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Theorem unxpwdom3 26609
Description: Weaker version of unxpwdom 7257 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
StepHypRef Expression
1 unxpwdom3.dv . . 3  |-  ( ph  ->  D  e.  X )
2 unxpwdom3.bv . . 3  |-  ( ph  ->  B  e.  W )
3 xpexg 4774 . . 3  |-  ( ( D  e.  X  /\  B  e.  W )  ->  ( D  X.  B
)  e.  _V )
41, 2, 3syl2anc 645 . 2  |-  ( ph  ->  ( D  X.  B
)  e.  _V )
5 unxpwdom3.ni . . . . . . 7  |-  ( ph  ->  -.  D  ~<_  A )
65adantr 453 . . . . . 6  |-  ( (
ph  /\  a  e.  C )  ->  -.  D  ~<_  A )
7 unxpwdom3.av . . . . . . . 8  |-  ( ph  ->  A  e.  V )
87ad2antrr 709 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  A  e.  V
)
9 oveq2 5786 . . . . . . . . . . . . . 14  |-  ( d  =  b  ->  (
a  .+  d )  =  ( a  .+  b ) )
109eleq1d 2322 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  (
( a  .+  d
)  e.  B  <->  ( a  .+  b )  e.  B
) )
1110notbid 287 . . . . . . . . . . . 12  |-  ( d  =  b  ->  ( -.  ( a  .+  d
)  e.  B  <->  -.  (
a  .+  b )  e.  B ) )
1211rcla4v 2848 . . . . . . . . . . 11  |-  ( b  e.  D  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  ->  -.  ( a  .+  b
)  e.  B ) )
1312adantl 454 . . . . . . . . . 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 1156 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
a  .+  b )  e.  ( A  u.  B
) )
16 elun 3277 . . . . . . . . . . . . 13  |-  ( ( a  .+  b )  e.  ( A  u.  B )  <->  ( (
a  .+  b )  e.  A  \/  (
a  .+  b )  e.  B ) )
1715, 16sylib 190 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
( a  .+  b
)  e.  A  \/  ( a  .+  b
)  e.  B ) )
1817orcomd 379 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
( a  .+  b
)  e.  B  \/  ( a  .+  b
)  e.  A ) )
1918ord 368 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  ( -.  ( a  .+  b
)  e.  B  -> 
( a  .+  b
)  e.  A ) )
2013, 19syld 42 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  -> 
( a  .+  b
)  e.  A ) )
2120impancom 429 . . . . . . . 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 425 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  C )  ->  (
( b  e.  D  /\  c  e.  D
)  ->  ( (
a  .+  b )  =  ( a  .+  c )  <->  b  =  c ) ) )
2423adantr 453 . . . . . . . 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 6856 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  ( A  e.  V  ->  D  ~<_  A ) )
268, 25mpd 16 . . . . . 6  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  D  ~<_  A )
276, 26mtand 643 . . . . 5  |-  ( (
ph  /\  a  e.  C )  ->  -.  A. d  e.  D  -.  ( a  .+  d
)  e.  B )
28 dfrex2 2529 . . . . 5  |-  ( E. d  e.  D  ( a  .+  d )  e.  B  <->  -.  A. d  e.  D  -.  (
a  .+  d )  e.  B )
2927, 28sylibr 205 . . . 4  |-  ( (
ph  /\  a  e.  C )  ->  E. d  e.  D  ( a  .+  d )  e.  B
)
30 simprr 736 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  ( a  .+  d )  e.  B
)
31 unxpwdom3.rc . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  d  e.  D )  /\  (
a  e.  C  /\  c  e.  C )
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
3231ancom1s 783 . . . . . . . . . . . . 13  |-  ( ( ( d  e.  D  /\  ph )  /\  (
a  e.  C  /\  c  e.  C )
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
3332adantllr 702 . . . . . . . . . . . 12  |-  ( ( ( ( d  e.  D  /\  ( a 
.+  d )  e.  B )  /\  ph )  /\  ( a  e.  C  /\  c  e.  C ) )  -> 
( ( c  .+  d )  =  ( a  .+  d )  <-> 
c  =  a ) )
34333impb 1152 . . . . . . . . . . 11  |-  ( ( ( ( d  e.  D  /\  ( a 
.+  d )  e.  B )  /\  ph )  /\  a  e.  C  /\  c  e.  C
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
3534riota5OLD 6285 . . . . . . . . . 10  |-  ( ( ( ( d  e.  D  /\  ( a 
.+  d )  e.  B )  /\  ph )  /\  a  e.  C
)  ->  ( iota_ c  e.  C ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
3635anasss 631 . . . . . . . . 9  |-  ( ( ( d  e.  D  /\  ( a  .+  d
)  e.  B )  /\  ( ph  /\  a  e.  C )
)  ->  ( iota_ c  e.  C ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
3736ancoms 441 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  ( iota_ c  e.  C ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
3837eqcomd 2261 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  ( a 
.+  d ) ) )
39 eqeq2 2265 . . . . . . . . . 10  |-  ( y  =  ( a  .+  d )  ->  (
( c  .+  d
)  =  y  <->  ( c  .+  d )  =  ( a  .+  d ) ) )
4039riotabidv 6260 . . . . . . . . 9  |-  ( y  =  ( a  .+  d )  ->  ( iota_ c  e.  C ( c  .+  d )  =  y )  =  ( iota_ c  e.  C
( c  .+  d
)  =  ( a 
.+  d ) ) )
4140eqeq2d 2267 . . . . . . . 8  |-  ( y  =  ( a  .+  d )  ->  (
a  =  ( iota_ c  e.  C ( c 
.+  d )  =  y )  <->  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  ( a 
.+  d ) ) ) )
4241rcla4ev 2852 . . . . . . 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 ) )
4330, 38, 42syl2anc 645 . . . . . 6  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  E. y  e.  B  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  y ) )
4443expr 601 . . . . 5  |-  ( ( ( ph  /\  a  e.  C )  /\  d  e.  D )  ->  (
( a  .+  d
)  e.  B  ->  E. y  e.  B  a  =  ( iota_ c  e.  C ( c 
.+  d )  =  y ) ) )
4544reximdva 2628 . . . 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 ) ) )
4629, 45mpd 16 . . 3  |-  ( (
ph  /\  a  e.  C )  ->  E. d  e.  D  E. y  e.  B  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  y ) )
47 vex 2760 . . . . . . . . 9  |-  d  e. 
_V
48 vex 2760 . . . . . . . . 9  |-  y  e. 
_V
4947, 48op1std 6050 . . . . . . . 8  |-  ( x  =  <. d ,  y
>.  ->  ( 1st `  x
)  =  d )
5049oveq2d 5794 . . . . . . 7  |-  ( x  =  <. d ,  y
>.  ->  ( c  .+  ( 1st `  x ) )  =  ( c 
.+  d ) )
5147, 48op2ndd 6051 . . . . . . 7  |-  ( x  =  <. d ,  y
>.  ->  ( 2nd `  x
)  =  y )
5250, 51eqeq12d 2270 . . . . . 6  |-  ( x  =  <. d ,  y
>.  ->  ( ( c 
.+  ( 1st `  x
) )  =  ( 2nd `  x )  <-> 
( c  .+  d
)  =  y ) )
5352riotabidv 6260 . . . . 5  |-  ( x  =  <. d ,  y
>.  ->  ( iota_ c  e.  C ( c  .+  ( 1st `  x ) )  =  ( 2nd `  x ) )  =  ( iota_ c  e.  C
( c  .+  d
)  =  y ) )
5453eqeq2d 2267 . . . 4  |-  ( x  =  <. d ,  y
>.  ->  ( a  =  ( iota_ c  e.  C
( c  .+  ( 1st `  x ) )  =  ( 2nd `  x
) )  <->  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  y ) ) )
5554rexxp 4802 . . 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 ) )
5646, 55sylibr 205 . 2  |-  ( (
ph  /\  a  e.  C )  ->  E. x  e.  ( D  X.  B
) a  =  (
iota_ c  e.  C
( c  .+  ( 1st `  x ) )  =  ( 2nd `  x
) ) )
574, 56wdomd 7249 1  |-  ( ph  ->  C  ~<_*  ( D  X.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517   _Vcvv 2757    u. cun 3111   <.cop 3603   class class class wbr 3983    X. cxp 4645   ` cfv 4659  (class class class)co 5778   1stc1st 6040   2ndc2nd 6041   iota_crio 6249    ~<_ cdom 6815    ~<_* cwdom 7225
This theorem is referenced by:  isnumbasgrplem2  26622
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-wdom 7227
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