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Theorem mulgpropd 16501
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 3438 . . . . . . . . . . 11  |-  ( B 
C_  K  ->  (
x  e.  B  ->  x  e.  K )
)
5 ssel 3438 . . . . . . . . . . 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 17 . . . . . . . . 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 16214 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
12113ad2ant1 1020 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
13 1zzd 10938 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  1  e.  ZZ )
14 vex 3064 . . . . . . . . . . . 12  |-  b  e. 
_V
1514fvconst2 6109 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
( NN  X.  {
b } ) `  x )  =  b )
16 nnuz 11164 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
1716eqcomi 2417 . . . . . . . . . . 11  |-  ( ZZ>= ` 
1 )  =  NN
1815, 17eleq2s 2512 . . . . . . . . . 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 1020 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  B  C_  K
)
21 simp3 1001 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  b  e.  B )
2220, 21sseldd 3445 . . . . . . . . . 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 2492 . . . . . . . 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 1161 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  (
x  e.  K  /\  y  e.  K )
)  ->  ( x
( +g  `  G ) y )  e.  K
)
2793ad2antl1 1161 . . . . . . . 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 12203 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { b } ) )  =  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) )
2928fveq1d 5853 . . . . . 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 16439 . . . . . . . 8  |-  ( ph  ->  ( invg `  G )  =  ( invg `  H
) )
31303ad2ant1 1020 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( invg `  G )  =  ( invg `  H ) )
3228fveq1d 5853 . . . . . . 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 5857 . . . . . 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 3907 . . . . 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 3907 . . . 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 6344 . . 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 2405 . . . 4  |-  ( ph  ->  ZZ  =  ZZ )
38 eqidd 2405 . . . 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 6342 . . 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 2405 . . . 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 6342 . . 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 2453 . 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 2404 . . 3  |-  ( Base `  G )  =  (
Base `  G )
44 eqid 2404 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
45 eqid 2404 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
46 eqid 2404 . . 3  |-  ( invg `  G )  =  ( invg `  G )
47 mulgpropd.m . . 3  |-  .x.  =  (.g
`  G )
4843, 44, 45, 46, 47mulgfval 16469 . 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 2404 . . 3  |-  ( Base `  H )  =  (
Base `  H )
50 eqid 2404 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
51 eqid 2404 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
52 eqid 2404 . . 3  |-  ( invg `  H )  =  ( invg `  H )
53 mulgpropd.n . . 3  |-  .X.  =  (.g
`  H )
5449, 50, 51, 52, 53mulgfval 16469 . 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 2470 1  |-  ( ph  ->  .x.  =  .X.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    C_ wss 3416   ifcif 3887   {csn 3974   class class class wbr 4397    X. cxp 4823   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282   0cc0 9524   1c1 9525    < clt 9660   -ucneg 9844   NNcn 10578   ZZcz 10907   ZZ>=cuz 11129    seqcseq 12153   Basecbs 14843   +g cplusg 14911   0gc0g 15056   invgcminusg 16380  .gcmg 16382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-seq 12154  df-0g 15058  df-minusg 16384  df-mulg 16386
This theorem is referenced by:  mulgass3  17608  coe1tm  18636  ply1coe  18659  ply1coeOLD  18660  evl1expd  18703
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