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Theorem supmullem2 9935
Description: Lemma for supmul 9936. (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 2923 . . . . 5  |-  w  e. 
_V
2 oveq1 6051 . . . . . . . . 9  |-  ( v  =  a  ->  (
v  x.  b )  =  ( a  x.  b ) )
32eqeq2d 2419 . . . . . . . 8  |-  ( v  =  a  ->  (
z  =  ( v  x.  b )  <->  z  =  ( a  x.  b
) ) )
43rexbidv 2691 . . . . . . 7  |-  ( v  =  a  ->  ( E. b  e.  B  z  =  ( v  x.  b )  <->  E. b  e.  B  z  =  ( a  x.  b
) ) )
54cbvrexv 2897 . . . . . 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 2414 . . . . . . 7  |-  ( z  =  w  ->  (
z  =  ( a  x.  b )  <->  w  =  ( a  x.  b
) ) )
762rexbidv 2713 . . . . . 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 249 . . . . 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 3049 . . . 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 973 . . . . . . . . . 10  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
1312simp1d 969 . . . . . . . . 9  |-  ( ph  ->  A  C_  RR )
1413sseld 3311 . . . . . . . 8  |-  ( ph  ->  ( a  e.  A  ->  a  e.  RR ) )
1511simp3bi 974 . . . . . . . . . 10  |-  ( ph  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) )
1615simp1d 969 . . . . . . . . 9  |-  ( ph  ->  B  C_  RR )
1716sseld 3311 . . . . . . . 8  |-  ( ph  ->  ( b  e.  B  ->  b  e.  RR ) )
1814, 17anim12d 547 . . . . . . 7  |-  ( ph  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
a  e.  RR  /\  b  e.  RR )
) )
19 remulcl 9035 . . . . . . 7  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( a  x.  b
)  e.  RR )
2018, 19syl6 31 . . . . . 6  |-  ( ph  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
a  x.  b )  e.  RR ) )
21 eleq1a 2477 . . . . . 6  |-  ( ( a  x.  b )  e.  RR  ->  (
w  =  ( a  x.  b )  ->  w  e.  RR )
)
2220, 21syl6 31 . . . . 5  |-  ( ph  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
w  =  ( a  x.  b )  ->  w  e.  RR )
) )
2322rexlimdvv 2800 . . . 4  |-  ( ph  ->  ( E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b )  ->  w  e.  RR ) )
2410, 23syl5bi 209 . . 3  |-  ( ph  ->  ( w  e.  C  ->  w  e.  RR ) )
2524ssrdv 3318 . 2  |-  ( ph  ->  C  C_  RR )
2612simp2d 970 . . . . 5  |-  ( ph  ->  A  =/=  (/) )
2715simp2d 970 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
28 ovex 6069 . . . . . . . . . 10  |-  ( a  x.  b )  e. 
_V
2928isseti 2926 . . . . . . . . 9  |-  E. w  w  =  ( a  x.  b )
3029rgenw 2737 . . . . . . . 8  |-  A. b  e.  B  E. w  w  =  ( a  x.  b )
31 r19.2z 3681 . . . . . . . 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 644 . . . . . . 7  |-  ( ph  ->  E. b  e.  B  E. w  w  =  ( a  x.  b
) )
33 rexcom4 2939 . . . . . . 7  |-  ( E. b  e.  B  E. w  w  =  (
a  x.  b )  <->  E. w E. b  e.  B  w  =  ( a  x.  b ) )
3432, 33sylib 189 . . . . . 6  |-  ( ph  ->  E. w E. b  e.  B  w  =  ( a  x.  b
) )
3534ralrimivw 2754 . . . . 5  |-  ( ph  ->  A. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b ) )
36 r19.2z 3681 . . . . 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 643 . . . 4  |-  ( ph  ->  E. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b ) )
38 rexcom4 2939 . . . 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 189 . . 3  |-  ( ph  ->  E. w E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) )
40 n0 3601 . . . 4  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
4110exbii 1589 . . . 4  |-  ( E. w  w  e.  C  <->  E. w E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b ) )
4240, 41bitri 241 . . 3  |-  ( C  =/=  (/)  <->  E. w E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) )
4339, 42sylibr 204 . 2  |-  ( ph  ->  C  =/=  (/) )
44 suprcl 9928 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
4512, 44syl 16 . . . 4  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
46 suprcl 9928 . . . . 5  |-  ( ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x
)  ->  sup ( B ,  RR ,  <  )  e.  RR )
4715, 46syl 16 . . . 4  |-  ( ph  ->  sup ( B ,  RR ,  <  )  e.  RR )
4845, 47remulcld 9076 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
499, 11supmullem1 9934 . . 3  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
50 breq2 4180 . . . . 5  |-  ( x  =  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  ->  (
w  <_  x  <->  w  <_  ( sup ( A ,  RR ,  <  )  x. 
sup ( B ,  RR ,  <  ) ) ) )
5150ralbidv 2690 . . . 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 3016 . . 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 643 . 2  |-  ( ph  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
5425, 43, 533jca 1134 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 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2394    =/= wne 2571   A.wral 2670   E.wrex 2671    C_ wss 3284   (/)c0 3592   class class class wbr 4176  (class class class)co 6044   supcsup 7407   RRcr 8949   0cc0 8950    x. cmul 8955    < clt 9080    <_ cle 9081
This theorem is referenced by:  supmul  9936  sqrlem5  12011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-riota 6512  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254
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