  /  A )  <  b )
114 rspe 2915 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  B  /\  w  =  ( A  x.  b ) )  ->  E. b  e.  B  w  =  ( A  x.  b ) )
115114, 9sylibr 212 . . . . . . . . . . . . . 14  |-  ( ( b  e.  B  /\  w  =  ( A  x.  b ) )  ->  w  e.  C )
116115adantl 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  B  /\  w  =  ( A  x.  b ) ) )  ->  w  e.  C
)
117 simplrr 762 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
b  e.  B  /\  w  =  ( A  x.  b ) ) )  /\  w  e.  C
)  ->  w  =  ( A  x.  b
) )
11891adantlr 714 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
b  e.  B  /\  w  =  ( A  x.  b ) ) )  /\  w  e.  C
)  ->  w  <_  sup ( C ,  RR ,  <  ) )
119117, 118eqbrtrrd 4478 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
b  e.  B  /\  w  =  ( A  x.  b ) ) )  /\  w  e.  C
)  ->  ( A  x.  b )  <_  sup ( C ,  RR ,  <  ) )
120116, 119mpdan 668 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  B  /\  w  =  ( A  x.  b ) ) )  ->  ( A  x.  b )  <_  sup ( C ,  RR ,  <  ) )
121120expr 615 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  B )  ->  (
w  =  ( A  x.  b )  -> 
( A  x.  b
)  <_  sup ( C ,  RR ,  <  ) ) )
122121exlimdv 1725 . . . . . . . . . 10  |-  ( (
ph  /\  b  e.  B )  ->  ( E. w  w  =  ( A  x.  b
)  ->  ( A  x.  b )  <_  sup ( C ,  RR ,  <  ) ) )
12343, 122mpi 17 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  B )  ->  ( A  x.  b )  <_  sup ( C ,  RR ,  <  ) )
124123adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  ( A  x.  b )  <_  sup ( C ,  RR ,  <  ) )
12534adantlr 714 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  ( A  x.  b )  e.  RR )
12663ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  sup ( C ,  RR ,  <  )  e.  RR )
127125, 126lenltd 9748 . . . . . . . 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 ) ) )
128124, 127mpbid 210 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  -.  sup ( C ,  RR ,  <  )  <  ( A  x.  b )
)
12914adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  b  e.  RR )
13019ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  A  e.  RR )
131104adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  0  <  A )
132 ltdivmul 10438 . . . . . . . 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 ) ) )
133126, 129, 130, 131, 132syl112anc 1232 . . . . . . 7  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  (
( sup ( C ,  RR ,  <  )  /  A )  < 
b  <->  sup ( C ,  RR ,  <  )  < 
( A  x.  b
) ) )
134128, 133mtbird 301 . . . . . 6  |-  ( ( ( ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  /\  b  e.  B )  ->  -.  ( sup ( C ,  RR ,  <  )  /  A )  <  b
)
135134nrexdv 2913 . . . . 5  |-  ( (
ph  /\  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )  ->  -.  E. b  e.  B  ( sup ( C ,  RR ,  <  )  /  A )  <  b
)
136113, 135pm2.65da 576 . . . 4  |-  ( ph  ->  -.  sup ( C ,  RR ,  <  )  <  ( A  x.  sup ( B ,  RR ,  <  ) ) )
13760, 136jca 532 . . 3  |-  ( ph  ->  ( sup ( C ,  RR ,  <  )  <_  ( A  x.  sup ( B ,  RR ,  <  ) )  /\  -.  sup ( C ,  RR ,  <  )  < 
( A  x.  sup ( B ,  RR ,  <  ) ) ) )
13863, 52eqleltd 9746 . . 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 ,  <  ) ) ) ) )
139137, 138mpbird 232 . 2  |-  ( ph  ->  sup ( C ,  RR ,  <  )  =  ( A  x.  sup ( B ,  RR ,  <  ) ) )
140139eqcomd 2465 1  |-  ( ph  ->  ( A  x.  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442    =/= wne 2652   A.wral 2807   E.wrex 2808    C_ wss 3471   (/)c0 3793   class class class wbr 4456  (class class class)co 6296   supcsup 7918   RRcr 9508   0cc0 9509    x. cmul 9514    < clt 9645    <_ cle 9646    / cdiv 10227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228
This theorem is referenced by:  supmul  10531
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