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Theorem supmullem2 9721
Description: Lemma for supmul 9722. (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 2791 . . . . 5  |-  w  e. 
_V
2 oveq1 5865 . . . . . . . . 9  |-  ( v  =  a  ->  (
v  x.  b )  =  ( a  x.  b ) )
32eqeq2d 2294 . . . . . . . 8  |-  ( v  =  a  ->  (
z  =  ( v  x.  b )  <->  z  =  ( a  x.  b
) ) )
43rexbidv 2564 . . . . . . 7  |-  ( v  =  a  ->  ( E. b  e.  B  z  =  ( v  x.  b )  <->  E. b  e.  B  z  =  ( a  x.  b
) ) )
54cbvrexv 2765 . . . . . 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 2289 . . . . . . 7  |-  ( z  =  w  ->  (
z  =  ( a  x.  b )  <->  w  =  ( a  x.  b
) ) )
762rexbidv 2586 . . . . . 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 2917 . . . 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 3179 . . . . . . . 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 3179 . . . . . . . 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 8822 . . . . . . 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 2352 . . . . . 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 2673 . . . 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 3185 . 2  |-  ( ph  ->  C  C_  RR )
2612simp2d 968 . . . . 5  |-  ( ph  ->  A  =/=  (/) )
2715simp2d 968 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
28 ovex 5883 . . . . . . . . . 10  |-  ( a  x.  b )  e. 
_V
2928isseti 2794 . . . . . . . . 9  |-  E. w  w  =  ( a  x.  b )
3029rgenw 2610 . . . . . . . 8  |-  A. b  e.  B  E. w  w  =  ( a  x.  b )
31 r19.2z 3543 . . . . . . . 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 2807 . . . . . . 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 2627 . . . . 5  |-  ( ph  ->  A. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b ) )
36 r19.2z 3543 . . . . 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 2807 . . . 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 3464 . . . 4  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
4110exbii 1569 . . . 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 9714 . . . . 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 9714 . . . . 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 8863 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
499, 11supmullem1 9720 . . 3  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
50 breq2 4027 . . . . 5  |-  ( x  =  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  ->  (
w  <_  x  <->  w  <_  ( sup ( A ,  RR ,  <  )  x. 
sup ( B ,  RR ,  <  ) ) ) )
5150ralbidv 2563 . . . 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 2884 . . 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 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   class class class wbr 4023  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737    x. cmul 8742    < clt 8867    <_ cle 8868
This theorem is referenced by:  supmul  9722  sqrlem5  11732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040
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