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Theorem unxpwdom3 26668
Description: Weaker version of unxpwdom 7303 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 4800 . . 3  |-  ( ( D  e.  X  /\  B  e.  W )  ->  ( D  X.  B
)  e.  _V )
41, 2, 3syl2anc 642 . 2  |-  ( ph  ->  ( D  X.  B
)  e.  _V )
5 unxpwdom3.ni . . . . . . 7  |-  ( ph  ->  -.  D  ~<_  A )
65adantr 451 . . . . . 6  |-  ( (
ph  /\  a  e.  C )  ->  -.  D  ~<_  A )
7 unxpwdom3.av . . . . . . . 8  |-  ( ph  ->  A  e.  V )
87ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  A  e.  V
)
9 oveq2 5866 . . . . . . . . . . . . . 14  |-  ( d  =  b  ->  (
a  .+  d )  =  ( a  .+  b ) )
109eleq1d 2349 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  (
( a  .+  d
)  e.  B  <->  ( a  .+  b )  e.  B
) )
1110notbid 285 . . . . . . . . . . . 12  |-  ( d  =  b  ->  ( -.  ( a  .+  d
)  e.  B  <->  -.  (
a  .+  b )  e.  B ) )
1211rspcv 2880 . . . . . . . . . . 11  |-  ( b  e.  D  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  ->  -.  ( a  .+  b
)  e.  B ) )
1312adantl 452 . . . . . . . . . 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 1151 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
a  .+  b )  e.  ( A  u.  B
) )
16 elun 3316 . . . . . . . . . . . . 13  |-  ( ( a  .+  b )  e.  ( A  u.  B )  <->  ( (
a  .+  b )  e.  A  \/  (
a  .+  b )  e.  B ) )
1715, 16sylib 188 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
( a  .+  b
)  e.  A  \/  ( a  .+  b
)  e.  B ) )
1817orcomd 377 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  (
( a  .+  b
)  e.  B  \/  ( a  .+  b
)  e.  A ) )
1918ord 366 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  ( -.  ( a  .+  b
)  e.  B  -> 
( a  .+  b
)  e.  A ) )
2013, 19syld 40 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  C )  /\  b  e.  D )  ->  ( A. d  e.  D  -.  ( a  .+  d
)  e.  B  -> 
( a  .+  b
)  e.  A ) )
2120impancom 427 . . . . . . . 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 423 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  C )  ->  (
( b  e.  D  /\  c  e.  D
)  ->  ( (
a  .+  b )  =  ( a  .+  c )  <->  b  =  c ) ) )
2423adantr 451 . . . . . . . 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 6902 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  ( A  e.  V  ->  D  ~<_  A ) )
268, 25mpd 14 . . . . . 6  |-  ( ( ( ph  /\  a  e.  C )  /\  A. d  e.  D  -.  ( a  .+  d
)  e.  B )  ->  D  ~<_  A )
276, 26mtand 640 . . . . 5  |-  ( (
ph  /\  a  e.  C )  ->  -.  A. d  e.  D  -.  ( a  .+  d
)  e.  B )
28 dfrex2 2556 . . . . 5  |-  ( E. d  e.  D  ( a  .+  d )  e.  B  <->  -.  A. d  e.  D  -.  (
a  .+  d )  e.  B )
2927, 28sylibr 203 . . . 4  |-  ( (
ph  /\  a  e.  C )  ->  E. d  e.  D  ( a  .+  d )  e.  B
)
30 simprr 733 . . . . . . 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 780 . . . . . . . . . . . . 13  |-  ( ( ( d  e.  D  /\  ph )  /\  (
a  e.  C  /\  c  e.  C )
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
3332adantllr 699 . . . . . . . . . . . 12  |-  ( ( ( ( d  e.  D  /\  ( a 
.+  d )  e.  B )  /\  ph )  /\  ( a  e.  C  /\  c  e.  C ) )  -> 
( ( c  .+  d )  =  ( a  .+  d )  <-> 
c  =  a ) )
34333impb 1147 . . . . . . . . . . 11  |-  ( ( ( ( d  e.  D  /\  ( a 
.+  d )  e.  B )  /\  ph )  /\  a  e.  C  /\  c  e.  C
)  ->  ( (
c  .+  d )  =  ( a  .+  d )  <->  c  =  a ) )
3534riota5OLD 6331 . . . . . . . . . 10  |-  ( ( ( ( d  e.  D  /\  ( a 
.+  d )  e.  B )  /\  ph )  /\  a  e.  C
)  ->  ( iota_ c  e.  C ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
3635anasss 628 . . . . . . . . 9  |-  ( ( ( d  e.  D  /\  ( a  .+  d
)  e.  B )  /\  ( ph  /\  a  e.  C )
)  ->  ( iota_ c  e.  C ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
3736ancoms 439 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  ( iota_ c  e.  C ( c 
.+  d )  =  ( a  .+  d
) )  =  a )
3837eqcomd 2288 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  C )  /\  (
d  e.  D  /\  ( a  .+  d
)  e.  B ) )  ->  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  ( a 
.+  d ) ) )
39 eqeq2 2292 . . . . . . . . . 10  |-  ( y  =  ( a  .+  d )  ->  (
( c  .+  d
)  =  y  <->  ( c  .+  d )  =  ( a  .+  d ) ) )
4039riotabidv 6306 . . . . . . . . 9  |-  ( y  =  ( a  .+  d )  ->  ( iota_ c  e.  C ( c  .+  d )  =  y )  =  ( iota_ c  e.  C
( c  .+  d
)  =  ( a 
.+  d ) ) )
4140eqeq2d 2294 . . . . . . . 8  |-  ( y  =  ( a  .+  d )  ->  (
a  =  ( iota_ c  e.  C ( c 
.+  d )  =  y )  <->  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  ( a 
.+  d ) ) ) )
4241rspcev 2884 . . . . . . 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 642 . . . . . 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 598 . . . . 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 2655 . . . 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 14 . . 3  |-  ( (
ph  /\  a  e.  C )  ->  E. d  e.  D  E. y  e.  B  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  y ) )
47 vex 2791 . . . . . . . . 9  |-  d  e. 
_V
48 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
4947, 48op1std 6130 . . . . . . . 8  |-  ( x  =  <. d ,  y
>.  ->  ( 1st `  x
)  =  d )
5049oveq2d 5874 . . . . . . 7  |-  ( x  =  <. d ,  y
>.  ->  ( c  .+  ( 1st `  x ) )  =  ( c 
.+  d ) )
5147, 48op2ndd 6131 . . . . . . 7  |-  ( x  =  <. d ,  y
>.  ->  ( 2nd `  x
)  =  y )
5250, 51eqeq12d 2297 . . . . . 6  |-  ( x  =  <. d ,  y
>.  ->  ( ( c 
.+  ( 1st `  x
) )  =  ( 2nd `  x )  <-> 
( c  .+  d
)  =  y ) )
5352riotabidv 6306 . . . . 5  |-  ( x  =  <. d ,  y
>.  ->  ( iota_ c  e.  C ( c  .+  ( 1st `  x ) )  =  ( 2nd `  x ) )  =  ( iota_ c  e.  C
( c  .+  d
)  =  y ) )
5453eqeq2d 2294 . . . 4  |-  ( x  =  <. d ,  y
>.  ->  ( a  =  ( iota_ c  e.  C
( c  .+  ( 1st `  x ) )  =  ( 2nd `  x
) )  <->  a  =  ( iota_ c  e.  C
( c  .+  d
)  =  y ) ) )
5554rexxp 4828 . . 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 203 . 2  |-  ( (
ph  /\  a  e.  C )  ->  E. x  e.  ( D  X.  B
) a  =  (
iota_ c  e.  C
( c  .+  ( 1st `  x ) )  =  ( 2nd `  x
) ) )
574, 56wdomd 7295 1  |-  ( ph  ->  C  ~<_*  ( D  X.  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    u. cun 3150   <.cop 3643   class class class wbr 4023    X. cxp 4687   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   iota_crio 6297    ~<_ cdom 6861    ~<_* cwdom 7271
This theorem is referenced by:  isnumbasgrplem2  26681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-wdom 7273
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