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Theorem pwssplit1 17576
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 17575 . 2  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
7 simp1 996 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  Mnd )
8 simp2 997 . . . . . . . . . 10  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
9 simp3 998 . . . . . . . . . 10  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
108, 9ssexd 4600 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
11 eqid 2467 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
122, 11, 4pwselbasb 14760 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  V  e.  _V )  ->  ( a  e.  C  <->  a : V --> ( Base `  W ) ) )
137, 10, 12syl2anc 661 . . . . . . . 8  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  -> 
( a  e.  C  <->  a : V --> ( Base `  W ) ) )
1413biimpa 484 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  a : V --> ( Base `  W ) )
15 fvex 5882 . . . . . . . . . 10  |-  ( 0g
`  W )  e. 
_V
1615fconst 5777 . . . . . . . . 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 999 . . . . . . . . . 10  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  W  e.  Mnd )
19 eqid 2467 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
2011, 19mndidcl 15811 . . . . . . . . . 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 4178 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  { ( 0g `  W ) }  C_  ( Base `  W )
)
23 fss 5745 . . . . . . . 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 661 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> ( Base `  W
) )
25 disjdif 3905 . . . . . . . 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 5754 . . . . . . 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 1227 . . . . . 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 1001 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  V  C_  U )
30 undif 3913 . . . . . . . 8  |-  ( V 
C_  U  <->  ( V  u.  ( U  \  V
) )  =  U )
3129, 30sylib 196 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( V  u.  ( U  \  V ) )  =  U )
32 unidm 3652 . . . . . . . 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 5732 . . . . . 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 210 . . . . 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 1000 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  U  e.  X )
371, 11, 3pwselbasb 14760 . . . . . 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 661 . . . . 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 232 . . . 4  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) )  e.  B )
405fvtresfn 5958 . . . . . 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 5294 . . . . . . 7  |-  ( ( a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) )  |`  V )  =  ( ( a  |`  V )  u.  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V ) )
43 ffn 5737 . . . . . . . . 9  |-  ( a : V --> ( Base `  W )  ->  a  Fn  V )
44 fnresdm 5696 . . . . . . . . 9  |-  ( a  Fn  V  ->  (
a  |`  V )  =  a )
4514, 43, 443syl 20 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  |`  V )  =  a )
46 incom 3696 . . . . . . . . . 10  |-  ( ( U  \  V )  i^i  V )  =  ( V  i^i  ( U  \  V ) )
4746, 25eqtri 2496 . . . . . . . . 9  |-  ( ( U  \  V )  i^i  V )  =  (/)
48 fnconstg 5779 . . . . . . . . . . 11  |-  ( ( 0g `  W )  e.  _V  ->  (
( U  \  V
)  X.  { ( 0g `  W ) } )  Fn  ( U  \  V ) )
4915, 48ax-mp 5 . . . . . . . . . 10  |-  ( ( U  \  V )  X.  { ( 0g
`  W ) } )  Fn  ( U 
\  V )
50 fnresdisj 5697 . . . . . . . . . 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 211 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( ( U 
\  V )  X. 
{ ( 0g `  W ) } )  |`  V )  =  (/) )
5345, 52uneq12d 3664 . . . . . . 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 2520 . . . . . 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 3815 . . . . . 6  |-  ( a  u.  (/) )  =  a
5654, 55syl6eq 2524 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V )  =  a )
5741, 56eqtr2d 2509 . . . 4  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  a  =  ( F `
 ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) ) )
58 fveq2 5872 . . . . . 6  |-  ( b  =  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  ->  ( F `  b )  =  ( F `  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) ) )
5958eqeq2d 2481 . . . . 5  |-  ( b  =  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  ->  ( a  =  ( F `  b
)  <->  a  =  ( F `  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) ) ) )
6059rspcev 3219 . . . 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 661 . . 3  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  E. b  e.  B  a  =  ( F `  b ) )
6261ralrimiva 2881 . 2  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  A. a  e.  C  E. b  e.  B  a  =  ( F `  b ) )
63 dffo3 6047 . 2  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  A. a  e.  C  E. b  e.  B  a  =  ( F `  b ) ) )
646, 62, 63sylanbrc 664 1  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B -onto-> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   _Vcvv 3118    \ cdif 3478    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   {csn 4033    |-> cmpt 4511    X. cxp 5003    |` cres 5007    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295   Basecbs 14507   0gc0g 14712    ^s cpws 14719   Mndcmnd 15793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-prds 14720  df-pws 14722  df-mgm 15746  df-sgrp 15785  df-mnd 15795
This theorem is referenced by:  pwslnmlem2  30967
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