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

Theorem lsmhash 15030
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 5899 . . . . 5  |-  ( T 
.(+)  U )  e.  _V
21a1i 10 . . . 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 4816 . . . . 5  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( T  X.  U )  e.  _V )
63, 4, 5syl2anc 642 . . . 4  |-  ( ph  ->  ( T  X.  U
)  e.  _V )
7 eqid 2296 . . . . . . . 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 2296 . . . . . . . 8  |-  ( proj
1 `  G )  =  ( proj 1 `  G )
147, 8, 9, 10, 3, 4, 11, 12, 13pj1f 15022 . . . . . . 7  |-  ( ph  ->  ( T ( proj
1 `  G ) U ) : ( T  .(+)  U ) --> T )
15 ffvelrn 5679 . . . . . . 7  |-  ( ( ( T ( proj
1 `  G ) U ) : ( T  .(+)  U ) --> T  /\  x  e.  ( T  .(+)  U )
)  ->  ( ( T ( proj 1 `  G ) U ) `
 x )  e.  T )
1614, 15sylan 457 . . . . . 6  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  ( ( T ( proj 1 `  G ) U ) `
 x )  e.  T )
177, 8, 9, 10, 3, 4, 11, 12, 13pj2f 15023 . . . . . . 7  |-  ( ph  ->  ( U ( proj
1 `  G ) T ) : ( T  .(+)  U ) --> U )
18 ffvelrn 5679 . . . . . . 7  |-  ( ( ( U ( proj
1 `  G ) T ) : ( T  .(+)  U ) --> U  /\  x  e.  ( T  .(+)  U )
)  ->  ( ( U ( proj 1 `  G ) T ) `
 x )  e.  U )
1917, 18sylan 457 . . . . . 6  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  ( ( U ( proj 1 `  G ) T ) `
 x )  e.  U )
20 opelxpi 4737 . . . . . 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 ) )
2116, 19, 20syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  <. ( ( T ( proj 1 `  G ) U ) `
 x ) ,  ( ( U (
proj 1 `  G ) T ) `  x
) >.  e.  ( T  X.  U ) )
2221ex 423 . . . 4  |-  ( ph  ->  ( x  e.  ( T  .(+)  U )  -> 
<. ( ( T (
proj 1 `  G ) U ) `  x
) ,  ( ( U ( proj 1 `  G ) T ) `
 x ) >.  e.  ( T  X.  U
) ) )
233, 4jca 518 . . . . . 6  |-  ( ph  ->  ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
) )
24 xp1st 6165 . . . . . . 7  |-  ( y  e.  ( T  X.  U )  ->  ( 1st `  y )  e.  T )
25 xp2nd 6166 . . . . . . 7  |-  ( y  e.  ( T  X.  U )  ->  ( 2nd `  y )  e.  U )
2624, 25jca 518 . . . . . 6  |-  ( y  e.  ( T  X.  U )  ->  (
( 1st `  y
)  e.  T  /\  ( 2nd `  y )  e.  U ) )
277, 8lsmelvali 14977 . . . . . 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 )
)
2823, 26, 27syl2an 463 . . . . 5  |-  ( (
ph  /\  y  e.  ( T  X.  U
) )  ->  (
( 1st `  y
) ( +g  `  G
) ( 2nd `  y
) )  e.  ( T  .(+)  U )
)
2928ex 423 . . . 4  |-  ( ph  ->  ( y  e.  ( T  X.  U )  ->  ( ( 1st `  y ) ( +g  `  G ) ( 2nd `  y ) )  e.  ( T  .(+)  U ) ) )
303adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  T  e.  (SubGrp `  G )
)
314adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  U  e.  (SubGrp `  G )
)
3211adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  ( T  i^i  U )  =  {  .0.  } )
3312adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  T  C_  ( Z `  U
) )
34 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  x  e.  ( T  .(+)  U ) )
3524ad2antll 709 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  ( 1st `  y )  e.  T )
3625ad2antll 709 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  ( 2nd `  y )  e.  U )
377, 8, 9, 10, 30, 31, 32, 33, 13, 34, 35, 36pj1eq 15025 . . . . . . 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 ) ) ) )
38 eqcom 2298 . . . . . . . 8  |-  ( ( ( T ( proj
1 `  G ) U ) `  x
)  =  ( 1st `  y )  <->  ( 1st `  y )  =  ( ( T ( proj
1 `  G ) U ) `  x
) )
39 eqcom 2298 . . . . . . . 8  |-  ( ( ( U ( proj
1 `  G ) T ) `  x
)  =  ( 2nd `  y )  <->  ( 2nd `  y )  =  ( ( U ( proj
1 `  G ) T ) `  x
) )
4038, 39anbi12i 678 . . . . . . 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
) ) )
4137, 40syl6bb 252 . . . . . 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
) ) ) )
42 eqop 6178 . . . . . . 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 ) ) ) )
4342ad2antll 709 . . . . . 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 ) ) ) )
4441, 43bitr4d 247 . . . . 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 ) >.
) )
4544ex 423 . . . 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 ) >.
) ) )
462, 6, 22, 29, 45en3d 6914 . . 3  |-  ( ph  ->  ( T  .(+)  U ) 
~~  ( T  X.  U ) )
47 hasheni 11363 . . 3  |-  ( ( T  .(+)  U )  ~~  ( T  X.  U
)  ->  ( # `  ( T  .(+)  U ) )  =  ( # `  ( T  X.  U ) ) )
4846, 47syl 15 . 2  |-  ( ph  ->  ( # `  ( T  .(+)  U ) )  =  ( # `  ( T  X.  U ) ) )
49 lsmhash.1 . . 3  |-  ( ph  ->  T  e.  Fin )
50 lsmhash.2 . . 3  |-  ( ph  ->  U  e.  Fin )
51 hashxp 11402 . . 3  |-  ( ( T  e.  Fin  /\  U  e.  Fin )  ->  ( # `  ( T  X.  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )
5249, 50, 51syl2anc 642 . 2  |-  ( ph  ->  ( # `  ( T  X.  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )
5348, 52eqtrd 2328 1  |-  ( ph  ->  ( # `  ( T  .(+)  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   {csn 3653   <.cop 3656   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    ~~ cen 6876   Fincfn 6879    x. cmul 8758   #chash 11353   +g cplusg 13224   0gc0g 13416  SubGrpcsubg 14631  Cntzccntz 14807   LSSumclsm 14961   proj
1cpj1 14962
This theorem is referenced by:  ablfacrp2  15318  ablfac1eulem  15323  ablfac1eu  15324  pgpfaclem2  15333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-pj1 14964
  Copyright terms: Public domain W3C validator