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Theorem unxpwdom3 35659
Description: Weaker version of unxpwdom 8104 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 6607 . . 3  |-  ( ( D  e.  X  /\  B  e.  W )  ->  ( D  X.  B
)  e.  _V )
41, 2, 3syl2anc 665 . 2  |-  ( ph  ->  ( D  X.  B
)  e.  _V )
5 simprr 764 . . . . 5  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  ( a  .+  d )  e.  B
)
6 simplr 760 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  a  e.  C )
7 unxpwdom3.rc . . . . . . . . . 10  |-  ( ( ( ph  /\  d  e.  D )  /\  (
a  e.  C  /\  c  e.  C )
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
87an4s 833 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  c  e.  C )
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
98anassrs 652 . . . . . . . 8  |-  ( ( ( ( ph  /\  a  e.  C )  /\  d  e.  D
)  /\  c  e.  C )  ->  (
( c  .+  d
)  =  ( a 
.+  d )  <->  c  =  a ) )
109adantlrr 725 . . . . . . 7  |-  ( ( ( ( ph  /\  a  e.  C )  /\  ( d  e.  D  /\  ( a  .+  d
)  e.  B ) )  /\  c  e.  C )  ->  (
( c  .+  d
)  =  ( a 
.+  d )  <->  c  =  a ) )
116, 10riota5 6292 . . . . . 6  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  ( iota_ c  e.  C  ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
1211eqcomd 2437 . . . . 5  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  ( a 
.+  d ) ) )
13 eqeq2 2444 . . . . . . . 8  |-  ( y  =  ( a  .+  d )  ->  (
( c  .+  d
)  =  y  <->  ( c  .+  d )  =  ( a  .+  d ) ) )
1413riotabidv 6269 . . . . . . 7  |-  ( y  =  ( a  .+  d )  ->  ( iota_ c  e.  C  ( c  .+  d )  =  y )  =  ( iota_ c  e.  C  ( c  .+  d
)  =  ( a 
.+  d ) ) )
1514eqeq2d 2443 . . . . . 6  |-  ( y  =  ( a  .+  d )  ->  (
a  =  ( iota_ c  e.  C  ( c 
.+  d )  =  y )  <->  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  ( a 
.+  d ) ) ) )
1615rspcev 3188 . . . . 5  |-  ( ( ( 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 ) )
175, 12, 16syl2anc 665 . . . 4  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  E. y  e.  B  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  y ) )
18 unxpwdom3.ni . . . . . . 7  |-  ( ph  ->  -.  D  ~<_  A )
1918adantr 466 . . . . . 6  |-  ( (
ph  /\  a  e.  C )  ->  -.  D  ~<_  A )
20 unxpwdom3.av . . . . . . . 8  |-  ( ph  ->  A  e.  V )
2120ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  A  e.  V
)
22 oveq2 6313 . . . . . . . . . . . . . 14  |-  ( d  =  b  ->  (
a  .+  d )  =  ( a  .+  b ) )
2322eleq1d 2498 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  (
( a  .+  d
)  e.  B  <->  ( a  .+  b )  e.  B
) )
2423notbid 295 . . . . . . . . . . . 12  |-  ( d  =  b  ->  ( -.  ( a  .+  d
)  e.  B  <->  -.  (
a  .+  b )  e.  B ) )
2524rspcv 3184 . . . . . . . . . . 11  |-  ( b  e.  D  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  ->  -.  ( a  .+  b
)  e.  B ) )
2625adantl 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  ->  -.  ( a  .+  b
)  e.  B ) )
27 unxpwdom3.ov . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  C  /\  b  e.  D
)  ->  ( a  .+  b )  e.  ( A  u.  B ) )
28273expa 1205 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
a  .+  b )  e.  ( A  u.  B
) )
29 elun 3612 . . . . . . . . . . . . 13  |-  ( ( a  .+  b )  e.  ( A  u.  B )  <->  ( (
a  .+  b )  e.  A  \/  (
a  .+  b )  e.  B ) )
3028, 29sylib 199 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
( a  .+  b
)  e.  A  \/  ( a  .+  b
)  e.  B ) )
3130orcomd 389 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
( a  .+  b
)  e.  B  \/  ( a  .+  b
)  e.  A ) )
3231ord 378 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  ( -.  ( a  .+  b
)  e.  B  -> 
( a  .+  b
)  e.  A ) )
3326, 32syld 45 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  -> 
( a  .+  b
)  e.  A ) )
3433impancom 441 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  ( b  e.  D  ->  ( a  .+  b )  e.  A
) )
35 unxpwdom3.lc . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  C )  /\  (
b  e.  D  /\  c  e.  D )
)  ->  ( (
a  .+  b )  =  ( a  .+  c )  <->  b  =  c ) )
3635ex 435 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  C )  ->  (
( b  e.  D  /\  c  e.  D
)  ->  ( (
a  .+  b )  =  ( a  .+  c )  <->  b  =  c ) ) )
3736adantr 466 . . . . . . . 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 ) ) )
3834, 37dom2d 7617 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  ( A  e.  V  ->  D  ~<_  A ) )
3921, 38mpd 15 . . . . . 6  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  D  ~<_  A )
4019, 39mtand 663 . . . . 5  |-  ( (
ph  /\  a  e.  C )  ->  -.  A. d  e.  D  -.  ( a  .+  d
)  e.  B )
41 dfrex2 2883 . . . . 5  |-  ( E. d  e.  D  ( a  .+  d )  e.  B  <->  -.  A. d  e.  D  -.  (
a  .+  d )  e.  B )
4240, 41sylibr 215 . . . 4  |-  ( (
ph  /\  a  e.  C )  ->  E. d  e.  D  ( a  .+  d )  e.  B
)
4317, 42reximddv 2908 . . 3  |-  ( (
ph  /\  a  e.  C )  ->  E. d  e.  D  E. y  e.  B  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  y ) )
44 vex 3090 . . . . . . . . 9  |-  d  e. 
_V
45 vex 3090 . . . . . . . . 9  |-  y  e. 
_V
4644, 45op1std 6817 . . . . . . . 8  |-  ( x  =  <. d ,  y
>.  ->  ( 1st `  x
)  =  d )
4746oveq2d 6321 . . . . . . 7  |-  ( x  =  <. d ,  y
>.  ->  ( c  .+  ( 1st `  x ) )  =  ( c 
.+  d ) )
4844, 45op2ndd 6818 . . . . . . 7  |-  ( x  =  <. d ,  y
>.  ->  ( 2nd `  x
)  =  y )
4947, 48eqeq12d 2451 . . . . . 6  |-  ( x  =  <. d ,  y
>.  ->  ( ( c 
.+  ( 1st `  x
) )  =  ( 2nd `  x )  <-> 
( c  .+  d
)  =  y ) )
5049riotabidv 6269 . . . . 5  |-  ( x  =  <. d ,  y
>.  ->  ( iota_ c  e.  C  ( c  .+  ( 1st `  x ) )  =  ( 2nd `  x ) )  =  ( iota_ c  e.  C  ( c  .+  d
)  =  y ) )
5150eqeq2d 2443 . . . 4  |-  ( x  =  <. d ,  y
>.  ->  ( a  =  ( iota_ c  e.  C  ( c  .+  ( 1st `  x ) )  =  ( 2nd `  x
) )  <->  a  =  ( iota_ c  e.  C  ( c  .+  d
)  =  y ) ) )
5251rexxp 4997 . . 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 ) )
5343, 52sylibr 215 . 2  |-  ( (
ph  /\  a  e.  C )  ->  E. x  e.  ( D  X.  B
) a  =  (
iota_ c  e.  C  ( c  .+  ( 1st `  x ) )  =  ( 2nd `  x
) ) )
544, 53wdomd 8096 1  |-  ( ph  ->  C  ~<_*  ( D  X.  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   _Vcvv 3087    u. cun 3440   <.cop 4008   class class class wbr 4426    X. cxp 4852   ` cfv 5601   iota_crio 6266  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806    ~<_ cdom 7575    ~<_* cwdom 8072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-wdom 8074
This theorem is referenced by:  isnumbasgrplem2  35669
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