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Theorem supmullem2 10530
Description: Lemma for supmul 10531. (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 3112 . . . . 5  |-  w  e. 
_V
2 oveq1 6303 . . . . . . . . 9  |-  ( v  =  a  ->  (
v  x.  b )  =  ( a  x.  b ) )
32eqeq2d 2471 . . . . . . . 8  |-  ( v  =  a  ->  (
z  =  ( v  x.  b )  <->  z  =  ( a  x.  b
) ) )
43rexbidv 2968 . . . . . . 7  |-  ( v  =  a  ->  ( E. b  e.  B  z  =  ( v  x.  b )  <->  E. b  e.  B  z  =  ( a  x.  b
) ) )
54cbvrexv 3085 . . . . . 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 2461 . . . . . . 7  |-  ( z  =  w  ->  (
z  =  ( a  x.  b )  <->  w  =  ( a  x.  b
) ) )
762rexbidv 2975 . . . . . 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 257 . . . . 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 3249 . . . 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 1012 . . . . . . . . . 10  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
1312simp1d 1008 . . . . . . . . 9  |-  ( ph  ->  A  C_  RR )
1413sseld 3498 . . . . . . . 8  |-  ( ph  ->  ( a  e.  A  ->  a  e.  RR ) )
1511simp3bi 1013 . . . . . . . . . 10  |-  ( ph  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) )
1615simp1d 1008 . . . . . . . . 9  |-  ( ph  ->  B  C_  RR )
1716sseld 3498 . . . . . . . 8  |-  ( ph  ->  ( b  e.  B  ->  b  e.  RR ) )
1814, 17anim12d 563 . . . . . . 7  |-  ( ph  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
a  e.  RR  /\  b  e.  RR )
) )
19 remulcl 9594 . . . . . . 7  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( a  x.  b
)  e.  RR )
2018, 19syl6 33 . . . . . 6  |-  ( ph  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
a  x.  b )  e.  RR ) )
21 eleq1a 2540 . . . . . 6  |-  ( ( a  x.  b )  e.  RR  ->  (
w  =  ( a  x.  b )  ->  w  e.  RR )
)
2220, 21syl6 33 . . . . 5  |-  ( ph  ->  ( ( a  e.  A  /\  b  e.  B )  ->  (
w  =  ( a  x.  b )  ->  w  e.  RR )
) )
2322rexlimdvv 2955 . . . 4  |-  ( ph  ->  ( E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b )  ->  w  e.  RR ) )
2410, 23syl5bi 217 . . 3  |-  ( ph  ->  ( w  e.  C  ->  w  e.  RR ) )
2524ssrdv 3505 . 2  |-  ( ph  ->  C  C_  RR )
2612simp2d 1009 . . . . 5  |-  ( ph  ->  A  =/=  (/) )
2715simp2d 1009 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
28 ovex 6324 . . . . . . . . . 10  |-  ( a  x.  b )  e. 
_V
2928isseti 3115 . . . . . . . . 9  |-  E. w  w  =  ( a  x.  b )
3029rgenw 2818 . . . . . . . 8  |-  A. b  e.  B  E. w  w  =  ( a  x.  b )
31 r19.2z 3921 . . . . . . . 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 662 . . . . . . 7  |-  ( ph  ->  E. b  e.  B  E. w  w  =  ( a  x.  b
) )
33 rexcom4 3129 . . . . . . 7  |-  ( E. b  e.  B  E. w  w  =  (
a  x.  b )  <->  E. w E. b  e.  B  w  =  ( a  x.  b ) )
3432, 33sylib 196 . . . . . 6  |-  ( ph  ->  E. w E. b  e.  B  w  =  ( a  x.  b
) )
3534ralrimivw 2872 . . . . 5  |-  ( ph  ->  A. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b ) )
36 r19.2z 3921 . . . . 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 661 . . . 4  |-  ( ph  ->  E. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b ) )
38 rexcom4 3129 . . . 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 196 . . 3  |-  ( ph  ->  E. w E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) )
40 n0 3803 . . . 4  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
4110exbii 1668 . . . 4  |-  ( E. w  w  e.  C  <->  E. w E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b ) )
4240, 41bitri 249 . . 3  |-  ( C  =/=  (/)  <->  E. w E. a  e.  A  E. b  e.  B  w  =  ( a  x.  b
) )
4339, 42sylibr 212 . 2  |-  ( ph  ->  C  =/=  (/) )
44 suprcl 10523 . . . . 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 10523 . . . . 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 9641 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
499, 11supmullem1 10529 . . 3  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
50 breq2 4460 . . . . 5  |-  ( x  =  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  ->  (
w  <_  x  <->  w  <_  ( sup ( A ,  RR ,  <  )  x. 
sup ( B ,  RR ,  <  ) ) ) )
5150ralbidv 2896 . . . 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 3210 . . 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 661 . 2  |-  ( ph  ->  E. x  e.  RR  A. w  e.  C  w  <_  x )
5425, 43, 533jca 1176 1  |-  ( ph  ->  ( C  C_  RR  /\  C  =/=  (/)  /\  E. x  e.  RR  A. w  e.  C  w  <_  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> 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
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
This theorem is referenced by:  supmul  10531  sqrlem5  13092
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