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Theorem lsmhash 15329
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 6098 . . . . 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 4981 . . . . 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 2435 . . . . . . . 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 2435 . . . . . . . 8  |-  ( proj
1 `  G )  =  ( proj 1 `  G )
147, 8, 9, 10, 3, 4, 11, 12, 13pj1f 15321 . . . . . . 7  |-  ( ph  ->  ( T ( proj
1 `  G ) U ) : ( T  .(+)  U ) --> T )
1514ffvelrnda 5862 . . . . . 6  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  ( ( T ( proj 1 `  G ) U ) `
 x )  e.  T )
167, 8, 9, 10, 3, 4, 11, 12, 13pj2f 15322 . . . . . . 7  |-  ( ph  ->  ( U ( proj
1 `  G ) T ) : ( T  .(+)  U ) --> U )
1716ffvelrnda 5862 . . . . . 6  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  ( ( U ( proj 1 `  G ) T ) `
 x )  e.  U )
18 opelxpi 4902 . . . . . 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 6368 . . . . . . 7  |-  ( y  e.  ( T  X.  U )  ->  ( 1st `  y )  e.  T )
23 xp2nd 6369 . . . . . . 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 15276 . . . . . 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 15324 . . . . . . 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 2437 . . . . . . . 8  |-  ( ( ( T ( proj
1 `  G ) U ) `  x
)  =  ( 1st `  y )  <->  ( 1st `  y )  =  ( ( T ( proj
1 `  G ) U ) `  x
) )
37 eqcom 2437 . . . . . . . 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 6381 . . . . . . 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 7136 . . 3  |-  ( ph  ->  ( T  .(+)  U ) 
~~  ( T  X.  U ) )
45 hasheni 11624 . . 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 11689 . . 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 2467 1  |-  ( ph  ->  ( # `  ( T  .(+)  U ) )  =  ( ( # `  T )  x.  ( # `
 U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   {csn 3806   <.cop 3809   class class class wbr 4204    X. cxp 4868   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340    ~~ cen 7098   Fincfn 7101    x. cmul 8987   #chash 11610   +g cplusg 13521   0gc0g 13715  SubGrpcsubg 14930  Cntzccntz 15106   LSSumclsm 15260   proj
1cpj1 15261
This theorem is referenced by:  ablfacrp2  15617  ablfac1eulem  15622  ablfac1eu  15623  pgpfaclem2  15632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-lsm 15262  df-pj1 15263
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