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Theorem unxpwdom3 29448
Description: Weaker version of unxpwdom 7804 where a function is required only to be cancellative, not an injection.  D and  B are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into  A, each row must hit an element of 
B; by column injectivity, each row can be identified in at least one way by the  B element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE
Hypotheses
Ref Expression
unxpwdom3.av  |-  ( ph  ->  A  e.  V )
unxpwdom3.bv  |-  ( ph  ->  B  e.  W )
unxpwdom3.dv  |-  ( ph  ->  D  e.  X )
unxpwdom3.ov  |-  ( (
ph  /\  a  e.  C  /\  b  e.  D
)  ->  ( a  .+  b )  e.  ( A  u.  B ) )
unxpwdom3.lc  |-  ( ( ( ph  /\  a  e.  C )  /\  (
b  e.  D  /\  c  e.  D )
)  ->  ( (
a  .+  b )  =  ( a  .+  c )  <->  b  =  c ) )
unxpwdom3.rc  |-  ( ( ( ph  /\  d  e.  D )  /\  (
a  e.  C  /\  c  e.  C )
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
unxpwdom3.ni  |-  ( ph  ->  -.  D  ~<_  A )
Assertion
Ref Expression
unxpwdom3  |-  ( ph  ->  C  ~<_*  ( D  X.  B
) )
Distinct variable groups:    a, b,
c, d, B    C, a, b, c, d    D, a, b, c, d    .+ , a,
b, c, d    ph, a,
b, c, d    A, b, c
Allowed substitution hints:    A( a, d)    V( a, b, c, d)    W( a, b, c, d)    X( a, b, c, d)

Proof of Theorem unxpwdom3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unxpwdom3.dv . . 3  |-  ( ph  ->  D  e.  X )
2 unxpwdom3.bv . . 3  |-  ( ph  ->  B  e.  W )
3 xpexg 6507 . . 3  |-  ( ( D  e.  X  /\  B  e.  W )  ->  ( D  X.  B
)  e.  _V )
41, 2, 3syl2anc 661 . 2  |-  ( ph  ->  ( D  X.  B
)  e.  _V )
5 unxpwdom3.ni . . . . . . 7  |-  ( ph  ->  -.  D  ~<_  A )
65adantr 465 . . . . . 6  |-  ( (
ph  /\  a  e.  C )  ->  -.  D  ~<_  A )
7 unxpwdom3.av . . . . . . . 8  |-  ( ph  ->  A  e.  V )
87ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  A  e.  V
)
9 oveq2 6099 . . . . . . . . . . . . . 14  |-  ( d  =  b  ->  (
a  .+  d )  =  ( a  .+  b ) )
109eleq1d 2509 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  (
( a  .+  d
)  e.  B  <->  ( a  .+  b )  e.  B
) )
1110notbid 294 . . . . . . . . . . . 12  |-  ( d  =  b  ->  ( -.  ( a  .+  d
)  e.  B  <->  -.  (
a  .+  b )  e.  B ) )
1211rspcv 3069 . . . . . . . . . . 11  |-  ( b  e.  D  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  ->  -.  ( a  .+  b
)  e.  B ) )
1312adantl 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  ->  -.  ( a  .+  b
)  e.  B ) )
14 unxpwdom3.ov . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  C  /\  b  e.  D
)  ->  ( a  .+  b )  e.  ( A  u.  B ) )
15143expa 1187 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
a  .+  b )  e.  ( A  u.  B
) )
16 elun 3497 . . . . . . . . . . . . 13  |-  ( ( a  .+  b )  e.  ( A  u.  B )  <->  ( (
a  .+  b )  e.  A  \/  (
a  .+  b )  e.  B ) )
1715, 16sylib 196 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
( a  .+  b
)  e.  A  \/  ( a  .+  b
)  e.  B ) )
1817orcomd 388 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
( a  .+  b
)  e.  B  \/  ( a  .+  b
)  e.  A ) )
1918ord 377 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  ( -.  ( a  .+  b
)  e.  B  -> 
( a  .+  b
)  e.  A ) )
2013, 19syld 44 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  -> 
( a  .+  b
)  e.  A ) )
2120impancom 440 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  ( b  e.  D  ->  ( a  .+  b )  e.  A
) )
22 unxpwdom3.lc . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  C )  /\  (
b  e.  D  /\  c  e.  D )
)  ->  ( (
a  .+  b )  =  ( a  .+  c )  <->  b  =  c ) )
2322ex 434 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  C )  ->  (
( b  e.  D  /\  c  e.  D
)  ->  ( (
a  .+  b )  =  ( a  .+  c )  <->  b  =  c ) ) )
2423adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  ( ( b  e.  D  /\  c  e.  D )  ->  (
( a  .+  b
)  =  ( a 
.+  c )  <->  b  =  c ) ) )
2521, 24dom2d 7350 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  ( A  e.  V  ->  D  ~<_  A ) )
268, 25mpd 15 . . . . . 6  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  D  ~<_  A )
276, 26mtand 659 . . . . 5  |-  ( (
ph  /\  a  e.  C )  ->  -.  A. d  e.  D  -.  ( a  .+  d
)  e.  B )
28 dfrex2 2728 . . . . 5  |-  ( E. d  e.  D  ( a  .+  d )  e.  B  <->  -.  A. d  e.  D  -.  (
a  .+  d )  e.  B )
2927, 28sylibr 212 . . . 4  |-  ( (
ph  /\  a  e.  C )  ->  E. d  e.  D  ( a  .+  d )  e.  B
)
30 simprr 756 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  ( a  .+  d )  e.  B
)
31 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  a  e.  C )
32 unxpwdom3.rc . . . . . . . . . . . 12  |-  ( ( ( ph  /\  d  e.  D )  /\  (
a  e.  C  /\  c  e.  C )
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
3332an4s 822 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  c  e.  C )
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
3433anassrs 648 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  a  e.  C )  /\  d  e.  D
)  /\  c  e.  C )  ->  (
( c  .+  d
)  =  ( a 
.+  d )  <->  c  =  a ) )
3534adantlrr 720 . . . . . . . . 9  |-  ( ( ( ( ph  /\  a  e.  C )  /\  ( d  e.  D  /\  ( a  .+  d
)  e.  B ) )  /\  c  e.  C )  ->  (
( c  .+  d
)  =  ( a 
.+  d )  <->  c  =  a ) )
3631, 35riota5 6078 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  ( iota_ c  e.  C  ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
3736eqcomd 2448 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  ( a 
.+  d ) ) )
38 eqeq2 2452 . . . . . . . . . 10  |-  ( y  =  ( a  .+  d )  ->  (
( c  .+  d
)  =  y  <->  ( c  .+  d )  =  ( a  .+  d ) ) )
3938riotabidv 6054 . . . . . . . . 9  |-  ( y  =  ( a  .+  d )  ->  ( iota_ c  e.  C  ( c  .+  d )  =  y )  =  ( iota_ c  e.  C  ( c  .+  d
)  =  ( a 
.+  d ) ) )
4039eqeq2d 2454 . . . . . . . 8  |-  ( y  =  ( a  .+  d )  ->  (
a  =  ( iota_ c  e.  C  ( c 
.+  d )  =  y )  <->  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  ( a 
.+  d ) ) ) )
4140rspcev 3073 . . . . . . 7  |-  ( ( ( a  .+  d
)  e.  B  /\  a  =  ( iota_ c  e.  C  ( c 
.+  d )  =  ( a  .+  d
) ) )  ->  E. y  e.  B  a  =  ( iota_ c  e.  C  ( c 
.+  d )  =  y ) )
4230, 37, 41syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  E. y  e.  B  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  y ) )
4342expr 615 . . . . 5  |-  ( ( ( ph  /\  a  e.  C )  /\  d  e.  D )  ->  (
( a  .+  d
)  e.  B  ->  E. y  e.  B  a  =  ( iota_ c  e.  C  ( c 
.+  d )  =  y ) ) )
4443reximdva 2828 . . . 4  |-  ( (
ph  /\  a  e.  C )  ->  ( E. d  e.  D  ( a  .+  d
)  e.  B  ->  E. d  e.  D  E. y  e.  B  a  =  ( iota_ c  e.  C  ( c 
.+  d )  =  y ) ) )
4529, 44mpd 15 . . 3  |-  ( (
ph  /\  a  e.  C )  ->  E. d  e.  D  E. y  e.  B  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  y ) )
46 vex 2975 . . . . . . . . 9  |-  d  e. 
_V
47 vex 2975 . . . . . . . . 9  |-  y  e. 
_V
4846, 47op1std 6587 . . . . . . . 8  |-  ( x  =  <. d ,  y
>.  ->  ( 1st `  x
)  =  d )
4948oveq2d 6107 . . . . . . 7  |-  ( x  =  <. d ,  y
>.  ->  ( c  .+  ( 1st `  x ) )  =  ( c 
.+  d ) )
5046, 47op2ndd 6588 . . . . . . 7  |-  ( x  =  <. d ,  y
>.  ->  ( 2nd `  x
)  =  y )
5149, 50eqeq12d 2457 . . . . . 6  |-  ( x  =  <. d ,  y
>.  ->  ( ( c 
.+  ( 1st `  x
) )  =  ( 2nd `  x )  <-> 
( c  .+  d
)  =  y ) )
5251riotabidv 6054 . . . . 5  |-  ( x  =  <. d ,  y
>.  ->  ( iota_ c  e.  C  ( c  .+  ( 1st `  x ) )  =  ( 2nd `  x ) )  =  ( iota_ c  e.  C  ( c  .+  d
)  =  y ) )
5352eqeq2d 2454 . . . 4  |-  ( x  =  <. d ,  y
>.  ->  ( a  =  ( iota_ c  e.  C  ( c  .+  ( 1st `  x ) )  =  ( 2nd `  x
) )  <->  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  y ) ) )
5453rexxp 4982 . . 3  |-  ( E. x  e.  ( D  X.  B ) a  =  ( iota_ c  e.  C  ( c  .+  ( 1st `  x ) )  =  ( 2nd `  x ) )  <->  E. d  e.  D  E. y  e.  B  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  y ) )
5545, 54sylibr 212 . 2  |-  ( (
ph  /\  a  e.  C )  ->  E. x  e.  ( D  X.  B
) a  =  (
iota_ c  e.  C  ( c  .+  ( 1st `  x ) )  =  ( 2nd `  x
) ) )
564, 55wdomd 7796 1  |-  ( ph  ->  C  ~<_*  ( D  X.  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   _Vcvv 2972    u. cun 3326   <.cop 3883   class class class wbr 4292    X. cxp 4838   ` cfv 5418   iota_crio 6051  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576    ~<_ cdom 7308    ~<_* cwdom 7772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-wdom 7774
This theorem is referenced by:  isnumbasgrplem2  29460
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