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Theorem supmullem2 10501
Description: Lemma for supmul 10502. (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 3111 . . . . 5  |-  w  e. 
_V
2 oveq1 6284 . . . . . . . . 9  |-  ( v  =  a  ->  (
v  x.  b )  =  ( a  x.  b ) )
32eqeq2d 2476 . . . . . . . 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 3084 . . . . . 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 2466 . . . . . . 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 3248 . . . 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 1007 . . . . . . . . . 10  |-  ( ph  ->  ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x ) )
1312simp1d 1003 . . . . . . . . 9  |-  ( ph  ->  A  C_  RR )
1413sseld 3498 . . . . . . . 8  |-  ( ph  ->  ( a  e.  A  ->  a  e.  RR ) )
1511simp3bi 1008 . . . . . . . . . 10  |-  ( ph  ->  ( B  C_  RR  /\  B  =/=  (/)  /\  E. x  e.  RR  A. y  e.  B  y  <_  x ) )
1615simp1d 1003 . . . . . . . . 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 9568 . . . . . . 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 2545 . . . . . 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 2956 . . . 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 1004 . . . . 5  |-  ( ph  ->  A  =/=  (/) )
2715simp2d 1004 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
28 ovex 6302 . . . . . . . . . 10  |-  ( a  x.  b )  e. 
_V
2928isseti 3114 . . . . . . . . 9  |-  E. w  w  =  ( a  x.  b )
3029rgenw 2820 . . . . . . . 8  |-  A. b  e.  B  E. w  w  =  ( a  x.  b )
31 r19.2z 3912 . . . . . . . 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 3128 . . . . . . 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 2874 . . . . 5  |-  ( ph  ->  A. a  e.  A  E. w E. b  e.  B  w  =  ( a  x.  b ) )
36 r19.2z 3912 . . . . 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 3128 . . . 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 3789 . . . 4  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
4110exbii 1639 . . . 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 10494 . . . . 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 10494 . . . . 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 9615 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  e.  RR )
499, 11supmullem1 10500 . . 3  |-  ( ph  ->  A. w  e.  C  w  <_  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) ) )
50 breq2 4446 . . . . 5  |-  ( x  =  ( sup ( A ,  RR ,  <  )  x.  sup ( B ,  RR ,  <  ) )  ->  (
w  <_  x  <->  w  <_  ( sup ( A ,  RR ,  <  )  x. 
sup ( B ,  RR ,  <  ) ) ) )
5150ralbidv 2898 . . . 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 3209 . . 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 1171 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 968    = wceq 1374   E.wex 1591    e. wcel 1762   {cab 2447    =/= wne 2657   A.wral 2809   E.wrex 2810    C_ wss 3471   (/)c0 3780   class class class wbr 4442  (class class class)co 6277   supcsup 7891   RRcr 9482   0cc0 9483    x. cmul 9488    < clt 9619    <_ cle 9620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799
This theorem is referenced by:  supmul  10502  sqrlem5  13032
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