MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgpropd Structured version   Unicode version

Theorem mulgpropd 15968
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 3491 . . . . . . . . . . 11  |-  ( B 
C_  K  ->  (
x  e.  B  ->  x  e.  K )
)
5 ssel 3491 . . . . . . . . . . 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 15753 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
12113ad2ant1 1012 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
13 1zzd 10884 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  1  e.  ZZ )
14 vex 3109 . . . . . . . . . . . 12  |-  b  e. 
_V
1514fvconst2 6107 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
( NN  X.  {
b } ) `  x )  =  b )
16 nnuz 11106 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
1716eqcomi 2473 . . . . . . . . . . 11  |-  ( ZZ>= ` 
1 )  =  NN
1815, 17eleq2s 2568 . . . . . . . . . 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 1012 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  B  C_  K
)
21 simp3 993 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  b  e.  B )
2220, 21sseldd 3498 . . . . . . . . . 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 2548 . . . . . . . 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 1153 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  (
x  e.  K  /\  y  e.  K )
)  ->  ( x
( +g  `  G ) y )  e.  K
)
2793ad2antl1 1153 . . . . . . . 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 12113 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { b } ) )  =  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) )
2928fveq1d 5859 . . . . . 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 15907 . . . . . . . 8  |-  ( ph  ->  ( invg `  G )  =  ( invg `  H
) )
31303ad2ant1 1012 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( invg `  G )  =  ( invg `  H ) )
3228fveq1d 5859 . . . . . . 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 5863 . . . . . 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 3952 . . . . 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 3952 . . . 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 6336 . . 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 2461 . . . 4  |-  ( ph  ->  ZZ  =  ZZ )
38 eqidd 2461 . . . 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 6334 . . 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 2461 . . . 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 6334 . . 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 2509 . 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 2460 . . 3  |-  ( Base `  G )  =  (
Base `  G )
44 eqid 2460 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
45 eqid 2460 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
46 eqid 2460 . . 3  |-  ( invg `  G )  =  ( invg `  G )
47 mulgpropd.m . . 3  |-  .x.  =  (.g
`  G )
4843, 44, 45, 46, 47mulgfval 15936 . 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 2460 . . 3  |-  ( Base `  H )  =  (
Base `  H )
50 eqid 2460 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
51 eqid 2460 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
52 eqid 2460 . . 3  |-  ( invg `  H )  =  ( invg `  H )
53 mulgpropd.n . . 3  |-  .X.  =  (.g
`  H )
5449, 50, 51, 52, 53mulgfval 15936 . 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 2526 1  |-  ( ph  ->  .x.  =  .X.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    C_ wss 3469   ifcif 3932   {csn 4020   class class class wbr 4440    X. cxp 4990   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   0cc0 9481   1c1 9482    < clt 9617   -ucneg 9795   NNcn 10525   ZZcz 10853   ZZ>=cuz 11071    seqcseq 12063   Basecbs 14479   +g cplusg 14544   0gc0g 14684   invgcminusg 15717  .gcmg 15720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-seq 12064  df-0g 14686  df-minusg 15852  df-mulg 15854
This theorem is referenced by:  mulgass3  17063  coe1tm  18078  ply1coe  18101  ply1coeOLD  18102  evl1expd  18145
  Copyright terms: Public domain W3C validator