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Theorem pwssplit1 17900
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 17899 . 2  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
7 simp1 994 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  Mnd )
8 simp2 995 . . . . . . . . . 10  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
9 simp3 996 . . . . . . . . . 10  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
108, 9ssexd 4584 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
11 eqid 2454 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
122, 11, 4pwselbasb 14977 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  V  e.  _V )  ->  ( a  e.  C  <->  a : V --> ( Base `  W ) ) )
137, 10, 12syl2anc 659 . . . . . . . 8  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  -> 
( a  e.  C  <->  a : V --> ( Base `  W ) ) )
1413biimpa 482 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  a : V --> ( Base `  W ) )
15 fvex 5858 . . . . . . . . . 10  |-  ( 0g
`  W )  e. 
_V
1615fconst 5753 . . . . . . . . 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 997 . . . . . . . . . 10  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  W  e.  Mnd )
19 eqid 2454 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
2011, 19mndidcl 16137 . . . . . . . . . 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 4161 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  { ( 0g `  W ) }  C_  ( Base `  W )
)
2317, 22fssd 5722 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> ( Base `  W
) )
24 disjdif 3888 . . . . . . . 8  |-  ( V  i^i  ( U  \  V ) )  =  (/)
2524a1i 11 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( V  i^i  ( U  \  V ) )  =  (/) )
26 fun 5730 . . . . . . 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 ) ) )
2714, 23, 25, 26syl21anc 1225 . . . . . 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 ) ) )
28 simpl3 999 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  V  C_  U )
29 undif 3896 . . . . . . . 8  |-  ( V 
C_  U  <->  ( V  u.  ( U  \  V
) )  =  U )
3028, 29sylib 196 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( V  u.  ( U  \  V ) )  =  U )
31 unidm 3633 . . . . . . . 8  |-  ( (
Base `  W )  u.  ( Base `  W
) )  =  (
Base `  W )
3231a1i 11 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( Base `  W
)  u.  ( Base `  W ) )  =  ( Base `  W
) )
3330, 32feq23d 5708 . . . . . 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
) ) )
3427, 33mpbid 210 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) ) : U --> ( Base `  W
) )
35 simpl2 998 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  U  e.  X )
361, 11, 3pwselbasb 14977 . . . . . 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 ) ) )
3718, 35, 36syl2anc 659 . . . . 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 ) ) )
3834, 37mpbird 232 . . . 4  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) )  e.  B )
395fvtresfn 5932 . . . . . 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 ) )
4038, 39syl 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 ) )
41 resundir 5276 . . . . . . 7  |-  ( ( a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) )  |`  V )  =  ( ( a  |`  V )  u.  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V ) )
42 ffn 5713 . . . . . . . . 9  |-  ( a : V --> ( Base `  W )  ->  a  Fn  V )
43 fnresdm 5672 . . . . . . . . 9  |-  ( a  Fn  V  ->  (
a  |`  V )  =  a )
4414, 42, 433syl 20 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  |`  V )  =  a )
45 incom 3677 . . . . . . . . . 10  |-  ( ( U  \  V )  i^i  V )  =  ( V  i^i  ( U  \  V ) )
4645, 24eqtri 2483 . . . . . . . . 9  |-  ( ( U  \  V )  i^i  V )  =  (/)
47 fnconstg 5755 . . . . . . . . . . 11  |-  ( ( 0g `  W )  e.  _V  ->  (
( U  \  V
)  X.  { ( 0g `  W ) } )  Fn  ( U  \  V ) )
4815, 47ax-mp 5 . . . . . . . . . 10  |-  ( ( U  \  V )  X.  { ( 0g
`  W ) } )  Fn  ( U 
\  V )
49 fnresdisj 5673 . . . . . . . . . 10  |-  ( ( ( U  \  V
)  X.  { ( 0g `  W ) } )  Fn  ( U  \  V )  -> 
( ( ( U 
\  V )  i^i 
V )  =  (/)  <->  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V )  =  (/) ) )
5048, 49mp1i 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 )  =  (/) ) )
5146, 50mpbii 211 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( ( U 
\  V )  X. 
{ ( 0g `  W ) } )  |`  V )  =  (/) )
5244, 51uneq12d 3645 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  |`  V )  u.  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V ) )  =  ( a  u.  (/) ) )
5341, 52syl5eq 2507 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V )  =  ( a  u.  (/) ) )
54 un0 3809 . . . . . 6  |-  ( a  u.  (/) )  =  a
5553, 54syl6eq 2511 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V )  =  a )
5640, 55eqtr2d 2496 . . . 4  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  a  =  ( F `
 ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) ) )
57 fveq2 5848 . . . . . 6  |-  ( b  =  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  ->  ( F `  b )  =  ( F `  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) ) )
5857eqeq2d 2468 . . . . 5  |-  ( b  =  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  ->  ( a  =  ( F `  b
)  <->  a  =  ( F `  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) ) ) )
5958rspcev 3207 . . . 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 ) )
6038, 56, 59syl2anc 659 . . 3  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  E. b  e.  B  a  =  ( F `  b ) )
6160ralrimiva 2868 . 2  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  A. a  e.  C  E. b  e.  B  a  =  ( F `  b ) )
62 dffo3 6022 . 2  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  A. a  e.  C  E. b  e.  B  a  =  ( F `  b ) ) )
636, 61, 62sylanbrc 662 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016    |-> cmpt 4497    X. cxp 4986    |` cres 4990    Fn wfn 5565   -->wf 5566   -onto->wfo 5568   ` cfv 5570  (class class class)co 6270   Basecbs 14716   0gc0g 14929    ^s cpws 14936   Mndcmnd 16118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-hom 14808  df-cco 14809  df-0g 14931  df-prds 14937  df-pws 14939  df-mgm 16071  df-sgrp 16110  df-mnd 16120
This theorem is referenced by:  pwslnmlem2  31278
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