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Theorem wdom2d2 27108
Description: Deduction for weak dominance by a cross product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
Hypotheses
Ref Expression
wdom2d2.a  |-  ( ph  ->  A  e.  V )
wdom2d2.b  |-  ( ph  ->  B  e.  W )
wdom2d2.c  |-  ( ph  ->  C  e.  X )
wdom2d2.o  |-  ( (
ph  /\  x  e.  A )  ->  E. y  e.  B  E. z  e.  C  x  =  X )
Assertion
Ref Expression
wdom2d2  |-  ( ph  ->  A  ~<_*  ( B  X.  C
) )
Distinct variable groups:    x, y,
z    x, X    x, A    x, B, y    x, C, y, z    ph, x
Allowed substitution hints:    ph( y, z)    A( y, z)    B( z)    V( x, y, z)    W( x, y, z)    X( y, z)

Proof of Theorem wdom2d2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 wdom2d2.a . 2  |-  ( ph  ->  A  e.  V )
2 wdom2d2.b . . 3  |-  ( ph  ->  B  e.  W )
3 wdom2d2.c . . 3  |-  ( ph  ->  C  e.  X )
4 xpexg 4991 . . 3  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  X.  C
)  e.  _V )
52, 3, 4syl2anc 644 . 2  |-  ( ph  ->  ( B  X.  C
)  e.  _V )
6 wdom2d2.o . . 3  |-  ( (
ph  /\  x  e.  A )  ->  E. y  e.  B  E. z  e.  C  x  =  X )
7 nfcsb1v 3285 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X
87nfeq2 2585 . . . 4  |-  F/ y  x  =  [_ ( 1st `  w )  / 
y ]_ [_ ( 2nd `  w )  /  z ]_ X
9 nfcv 2574 . . . . . 6  |-  F/_ z
( 1st `  w
)
10 nfcsb1v 3285 . . . . . 6  |-  F/_ z [_ ( 2nd `  w
)  /  z ]_ X
119, 10nfcsb 3287 . . . . 5  |-  F/_ z [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X
1211nfeq2 2585 . . . 4  |-  F/ z  x  =  [_ ( 1st `  w )  / 
y ]_ [_ ( 2nd `  w )  /  z ]_ X
13 nfv 1630 . . . 4  |-  F/ w  x  =  X
14 csbopeq1a 6402 . . . . 5  |-  ( w  =  <. y ,  z
>.  ->  [_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X  =  X )
1514eqeq2d 2449 . . . 4  |-  ( w  =  <. y ,  z
>.  ->  ( x  = 
[_ ( 1st `  w
)  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X 
<->  x  =  X ) )
168, 12, 13, 15rexxpf 5022 . . 3  |-  ( E. w  e.  ( B  X.  C ) x  =  [_ ( 1st `  w )  /  y ]_ [_ ( 2nd `  w
)  /  z ]_ X 
<->  E. y  e.  B  E. z  e.  C  x  =  X )
176, 16sylibr 205 . 2  |-  ( (
ph  /\  x  e.  A )  ->  E. w  e.  ( B  X.  C
) x  =  [_ ( 1st `  w )  /  y ]_ [_ ( 2nd `  w )  / 
z ]_ X )
181, 5, 17wdom2d 7550 1  |-  ( ph  ->  A  ~<_*  ( B  X.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958   [_csb 3253   <.cop 3819   class class class wbr 4214    X. cxp 4878   ` cfv 5456   1stc1st 6349   2ndc2nd 6350    ~<_* cwdom 7527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1st 6351  df-2nd 6352  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-wdom 7529
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