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Theorem mulgpropd 15651
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 3345 . . . . . . . . . . 11  |-  ( B 
C_  K  ->  (
x  e.  B  ->  x  e.  K )
)
5 ssel 3345 . . . . . . . . . . 11  |-  ( B 
C_  K  ->  (
y  e.  B  -> 
y  e.  K ) )
64, 5anim12d 563 . . . . . . . . . 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 429 . . . . . . . 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 470 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
111, 2, 10grpidpropd 15439 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
12113ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
13 1zzd 10669 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  1  e.  ZZ )
14 vex 2970 . . . . . . . . . . . 12  |-  b  e. 
_V
1514fvconst2 5928 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
( NN  X.  {
b } ) `  x )  =  b )
16 nnuz 10888 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
1716eqcomi 2442 . . . . . . . . . . 11  |-  ( ZZ>= ` 
1 )  =  NN
1815, 17eleq2s 2530 . . . . . . . . . 10  |-  ( x  e.  ( ZZ>= `  1
)  ->  ( ( NN  X.  { b } ) `  x )  =  b )
1918adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { b } ) `  x )  =  b )
2033ad2ant1 1009 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  B  C_  K
)
21 simp3 990 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  b  e.  B )
2220, 21sseldd 3352 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  b  e.  K )
2322adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  x  e.  ( ZZ>= `  1 )
)  ->  b  e.  K )
2419, 23eqeltrd 2512 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { b } ) `  x )  e.  K )
25 mulgpropd.k . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K ) )  -> 
( x ( +g  `  G ) y )  e.  K )
26253ad2antl1 1150 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  (
x  e.  K  /\  y  e.  K )
)  ->  ( x
( +g  `  G ) y )  e.  K
)
2793ad2antl1 1150 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  (
x  e.  K  /\  y  e.  K )
)  ->  ( x
( +g  `  G ) y )  =  ( x ( +g  `  H
) y ) )
2813, 24, 26, 27seqfeq3 11848 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { b } ) )  =  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) )
2928fveq1d 5688 . . . . . 6  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  a )  =  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  a ) )
301, 2, 10grpinvpropd 15592 . . . . . . . 8  |-  ( ph  ->  ( invg `  G )  =  ( invg `  H
) )
31303ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( invg `  G )  =  ( invg `  H ) )
3228fveq1d 5688 . . . . . . 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
) )
3331, 32fveq12d 5692 . . . . . 6  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( ( invg `  G ) `
 (  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { b } ) ) `  -u a
) )  =  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) )
3429, 33ifeq12d 3804 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  if (
0  <  a , 
(  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( invg `  G ) `
 (  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { b } ) ) `  -u a
) ) )  =  if ( 0  < 
a ,  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) )
3512, 34ifeq12d 3804 . . . 4  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  if (
a  =  0 ,  ( 0g `  G
) ,  if ( 0  <  a ,  (  seq 1 ( ( +g  `  G
) ,  ( NN 
X.  { b } ) ) `  a
) ,  ( ( invg `  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
) ,  ( ( invg `  H
) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )
3635mpt2eq3dva 6145 . . 3  |-  ( ph  ->  ( a  e.  ZZ ,  b  e.  B  |->  if ( a  =  0 ,  ( 0g
`  G ) ,  if ( 0  < 
a ,  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( invg `  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
) ,  ( ( invg `  H
) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) ) )
37 eqidd 2439 . . . 4  |-  ( ph  ->  ZZ  =  ZZ )
38 eqidd 2439 . . . 4  |-  ( ph  ->  if ( a  =  0 ,  ( 0g
`  G ) ,  if ( 0  < 
a ,  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( invg `  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
) ,  ( ( invg `  G
) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )
3937, 1, 38mpt2eq123dv 6143 . . 3  |-  ( ph  ->  ( a  e.  ZZ ,  b  e.  B  |->  if ( a  =  0 ,  ( 0g
`  G ) ,  if ( 0  < 
a ,  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( invg `  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
) ,  ( ( invg `  G
) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) ) )
40 eqidd 2439 . . . 4  |-  ( ph  ->  if ( a  =  0 ,  ( 0g
`  H ) ,  if ( 0  < 
a ,  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( invg `  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
) ,  ( ( invg `  H
) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )
4137, 2, 40mpt2eq123dv 6143 . . 3  |-  ( ph  ->  ( a  e.  ZZ ,  b  e.  B  |->  if ( a  =  0 ,  ( 0g
`  H ) ,  if ( 0  < 
a ,  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  a ) ,  ( ( invg `  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
) ,  ( ( invg `  H
) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) ) )
4236, 39, 413eqtr3d 2478 . 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
) ,  ( ( invg `  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
) ,  ( ( invg `  H
) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) ) )
43 eqid 2438 . . 3  |-  ( Base `  G )  =  (
Base `  G )
44 eqid 2438 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
45 eqid 2438 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
46 eqid 2438 . . 3  |-  ( invg `  G )  =  ( invg `  G )
47 mulgpropd.m . . 3  |-  .x.  =  (.g
`  G )
4843, 44, 45, 46, 47mulgfval 15619 . 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
) ,  ( ( invg `  G
) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )
49 eqid 2438 . . 3  |-  ( Base `  H )  =  (
Base `  H )
50 eqid 2438 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
51 eqid 2438 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
52 eqid 2438 . . 3  |-  ( invg `  H )  =  ( invg `  H )
53 mulgpropd.n . . 3  |-  .X.  =  (.g
`  H )
5449, 50, 51, 52, 53mulgfval 15619 . 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
) ,  ( ( invg `  H
) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  -u a ) ) ) ) )
5542, 48, 543eqtr4g 2495 1  |-  ( ph  ->  .x.  =  .X.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3323   ifcif 3786   {csn 3872   class class class wbr 4287    X. cxp 4833   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   0cc0 9274   1c1 9275    < clt 9410   -ucneg 9588   NNcn 10314   ZZcz 10638   ZZ>=cuz 10853    seqcseq 11798   Basecbs 14166   +g cplusg 14230   0gc0g 14370   invgcminusg 15403  .gcmg 15406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-seq 11799  df-0g 14372  df-minusg 15537  df-mulg 15539
This theorem is referenced by:  mulgass3  16717  coe1tm  17701  ply1coe  17721  ply1coeOLD  17722  evl1expd  17754
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