MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lsmhash Unicode version

Theorem lsmhash 15266
Description: The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
lsmhash.p  |-  .(+)  =  (
LSSum `  G )
lsmhash.o  |-  .0.  =  ( 0g `  G )
lsmhash.z  |-  Z  =  (Cntz `  G )
lsmhash.t  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
lsmhash.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
lsmhash.i  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
lsmhash.s  |-  ( ph  ->  T  C_  ( Z `  U ) )
lsmhash.1  |-  ( ph  ->  T  e.  Fin )
lsmhash.2  |-  ( ph  ->  U  e.  Fin )
Assertion
Ref Expression
lsmhash  |-  ( ph  ->  ( # `  ( T  .(+)  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )

Proof of Theorem lsmhash
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6047 . . . . 5  |-  ( T 
.(+)  U )  e.  _V
21a1i 11 . . . 4  |-  ( ph  ->  ( T  .(+)  U )  e.  _V )
3 lsmhash.t . . . . 5  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
4 lsmhash.u . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
5 xpexg 4931 . . . . 5  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( T  X.  U )  e.  _V )
63, 4, 5syl2anc 643 . . . 4  |-  ( ph  ->  ( T  X.  U
)  e.  _V )
7 eqid 2389 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
8 lsmhash.p . . . . . . . 8  |-  .(+)  =  (
LSSum `  G )
9 lsmhash.o . . . . . . . 8  |-  .0.  =  ( 0g `  G )
10 lsmhash.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
11 lsmhash.i . . . . . . . 8  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
12 lsmhash.s . . . . . . . 8  |-  ( ph  ->  T  C_  ( Z `  U ) )
13 eqid 2389 . . . . . . . 8  |-  ( proj
1 `  G )  =  ( proj 1 `  G )
147, 8, 9, 10, 3, 4, 11, 12, 13pj1f 15258 . . . . . . 7  |-  ( ph  ->  ( T ( proj
1 `  G ) U ) : ( T  .(+)  U ) --> T )
1514ffvelrnda 5811 . . . . . 6  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  ( ( T ( proj 1 `  G ) U ) `
 x )  e.  T )
167, 8, 9, 10, 3, 4, 11, 12, 13pj2f 15259 . . . . . . 7  |-  ( ph  ->  ( U ( proj
1 `  G ) T ) : ( T  .(+)  U ) --> U )
1716ffvelrnda 5811 . . . . . 6  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  ( ( U ( proj 1 `  G ) T ) `
 x )  e.  U )
18 opelxpi 4852 . . . . . 6  |-  ( ( ( ( T (
proj 1 `  G ) U ) `  x
)  e.  T  /\  ( ( U (
proj 1 `  G ) T ) `  x
)  e.  U )  ->  <. ( ( T ( proj 1 `  G ) U ) `
 x ) ,  ( ( U (
proj 1 `  G ) T ) `  x
) >.  e.  ( T  X.  U ) )
1915, 17, 18syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  <. ( ( T ( proj 1 `  G ) U ) `
 x ) ,  ( ( U (
proj 1 `  G ) T ) `  x
) >.  e.  ( T  X.  U ) )
2019ex 424 . . . 4  |-  ( ph  ->  ( x  e.  ( T  .(+)  U )  -> 
<. ( ( T (
proj 1 `  G ) U ) `  x
) ,  ( ( U ( proj 1 `  G ) T ) `
 x ) >.  e.  ( T  X.  U
) ) )
213, 4jca 519 . . . . . 6  |-  ( ph  ->  ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
) )
22 xp1st 6317 . . . . . . 7  |-  ( y  e.  ( T  X.  U )  ->  ( 1st `  y )  e.  T )
23 xp2nd 6318 . . . . . . 7  |-  ( y  e.  ( T  X.  U )  ->  ( 2nd `  y )  e.  U )
2422, 23jca 519 . . . . . 6  |-  ( y  e.  ( T  X.  U )  ->  (
( 1st `  y
)  e.  T  /\  ( 2nd `  y )  e.  U ) )
257, 8lsmelvali 15213 . . . . . 6  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( ( 1st `  y )  e.  T  /\  ( 2nd `  y )  e.  U
) )  ->  (
( 1st `  y
) ( +g  `  G
) ( 2nd `  y
) )  e.  ( T  .(+)  U )
)
2621, 24, 25syl2an 464 . . . . 5  |-  ( (
ph  /\  y  e.  ( T  X.  U
) )  ->  (
( 1st `  y
) ( +g  `  G
) ( 2nd `  y
) )  e.  ( T  .(+)  U )
)
2726ex 424 . . . 4  |-  ( ph  ->  ( y  e.  ( T  X.  U )  ->  ( ( 1st `  y ) ( +g  `  G ) ( 2nd `  y ) )  e.  ( T  .(+)  U ) ) )
283adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  T  e.  (SubGrp `  G )
)
294adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  U  e.  (SubGrp `  G )
)
3011adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  ( T  i^i  U )  =  {  .0.  } )
3112adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  T  C_  ( Z `  U
) )
32 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  x  e.  ( T  .(+)  U ) )
3322ad2antll 710 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  ( 1st `  y )  e.  T )
3423ad2antll 710 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  ( 2nd `  y )  e.  U )
357, 8, 9, 10, 28, 29, 30, 31, 13, 32, 33, 34pj1eq 15261 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  (
x  =  ( ( 1st `  y ) ( +g  `  G
) ( 2nd `  y
) )  <->  ( (
( T ( proj
1 `  G ) U ) `  x
)  =  ( 1st `  y )  /\  (
( U ( proj
1 `  G ) T ) `  x
)  =  ( 2nd `  y ) ) ) )
36 eqcom 2391 . . . . . . . 8  |-  ( ( ( T ( proj
1 `  G ) U ) `  x
)  =  ( 1st `  y )  <->  ( 1st `  y )  =  ( ( T ( proj
1 `  G ) U ) `  x
) )
37 eqcom 2391 . . . . . . . 8  |-  ( ( ( U ( proj
1 `  G ) T ) `  x
)  =  ( 2nd `  y )  <->  ( 2nd `  y )  =  ( ( U ( proj
1 `  G ) T ) `  x
) )
3836, 37anbi12i 679 . . . . . . 7  |-  ( ( ( ( T (
proj 1 `  G ) U ) `  x
)  =  ( 1st `  y )  /\  (
( U ( proj
1 `  G ) T ) `  x
)  =  ( 2nd `  y ) )  <->  ( ( 1st `  y )  =  ( ( T (
proj 1 `  G ) U ) `  x
)  /\  ( 2nd `  y )  =  ( ( U ( proj
1 `  G ) T ) `  x
) ) )
3935, 38syl6bb 253 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  (
x  =  ( ( 1st `  y ) ( +g  `  G
) ( 2nd `  y
) )  <->  ( ( 1st `  y )  =  ( ( T (
proj 1 `  G ) U ) `  x
)  /\  ( 2nd `  y )  =  ( ( U ( proj
1 `  G ) T ) `  x
) ) ) )
40 eqop 6330 . . . . . . 7  |-  ( y  e.  ( T  X.  U )  ->  (
y  =  <. (
( T ( proj
1 `  G ) U ) `  x
) ,  ( ( U ( proj 1 `  G ) T ) `
 x ) >.  <->  ( ( 1st `  y
)  =  ( ( T ( proj 1 `  G ) U ) `
 x )  /\  ( 2nd `  y )  =  ( ( U ( proj 1 `  G ) T ) `
 x ) ) ) )
4140ad2antll 710 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  (
y  =  <. (
( T ( proj
1 `  G ) U ) `  x
) ,  ( ( U ( proj 1 `  G ) T ) `
 x ) >.  <->  ( ( 1st `  y
)  =  ( ( T ( proj 1 `  G ) U ) `
 x )  /\  ( 2nd `  y )  =  ( ( U ( proj 1 `  G ) T ) `
 x ) ) ) )
4239, 41bitr4d 248 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  (
x  =  ( ( 1st `  y ) ( +g  `  G
) ( 2nd `  y
) )  <->  y  =  <. ( ( T (
proj 1 `  G ) U ) `  x
) ,  ( ( U ( proj 1 `  G ) T ) `
 x ) >.
) )
4342ex 424 . . . 4  |-  ( ph  ->  ( ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) )  ->  ( x  =  ( ( 1st `  y ) ( +g  `  G ) ( 2nd `  y ) )  <->  y  =  <. ( ( T (
proj 1 `  G ) U ) `  x
) ,  ( ( U ( proj 1 `  G ) T ) `
 x ) >.
) ) )
442, 6, 20, 27, 43en3d 7082 . . 3  |-  ( ph  ->  ( T  .(+)  U ) 
~~  ( T  X.  U ) )
45 hasheni 11561 . . 3  |-  ( ( T  .(+)  U )  ~~  ( T  X.  U
)  ->  ( # `  ( T  .(+)  U ) )  =  ( # `  ( T  X.  U ) ) )
4644, 45syl 16 . 2  |-  ( ph  ->  ( # `  ( T  .(+)  U ) )  =  ( # `  ( T  X.  U ) ) )
47 lsmhash.1 . . 3  |-  ( ph  ->  T  e.  Fin )
48 lsmhash.2 . . 3  |-  ( ph  ->  U  e.  Fin )
49 hashxp 11626 . . 3  |-  ( ( T  e.  Fin  /\  U  e.  Fin )  ->  ( # `  ( T  X.  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )
5047, 48, 49syl2anc 643 . 2  |-  ( ph  ->  ( # `  ( T  X.  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )
5146, 50eqtrd 2421 1  |-  ( ph  ->  ( # `  ( T  .(+)  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901    i^i cin 3264    C_ wss 3265   {csn 3759   <.cop 3762   class class class wbr 4155    X. cxp 4818   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289    ~~ cen 7044   Fincfn 7047    x. cmul 8930   #chash 11547   +g cplusg 13458   0gc0g 13652  SubGrpcsubg 14867  Cntzccntz 15043   LSSumclsm 15197   proj
1cpj1 15198
This theorem is referenced by:  ablfacrp2  15554  ablfac1eulem  15559  ablfac1eu  15560  pgpfaclem2  15569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-hash 11548  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742  df-sbg 14743  df-subg 14870  df-cntz 15045  df-lsm 15199  df-pj1 15200
  Copyright terms: Public domain W3C validator