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Theorem mulgpropd 14850
Description: Two structures with the same group-nature have the same group multiple function.  K is expected to either be  _V (when strong equality is available) or  B (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
mulgpropd.m  |-  .x.  =  (.g
`  G )
mulgpropd.n  |-  .X.  =  (.g
`  H )
mulgpropd.b1  |-  ( ph  ->  B  =  ( Base `  G ) )
mulgpropd.b2  |-  ( ph  ->  B  =  ( Base `  H ) )
mulgpropd.i  |-  ( ph  ->  B  C_  K )
mulgpropd.k  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K ) )  -> 
( x ( +g  `  G ) y )  e.  K )
mulgpropd.e  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
Assertion
Ref Expression
mulgpropd  |-  ( ph  ->  .x.  =  .X.  )
Distinct variable groups:    ph, x, y   
x, B, y    x, G, y    x, H, y   
x, K, y
Allowed substitution hints:    .x. ( x, y)    .X. ( x, y)

Proof of Theorem mulgpropd
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgpropd.b1 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  G ) )
2 mulgpropd.b2 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  H ) )
3 mulgpropd.i . . . . . . . . . 10  |-  ( ph  ->  B  C_  K )
4 ssel 3285 . . . . . . . . . . 11  |-  ( B 
C_  K  ->  (
x  e.  B  ->  x  e.  K )
)
5 ssel 3285 . . . . . . . . . . 11  |-  ( B 
C_  K  ->  (
y  e.  B  -> 
y  e.  K ) )
64, 5anim12d 547 . . . . . . . . . 10  |-  ( B 
C_  K  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  e.  K  /\  y  e.  K ) ) )
73, 6syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  ->  (
x  e.  K  /\  y  e.  K )
) )
87imp 419 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  e.  K  /\  y  e.  K
) )
9 mulgpropd.e . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
108, 9syldan 457 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
111, 2, 10grpidpropd 14649 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
12113ad2ant1 978 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
13 1z 10243 . . . . . . . . 9  |-  1  e.  ZZ
1413a1i 11 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  1  e.  ZZ )
15 vex 2902 . . . . . . . . . . . 12  |-  b  e. 
_V
1615fvconst2 5886 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
( NN  X.  {
b } ) `  x )  =  b )
17 nnuz 10453 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
1817eqcomi 2391 . . . . . . . . . . 11  |-  ( ZZ>= ` 
1 )  =  NN
1916, 18eleq2s 2479 . . . . . . . . . 10  |-  ( x  e.  ( ZZ>= `  1
)  ->  ( ( NN  X.  { b } ) `  x )  =  b )
2019adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { b } ) `  x )  =  b )
2133ad2ant1 978 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  B  C_  K
)
22 simp3 959 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  b  e.  B )
2321, 22sseldd 3292 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  b  e.  K )
2423adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  x  e.  ( ZZ>= `  1 )
)  ->  b  e.  K )
2520, 24eqeltrd 2461 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { b } ) `  x )  e.  K )
26 mulgpropd.k . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K ) )  -> 
( x ( +g  `  G ) y )  e.  K )
27263ad2antl1 1119 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  (
x  e.  K  /\  y  e.  K )
)  ->  ( x
( +g  `  G ) y )  e.  K
)
2893ad2antl1 1119 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  (
x  e.  K  /\  y  e.  K )
)  ->  ( x
( +g  `  G ) y )  =  ( x ( +g  `  H
) y ) )
2914, 25, 27, 28seqfeq3 11300 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { b } ) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) )
3029fveq1d 5670 . . . . . 6  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  a )  =  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  a ) )
311, 2, 10grpinvpropd 14793 . . . . . . . 8  |-  ( ph  ->  ( inv g `  G )  =  ( inv g `  H
) )
32313ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( inv g `  G )  =  ( inv g `  H ) )
3329fveq1d 5670 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a )  =  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a
) )
3432, 33fveq12d 5674 . . . . . 6  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( ( inv g `  G ) `
 (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { b } ) ) `  -u a
) )  =  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) )
3530, 34ifeq12d 3698 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  if (
0  <  a , 
(  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) )  =  if ( 0  <  a ,  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( inv g `  H ) `
 (  seq  1
( ( +g  `  H
) ,  ( NN 
X.  { b } ) ) `  -u a
) ) ) )
3612, 35ifeq12d 3698 . . . 4  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  if (
a  =  0 ,  ( 0g `  G
) ,  if ( 0  <  a ,  (  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) )  =  if ( a  =  0 ,  ( 0g `  H
) ,  if ( 0  <  a ,  (  seq  1 ( ( +g  `  H
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  H
) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )
3736mpt2eq3dva 6077 . . 3  |-  ( ph  ->  ( a  e.  ZZ ,  b  e.  B  |->  if ( a  =  0 ,  ( 0g
`  G ) ,  if ( 0  < 
a ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )  =  ( a  e.  ZZ , 
b  e.  B  |->  if ( a  =  0 ,  ( 0g `  H ) ,  if ( 0  <  a ,  (  seq  1
( ( +g  `  H
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  H
) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) ) )
38 eqidd 2388 . . . 4  |-  ( ph  ->  ZZ  =  ZZ )
39 eqidd 2388 . . . 4  |-  ( ph  ->  if ( a  =  0 ,  ( 0g
`  G ) ,  if ( 0  < 
a ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) )  =  if ( a  =  0 ,  ( 0g `  G
) ,  if ( 0  <  a ,  (  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )
4038, 1, 39mpt2eq123dv 6075 . . 3  |-  ( ph  ->  ( a  e.  ZZ ,  b  e.  B  |->  if ( a  =  0 ,  ( 0g
`  G ) ,  if ( 0  < 
a ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )  =  ( a  e.  ZZ , 
b  e.  ( Base `  G )  |->  if ( a  =  0 ,  ( 0g `  G
) ,  if ( 0  <  a ,  (  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) ) )
41 eqidd 2388 . . . 4  |-  ( ph  ->  if ( a  =  0 ,  ( 0g
`  H ) ,  if ( 0  < 
a ,  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) )  =  if ( a  =  0 ,  ( 0g `  H
) ,  if ( 0  <  a ,  (  seq  1 ( ( +g  `  H
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  H
) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )
4238, 2, 41mpt2eq123dv 6075 . . 3  |-  ( ph  ->  ( a  e.  ZZ ,  b  e.  B  |->  if ( a  =  0 ,  ( 0g
`  H ) ,  if ( 0  < 
a ,  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )  =  ( a  e.  ZZ , 
b  e.  ( Base `  H )  |->  if ( a  =  0 ,  ( 0g `  H
) ,  if ( 0  <  a ,  (  seq  1 ( ( +g  `  H
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  H
) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) ) )
4337, 40, 423eqtr3d 2427 . 2  |-  ( ph  ->  ( a  e.  ZZ ,  b  e.  ( Base `  G )  |->  if ( a  =  0 ,  ( 0g `  G ) ,  if ( 0  <  a ,  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )  =  ( a  e.  ZZ , 
b  e.  ( Base `  H )  |->  if ( a  =  0 ,  ( 0g `  H
) ,  if ( 0  <  a ,  (  seq  1 ( ( +g  `  H
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  H
) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) ) )
44 eqid 2387 . . 3  |-  ( Base `  G )  =  (
Base `  G )
45 eqid 2387 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
46 eqid 2387 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
47 eqid 2387 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
48 mulgpropd.m . . 3  |-  .x.  =  (.g
`  G )
4944, 45, 46, 47, 48mulgfval 14818 . 2  |-  .x.  =  ( a  e.  ZZ ,  b  e.  ( Base `  G )  |->  if ( a  =  0 ,  ( 0g `  G ) ,  if ( 0  <  a ,  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )
50 eqid 2387 . . 3  |-  ( Base `  H )  =  (
Base `  H )
51 eqid 2387 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
52 eqid 2387 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
53 eqid 2387 . . 3  |-  ( inv g `  H )  =  ( inv g `  H )
54 mulgpropd.n . . 3  |-  .X.  =  (.g
`  H )
5550, 51, 52, 53, 54mulgfval 14818 . 2  |-  .X.  =  ( a  e.  ZZ ,  b  e.  ( Base `  H )  |->  if ( a  =  0 ,  ( 0g `  H ) ,  if ( 0  <  a ,  (  seq  1
( ( +g  `  H
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( inv g `  H
) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )
5643, 49, 553eqtr4g 2444 1  |-  ( ph  ->  .x.  =  .X.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3263   ifcif 3682   {csn 3757   class class class wbr 4153    X. cxp 4816   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   0cc0 8923   1c1 8924    < clt 9053   -ucneg 9224   NNcn 9932   ZZcz 10214   ZZ>=cuz 10420    seq cseq 11250   Basecbs 13396   +g cplusg 13456   0gc0g 13650   inv gcminusg 14613  .gcmg 14616
This theorem is referenced by:  mulgass3  15669  coe1tm  16592  ply1coe  16611  evl1expd  19825
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-seq 11251  df-0g 13654  df-minusg 14740  df-mulg 14742
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