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Theorem lsmhash 16596
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 6320 . . . . 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 6597 . . . . 5  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( T  X.  U )  e.  _V )
63, 4, 5syl2anc 661 . . . 4  |-  ( ph  ->  ( T  X.  U
)  e.  _V )
7 eqid 2467 . . . . . . . 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 2467 . . . . . . . 8  |-  ( proj1 `  G )  =  ( proj1 `  G )
147, 8, 9, 10, 3, 4, 11, 12, 13pj1f 16588 . . . . . . 7  |-  ( ph  ->  ( T ( proj1 `  G ) U ) : ( T  .(+)  U ) --> T )
1514ffvelrnda 6032 . . . . . 6  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  ( ( T ( proj1 `  G ) U ) `
 x )  e.  T )
167, 8, 9, 10, 3, 4, 11, 12, 13pj2f 16589 . . . . . . 7  |-  ( ph  ->  ( U ( proj1 `  G ) T ) : ( T  .(+)  U ) --> U )
1716ffvelrnda 6032 . . . . . 6  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  ( ( U ( proj1 `  G ) T ) `
 x )  e.  U )
18 opelxpi 5037 . . . . . 6  |-  ( ( ( ( T (
proj1 `  G
) U ) `  x )  e.  T  /\  ( ( U (
proj1 `  G
) T ) `  x )  e.  U
)  ->  <. ( ( T ( proj1 `  G ) U ) `
 x ) ,  ( ( U (
proj1 `  G
) T ) `  x ) >.  e.  ( T  X.  U ) )
1915, 17, 18syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  <. ( ( T ( proj1 `  G ) U ) `
 x ) ,  ( ( U (
proj1 `  G
) T ) `  x ) >.  e.  ( T  X.  U ) )
2019ex 434 . . . 4  |-  ( ph  ->  ( x  e.  ( T  .(+)  U )  -> 
<. ( ( T (
proj1 `  G
) U ) `  x ) ,  ( ( U ( proj1 `  G ) T ) `  x
) >.  e.  ( T  X.  U ) ) )
213, 4jca 532 . . . . . 6  |-  ( ph  ->  ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
) )
22 xp1st 6825 . . . . . . 7  |-  ( y  e.  ( T  X.  U )  ->  ( 1st `  y )  e.  T )
23 xp2nd 6826 . . . . . . 7  |-  ( y  e.  ( T  X.  U )  ->  ( 2nd `  y )  e.  U )
2422, 23jca 532 . . . . . 6  |-  ( y  e.  ( T  X.  U )  ->  (
( 1st `  y
)  e.  T  /\  ( 2nd `  y )  e.  U ) )
257, 8lsmelvali 16543 . . . . . 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 477 . . . . 5  |-  ( (
ph  /\  y  e.  ( T  X.  U
) )  ->  (
( 1st `  y
) ( +g  `  G
) ( 2nd `  y
) )  e.  ( T  .(+)  U )
)
2726ex 434 . . . 4  |-  ( ph  ->  ( y  e.  ( T  X.  U )  ->  ( ( 1st `  y ) ( +g  `  G ) ( 2nd `  y ) )  e.  ( T  .(+)  U ) ) )
283adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  T  e.  (SubGrp `  G )
)
294adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  U  e.  (SubGrp `  G )
)
3011adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  ( T  i^i  U )  =  {  .0.  } )
3112adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  T  C_  ( Z `  U
) )
32 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  x  e.  ( T  .(+)  U ) )
3322ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  ( 1st `  y )  e.  T )
3423ad2antll 728 . . . . . . . 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 16591 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  (
x  =  ( ( 1st `  y ) ( +g  `  G
) ( 2nd `  y
) )  <->  ( (
( T ( proj1 `  G ) U ) `  x
)  =  ( 1st `  y )  /\  (
( U ( proj1 `  G ) T ) `  x
)  =  ( 2nd `  y ) ) ) )
36 eqcom 2476 . . . . . . . 8  |-  ( ( ( T ( proj1 `  G ) U ) `  x
)  =  ( 1st `  y )  <->  ( 1st `  y )  =  ( ( T ( proj1 `  G ) U ) `  x
) )
37 eqcom 2476 . . . . . . . 8  |-  ( ( ( U ( proj1 `  G ) T ) `  x
)  =  ( 2nd `  y )  <->  ( 2nd `  y )  =  ( ( U ( proj1 `  G ) T ) `  x
) )
3836, 37anbi12i 697 . . . . . . 7  |-  ( ( ( ( T (
proj1 `  G
) U ) `  x )  =  ( 1st `  y )  /\  ( ( U ( proj1 `  G ) T ) `
 x )  =  ( 2nd `  y
) )  <->  ( ( 1st `  y )  =  ( ( T (
proj1 `  G
) U ) `  x )  /\  ( 2nd `  y )  =  ( ( U (
proj1 `  G
) T ) `  x ) ) )
3935, 38syl6bb 261 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  (
x  =  ( ( 1st `  y ) ( +g  `  G
) ( 2nd `  y
) )  <->  ( ( 1st `  y )  =  ( ( T (
proj1 `  G
) U ) `  x )  /\  ( 2nd `  y )  =  ( ( U (
proj1 `  G
) T ) `  x ) ) ) )
40 eqop 6835 . . . . . . 7  |-  ( y  e.  ( T  X.  U )  ->  (
y  =  <. (
( T ( proj1 `  G ) U ) `  x
) ,  ( ( U ( proj1 `  G ) T ) `
 x ) >.  <->  ( ( 1st `  y
)  =  ( ( T ( proj1 `  G ) U ) `
 x )  /\  ( 2nd `  y )  =  ( ( U ( proj1 `  G ) T ) `
 x ) ) ) )
4140ad2antll 728 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  (
y  =  <. (
( T ( proj1 `  G ) U ) `  x
) ,  ( ( U ( proj1 `  G ) T ) `
 x ) >.  <->  ( ( 1st `  y
)  =  ( ( T ( proj1 `  G ) U ) `
 x )  /\  ( 2nd `  y )  =  ( ( U ( proj1 `  G ) T ) `
 x ) ) ) )
4239, 41bitr4d 256 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) ) )  ->  (
x  =  ( ( 1st `  y ) ( +g  `  G
) ( 2nd `  y
) )  <->  y  =  <. ( ( T (
proj1 `  G
) U ) `  x ) ,  ( ( U ( proj1 `  G ) T ) `  x
) >. ) )
4342ex 434 . . . 4  |-  ( ph  ->  ( ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  X.  U ) )  ->  ( x  =  ( ( 1st `  y ) ( +g  `  G ) ( 2nd `  y ) )  <->  y  =  <. ( ( T (
proj1 `  G
) U ) `  x ) ,  ( ( U ( proj1 `  G ) T ) `  x
) >. ) ) )
442, 6, 20, 27, 43en3d 7564 . . 3  |-  ( ph  ->  ( T  .(+)  U ) 
~~  ( T  X.  U ) )
45 hasheni 12401 . . 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 12473 . . 3  |-  ( ( T  e.  Fin  /\  U  e.  Fin )  ->  ( # `  ( T  X.  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )
5047, 48, 49syl2anc 661 . 2  |-  ( ph  ->  ( # `  ( T  X.  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )
5146, 50eqtrd 2508 1  |-  ( ph  ->  ( # `  ( T  .(+)  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480    C_ wss 3481   {csn 4033   <.cop 4039   class class class wbr 4453    X. cxp 5003   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794    ~~ cen 7525   Fincfn 7528    x. cmul 9509   #chash 12385   +g cplusg 14572   0gc0g 14712  SubGrpcsubg 16067  Cntzccntz 16225   LSSumclsm 16527   proj1cpj1 16528
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-rep 4564  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-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  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-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-hash 12386  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-pj1 16530
This theorem is referenced by:  ablfacrp2  16990  ablfac1eulem  16995  ablfac1eu  16996  pgpfaclem2  17005
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