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Theorem supmullem2 9811
Description: Lemma for supmul 9812. (Contributed by Mario Carneiro, 5-Jul-2013.)
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
supmullem2  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
Distinct variable groups:    A, b,
v, x, y, w, z    B, b, v, x, y, w, z    x, C, w    ph, b, w, z
Allowed substitution hints:    ph( x, y, v)    C( y, z, v, b)

Proof of Theorem supmullem2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 vex 2867 . . . . 5  |-  w  e. 
_V
2 oveq1 5952 . . . . . . . . 9  |-  ( v  =  a  ->  (
v  x.  b )  =  ( a  x.  b ) )
32eqeq2d 2369 . . . . . . . 8  |-  ( v  =  a  ->  (
z  =  ( v  x.  b )  <->  z  =  ( a  x.  b
) ) )
43rexbidv 2640 . . . . . . 7  |-  ( v  =  a  ->  ( E. b  e.  B  z  =  ( v  x.  b )  <->  E. b  e.  B  z  =  ( a  x.  b
) ) )
54cbvrexv 2841 . . . . . 6  |-  ( E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b )  <->  E. a  e.  A  E. b  e.  B  z  =  ( a  x.  b
) )
6 eqeq1 2364 . . . . . . 7  |-  ( z  =  w  ->  (
z  =  ( a  x.  b )  <->  w  =  ( a  x.  b
) ) )
762rexbidv 2662 . . . . . 6  |-  ( 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
) ) )
85, 7syl5bb 248 . . . . 5  |-  ( 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
) ) )
9 supmul.1 . . . . 5  |-  C  =  { z  |  E. v  e.  A  E. b  e.  B  z  =  ( v  x.  b ) }
101, 8, 9elab2 2993 . . . 4  |-  ( w  e.  C  <->  E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) )
11 supmul.2 . . . . . . . . . . 11  |-  ( 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
) ) )
1211simp2bi 971 . . . . . . . . . 10  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
1312simp1d 967 . . . . . . . . 9  |-  ( ph  ->  A  C_  RR )
1413sseld 3255 . . . . . . . 8  |-  ( ph  ->  ( a  e.  A  ->  a  e.  RR ) )
1511simp3bi 972 . . . . . . . . . 10  |-  ( ph  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) )
1615simp1d 967 . . . . . . . . 9  |-  ( ph  ->  B  C_  RR )
1716sseld 3255 . . . . . . . 8  |-  ( ph  ->  ( b  e.  B  ->  b  e.  RR ) )
1814, 17anim12d 546 . . . . . . 7  |-  ( ph  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
a  e.  RR  /\  b  e.  RR )
) )
19 remulcl 8912 . . . . . . 7  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( a  x.  b
)  e.  RR )
2018, 19syl6 29 . . . . . 6  |-  ( ph  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
a  x.  b )  e.  RR ) )
21 eleq1a 2427 . . . . . 6  |-  ( ( a  x.  b )  e.  RR  ->  (
w  =  ( a  x.  b )  ->  w  e.  RR )
)
2220, 21syl6 29 . . . . 5  |-  ( ph  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
w  =  ( a  x.  b )  ->  w  e.  RR )
) )
2322rexlimdvv 2749 . . . 4  |-  ( ph  ->  ( E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b )  ->  w  e.  RR ) )
2410, 23syl5bi 208 . . 3  |-  ( ph  ->  ( w  e.  C  ->  w  e.  RR ) )
2524ssrdv 3261 . 2  |-  ( ph  ->  C  C_  RR )
2612simp2d 968 . . . . 5  |-  ( ph  ->  A  =/=  (/) )
2715simp2d 968 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
28 ovex 5970 . . . . . . . . . 10  |-  ( a  x.  b )  e. 
_V
2928isseti 2870 . . . . . . . . 9  |-  E. w  w  =  ( a  x.  b )
3029rgenw 2686 . . . . . . . 8  |-  A. b  e.  B  E. w  w  =  ( a  x.  b )
31 r19.2z 3619 . . . . . . . 8  |-  ( ( B  =/=  (/)  /\  A. b  e.  B  E. w  w  =  (
a  x.  b ) )  ->  E. b  e.  B  E. w  w  =  ( a  x.  b ) )
3227, 30, 31sylancl 643 . . . . . . 7  |-  ( ph  ->  E. b  e.  B  E. w  w  =  ( a  x.  b
) )
33 rexcom4 2883 . . . . . . 7  |-  ( E. b  e.  B  E. w  w  =  (
a  x.  b )  <->  E. w E. b  e.  B  w  =  ( a  x.  b ) )
3432, 33sylib 188 . . . . . 6  |-  ( ph  ->  E. w E. b  e.  B  w  =  ( a  x.  b
) )
3534ralrimivw 2703 . . . . 5  |-  ( ph  ->  A. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b ) )
36 r19.2z 3619 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b ) )  ->  E. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b ) )
3726, 35, 36syl2anc 642 . . . 4  |-  ( ph  ->  E. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b ) )
38 rexcom4 2883 . . . 4  |-  ( E. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b )  <->  E. w E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b ) )
3937, 38sylib 188 . . 3  |-  ( ph  ->  E. w E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) )
40 n0 3540 . . . 4  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
4110exbii 1582 . . . 4  |-  ( E. w  w  e.  C  <->  E. w E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b ) )
4240, 41bitri 240 . . 3  |-  ( C  =/=  (/)  <->  E. w E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) )
4339, 42sylibr 203 . 2  |-  ( ph  ->  C  =/=  (/) )
44 suprcl 9804 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
4512, 44syl 15 . . . 4  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
46 suprcl 9804 . . . . 5  |-  ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
)  ->  sup ( B ,  RR ,  <  )  e.  RR )
4715, 46syl 15 . . . 4  |-  ( ph  ->  sup ( B ,  RR ,  <  )  e.  RR )
4845, 47remulcld 8953 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
499, 11supmullem1 9810 . . 3  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
50 breq2 4108 . . . . 5  |-  ( x  =  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  ->  (
w  <_  x  <->  w  <_  ( sup ( A ,  RR ,  <  )  x. 
sup ( B ,  RR ,  <  ) ) ) )
5150ralbidv 2639 . . . 4  |-  ( x  =  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  ->  ( A. w  e.  C  w  <_  x  <->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x. 
sup ( B ,  RR ,  <  ) ) ) )
5251rspcev 2960 . . 3  |-  ( ( ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR  /\  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
5348, 49, 52syl2anc 642 . 2  |-  ( ph  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
5425, 43, 533jca 1132 1  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
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   RRcr 8826   0cc0 8827    x. cmul 8832    < clt 8957    <_ cle 8958
This theorem is referenced by:  supmul  9812  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
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