Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwssplit1 Structured version   Unicode version

Theorem pwssplit1 27165
Description: Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
pwssplit1.y  |-  Y  =  ( W  ^s  U )
pwssplit1.z  |-  Z  =  ( W  ^s  V )
pwssplit1.b  |-  B  =  ( Base `  Y
)
pwssplit1.c  |-  C  =  ( Base `  Z
)
pwssplit1.f  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
Assertion
Ref Expression
pwssplit1  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B -onto-> C )
Distinct variable groups:    x, Y    x, W    x, U    x, Z    x, V    x, B    x, C    x, X
Allowed substitution hint:    F( x)

Proof of Theorem pwssplit1
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwssplit1.y . . 3  |-  Y  =  ( W  ^s  U )
2 pwssplit1.z . . 3  |-  Z  =  ( W  ^s  V )
3 pwssplit1.b . . 3  |-  B  =  ( Base `  Y
)
4 pwssplit1.c . . 3  |-  C  =  ( Base `  Z
)
5 pwssplit1.f . . 3  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
61, 2, 3, 4, 5pwssplit0 27164 . 2  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
7 simp1 957 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  Mnd )
8 simp2 958 . . . . . . . . . 10  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
9 simp3 959 . . . . . . . . . 10  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
108, 9ssexd 4350 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
11 eqid 2436 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
122, 11, 4pwselbasb 13710 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  V  e.  _V )  ->  ( a  e.  C  <->  a : V --> ( Base `  W ) ) )
137, 10, 12syl2anc 643 . . . . . . . 8  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  -> 
( a  e.  C  <->  a : V --> ( Base `  W ) ) )
1413biimpa 471 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  a : V --> ( Base `  W ) )
15 fvex 5742 . . . . . . . . . 10  |-  ( 0g
`  W )  e. 
_V
1615fconst 5629 . . . . . . . . 9  |-  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) : ( U 
\  V ) --> { ( 0g `  W
) }
1716a1i 11 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> { ( 0g
`  W ) } )
18 simpl1 960 . . . . . . . . . 10  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  W  e.  Mnd )
19 eqid 2436 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
2011, 19mndidcl 14714 . . . . . . . . . 10  |-  ( W  e.  Mnd  ->  ( 0g `  W )  e.  ( Base `  W
) )
2118, 20syl 16 . . . . . . . . 9  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( 0g `  W
)  e.  ( Base `  W ) )
2221snssd 3943 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  { ( 0g `  W ) }  C_  ( Base `  W )
)
23 fss 5599 . . . . . . . 8  |-  ( ( ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> { ( 0g
`  W ) }  /\  { ( 0g
`  W ) } 
C_  ( Base `  W
) )  ->  (
( U  \  V
)  X.  { ( 0g `  W ) } ) : ( U  \  V ) --> ( Base `  W
) )
2417, 22, 23syl2anc 643 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> ( Base `  W
) )
25 disjdif 3700 . . . . . . . 8  |-  ( V  i^i  ( U  \  V ) )  =  (/)
2625a1i 11 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( V  i^i  ( U  \  V ) )  =  (/) )
27 fun 5607 . . . . . . 7  |-  ( ( ( a : V --> ( Base `  W )  /\  ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> ( Base `  W
) )  /\  ( V  i^i  ( U  \  V ) )  =  (/) )  ->  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) : ( V  u.  ( U  \  V ) ) --> ( ( Base `  W
)  u.  ( Base `  W ) ) )
2814, 24, 26, 27syl21anc 1183 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) ) : ( V  u.  ( U  \  V ) ) --> ( ( Base `  W
)  u.  ( Base `  W ) ) )
29 simpl3 962 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  V  C_  U )
30 undif 3708 . . . . . . . 8  |-  ( V 
C_  U  <->  ( V  u.  ( U  \  V
) )  =  U )
3129, 30sylib 189 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( V  u.  ( U  \  V ) )  =  U )
32 unidm 3490 . . . . . . . 8  |-  ( (
Base `  W )  u.  ( Base `  W
) )  =  (
Base `  W )
3332a1i 11 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( Base `  W
)  u.  ( Base `  W ) )  =  ( Base `  W
) )
3431, 33feq23d 5588 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) : ( V  u.  ( U  \  V ) ) --> ( ( Base `  W )  u.  ( Base `  W ) )  <-> 
( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) ) : U --> ( Base `  W
) ) )
3528, 34mpbid 202 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) ) : U --> ( Base `  W
) )
36 simpl2 961 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  U  e.  X )
371, 11, 3pwselbasb 13710 . . . . . 6  |-  ( ( W  e.  Mnd  /\  U  e.  X )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  e.  B  <->  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) : U --> ( Base `  W ) ) )
3818, 36, 37syl2anc 643 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  e.  B  <->  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) : U --> ( Base `  W ) ) )
3935, 38mpbird 224 . . . 4  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) )  e.  B )
405fvtresfn 26744 . . . . . 6  |-  ( ( a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) )  e.  B  ->  ( F `  (
a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) ) )  =  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V ) )
4139, 40syl 16 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( F `  (
a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) ) )  =  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V ) )
42 resundir 5161 . . . . . . 7  |-  ( ( a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) )  |`  V )  =  ( ( a  |`  V )  u.  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V ) )
43 ffn 5591 . . . . . . . . 9  |-  ( a : V --> ( Base `  W )  ->  a  Fn  V )
44 fnresdm 5554 . . . . . . . . 9  |-  ( a  Fn  V  ->  (
a  |`  V )  =  a )
4514, 43, 443syl 19 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  |`  V )  =  a )
46 incom 3533 . . . . . . . . . 10  |-  ( ( U  \  V )  i^i  V )  =  ( V  i^i  ( U  \  V ) )
4746, 25eqtri 2456 . . . . . . . . 9  |-  ( ( U  \  V )  i^i  V )  =  (/)
48 fnconstg 5631 . . . . . . . . . . 11  |-  ( ( 0g `  W )  e.  _V  ->  (
( U  \  V
)  X.  { ( 0g `  W ) } )  Fn  ( U  \  V ) )
4915, 48ax-mp 8 . . . . . . . . . 10  |-  ( ( U  \  V )  X.  { ( 0g
`  W ) } )  Fn  ( U 
\  V )
50 fnresdisj 5555 . . . . . . . . . 10  |-  ( ( ( U  \  V
)  X.  { ( 0g `  W ) } )  Fn  ( U  \  V )  -> 
( ( ( U 
\  V )  i^i 
V )  =  (/)  <->  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V )  =  (/) ) )
5149, 50mp1i 12 . . . . . . . . 9  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( ( U 
\  V )  i^i 
V )  =  (/)  <->  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V )  =  (/) ) )
5247, 51mpbii 203 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( ( U 
\  V )  X. 
{ ( 0g `  W ) } )  |`  V )  =  (/) )
5345, 52uneq12d 3502 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  |`  V )  u.  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V ) )  =  ( a  u.  (/) ) )
5442, 53syl5eq 2480 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V )  =  ( a  u.  (/) ) )
55 un0 3652 . . . . . 6  |-  ( a  u.  (/) )  =  a
5654, 55syl6eq 2484 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V )  =  a )
5741, 56eqtr2d 2469 . . . 4  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  a  =  ( F `
 ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) ) )
58 fveq2 5728 . . . . . 6  |-  ( b  =  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  ->  ( F `  b )  =  ( F `  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) ) )
5958eqeq2d 2447 . . . . 5  |-  ( b  =  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  ->  ( a  =  ( F `  b
)  <->  a  =  ( F `  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) ) ) )
6059rspcev 3052 . . . 4  |-  ( ( ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) )  e.  B  /\  a  =  ( F `  (
a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) ) ) )  ->  E. b  e.  B  a  =  ( F `  b ) )
6139, 57, 60syl2anc 643 . . 3  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  E. b  e.  B  a  =  ( F `  b ) )
6261ralrimiva 2789 . 2  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  A. a  e.  C  E. b  e.  B  a  =  ( F `  b ) )
63 dffo3 5884 . 2  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  A. a  e.  C  E. b  e.  B  a  =  ( F `  b ) ) )
646, 62, 63sylanbrc 646 1  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B -onto-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   _Vcvv 2956    \ cdif 3317    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   {csn 3814    e. cmpt 4266    X. cxp 4876    |` cres 4880    Fn wfn 5449   -->wf 5450   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081   Basecbs 13469    ^s cpws 13670   0gc0g 13723   Mndcmnd 14684
This theorem is referenced by:  pwslnmlem2  27172
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-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  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-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  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-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-hom 13553  df-cco 13554  df-prds 13671  df-pws 13673  df-0g 13727  df-mnd 14690
  Copyright terms: Public domain W3C validator