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Theorem unxpwdom3 27233
Description: Weaker version of unxpwdom 7557 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 4989 . . 3  |-  ( ( D  e.  X  /\  B  e.  W )  ->  ( D  X.  B
)  e.  _V )
41, 2, 3syl2anc 643 . 2  |-  ( ph  ->  ( D  X.  B
)  e.  _V )
5 unxpwdom3.ni . . . . . . 7  |-  ( ph  ->  -.  D  ~<_  A )
65adantr 452 . . . . . 6  |-  ( (
ph  /\  a  e.  C )  ->  -.  D  ~<_  A )
7 unxpwdom3.av . . . . . . . 8  |-  ( ph  ->  A  e.  V )
87ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  A  e.  V
)
9 oveq2 6089 . . . . . . . . . . . . . 14  |-  ( d  =  b  ->  (
a  .+  d )  =  ( a  .+  b ) )
109eleq1d 2502 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  (
( a  .+  d
)  e.  B  <->  ( a  .+  b )  e.  B
) )
1110notbid 286 . . . . . . . . . . . 12  |-  ( d  =  b  ->  ( -.  ( a  .+  d
)  e.  B  <->  -.  (
a  .+  b )  e.  B ) )
1211rspcv 3048 . . . . . . . . . . 11  |-  ( b  e.  D  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  ->  -.  ( a  .+  b
)  e.  B ) )
1312adantl 453 . . . . . . . . . 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 1153 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
a  .+  b )  e.  ( A  u.  B
) )
16 elun 3488 . . . . . . . . . . . . 13  |-  ( ( a  .+  b )  e.  ( A  u.  B )  <->  ( (
a  .+  b )  e.  A  \/  (
a  .+  b )  e.  B ) )
1715, 16sylib 189 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
( a  .+  b
)  e.  A  \/  ( a  .+  b
)  e.  B ) )
1817orcomd 378 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
( a  .+  b
)  e.  B  \/  ( a  .+  b
)  e.  A ) )
1918ord 367 . . . . . . . . . 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 428 . . . . . . . 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 424 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  C )  ->  (
( b  e.  D  /\  c  e.  D
)  ->  ( (
a  .+  b )  =  ( a  .+  c )  <->  b  =  c ) ) )
2423adantr 452 . . . . . . . 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 7148 . . . . . . 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 641 . . . . 5  |-  ( (
ph  /\  a  e.  C )  ->  -.  A. d  e.  D  -.  ( a  .+  d
)  e.  B )
28 dfrex2 2718 . . . . 5  |-  ( E. d  e.  D  ( a  .+  d )  e.  B  <->  -.  A. d  e.  D  -.  (
a  .+  d )  e.  B )
2927, 28sylibr 204 . . . 4  |-  ( (
ph  /\  a  e.  C )  ->  E. d  e.  D  ( a  .+  d )  e.  B
)
30 simprr 734 . . . . . . 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 781 . . . . . . . . . . . . 13  |-  ( ( ( d  e.  D  /\  ph )  /\  (
a  e.  C  /\  c  e.  C )
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
3332adantllr 700 . . . . . . . . . . . 12  |-  ( ( ( ( d  e.  D  /\  ( a 
.+  d )  e.  B )  /\  ph )  /\  ( a  e.  C  /\  c  e.  C ) )  -> 
( ( c  .+  d )  =  ( a  .+  d )  <-> 
c  =  a ) )
34333impb 1149 . . . . . . . . . . 11  |-  ( ( ( ( d  e.  D  /\  ( a 
.+  d )  e.  B )  /\  ph )  /\  a  e.  C  /\  c  e.  C
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
3534riota5OLD 6576 . . . . . . . . . 10  |-  ( ( ( ( d  e.  D  /\  ( a 
.+  d )  e.  B )  /\  ph )  /\  a  e.  C
)  ->  ( iota_ c  e.  C ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
3635anasss 629 . . . . . . . . 9  |-  ( ( ( d  e.  D  /\  ( a  .+  d
)  e.  B )  /\  ( ph  /\  a  e.  C )
)  ->  ( iota_ c  e.  C ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
3736ancoms 440 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  ( iota_ c  e.  C ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
3837eqcomd 2441 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  ( a 
.+  d ) ) )
39 eqeq2 2445 . . . . . . . . . 10  |-  ( y  =  ( a  .+  d )  ->  (
( c  .+  d
)  =  y  <->  ( c  .+  d )  =  ( a  .+  d ) ) )
4039riotabidv 6551 . . . . . . . . 9  |-  ( y  =  ( a  .+  d )  ->  ( iota_ c  e.  C ( c  .+  d )  =  y )  =  ( iota_ c  e.  C
( c  .+  d
)  =  ( a 
.+  d ) ) )
4140eqeq2d 2447 . . . . . . . 8  |-  ( y  =  ( a  .+  d )  ->  (
a  =  ( iota_ c  e.  C ( c 
.+  d )  =  y )  <->  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  ( a 
.+  d ) ) ) )
4241rspcev 3052 . . . . . . 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 643 . . . . . 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 599 . . . . 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 2818 . . . 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 15 . . 3  |-  ( (
ph  /\  a  e.  C )  ->  E. d  e.  D  E. y  e.  B  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  y ) )
47 vex 2959 . . . . . . . . 9  |-  d  e. 
_V
48 vex 2959 . . . . . . . . 9  |-  y  e. 
_V
4947, 48op1std 6357 . . . . . . . 8  |-  ( x  =  <. d ,  y
>.  ->  ( 1st `  x
)  =  d )
5049oveq2d 6097 . . . . . . 7  |-  ( x  =  <. d ,  y
>.  ->  ( c  .+  ( 1st `  x ) )  =  ( c 
.+  d ) )
5147, 48op2ndd 6358 . . . . . . 7  |-  ( x  =  <. d ,  y
>.  ->  ( 2nd `  x
)  =  y )
5250, 51eqeq12d 2450 . . . . . 6  |-  ( x  =  <. d ,  y
>.  ->  ( ( c 
.+  ( 1st `  x
) )  =  ( 2nd `  x )  <-> 
( c  .+  d
)  =  y ) )
5352riotabidv 6551 . . . . 5  |-  ( x  =  <. d ,  y
>.  ->  ( iota_ c  e.  C ( c  .+  ( 1st `  x ) )  =  ( 2nd `  x ) )  =  ( iota_ c  e.  C
( c  .+  d
)  =  y ) )
5453eqeq2d 2447 . . . 4  |-  ( x  =  <. d ,  y
>.  ->  ( a  =  ( iota_ c  e.  C
( c  .+  ( 1st `  x ) )  =  ( 2nd `  x
) )  <->  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  y ) ) )
5554rexxp 5017 . . 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 204 . 2  |-  ( (
ph  /\  a  e.  C )  ->  E. x  e.  ( D  X.  B
) a  =  (
iota_ c  e.  C
( c  .+  ( 1st `  x ) )  =  ( 2nd `  x
) ) )
574, 56wdomd 7549 1  |-  ( ph  ->  C  ~<_*  ( D  X.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   _Vcvv 2956    u. cun 3318   <.cop 3817   class class class wbr 4212    X. cxp 4876   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   iota_crio 6542    ~<_ cdom 7107    ~<_* cwdom 7525
This theorem is referenced by:  isnumbasgrplem2  27246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-wdom 7527
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