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Theorem mulgpropdg 12900
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
mulgpropdg.m  |-  ( ph  ->  .x.  =  (.g `  G
) )
mulgpropdg.n  |-  ( ph  ->  .X.  =  (.g `  H
) )
mulgpropdg.g  |-  ( ph  ->  G  e.  V )
mulgpropdg.h  |-  ( ph  ->  H  e.  W )
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
mulgpropdg  |-  ( 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)    V( x, y)    W( x, y)

Proof of Theorem mulgpropdg
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 mulgpropdg.g . . . . . . 7  |-  ( ph  ->  G  e.  V )
4 mulgpropdg.h . . . . . . 7  |-  ( ph  ->  H  e.  W )
5 mulgpropd.i . . . . . . . . . 10  |-  ( ph  ->  B  C_  K )
6 ssel 3149 . . . . . . . . . . 11  |-  ( B 
C_  K  ->  (
x  e.  B  ->  x  e.  K )
)
7 ssel 3149 . . . . . . . . . . 11  |-  ( B 
C_  K  ->  (
y  e.  B  -> 
y  e.  K ) )
86, 7anim12d 335 . . . . . . . . . 10  |-  ( B 
C_  K  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  e.  K  /\  y  e.  K ) ) )
95, 8syl 14 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  ->  (
x  e.  K  /\  y  e.  K )
) )
109imp 124 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  e.  K  /\  y  e.  K
) )
11 mulgpropd.e . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
1210, 11syldan 282 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
131, 2, 3, 4, 12grpidpropdg 12672 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
14133ad2ant1 1018 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
15 1zzd 9256 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  1  e.  ZZ )
16 nnuz 9539 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
1753ad2ant1 1018 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  B  C_  K
)
18 simp3 999 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  b  e.  B )
1917, 18sseldd 3156 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  b  e.  K )
2016, 19ialgrlemconst 12013 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  x  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { b } ) `  x )  e.  K )
21 mulgpropd.k . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  K ) )  -> 
( x ( +g  `  G ) y )  e.  K )
22213ad2antl1 1159 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  (
x  e.  K  /\  y  e.  K )
)  ->  ( x
( +g  `  G ) y )  e.  K
)
23113ad2antl1 1159 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ  /\  b  e.  B )  /\  (
x  e.  K  /\  y  e.  K )
)  ->  ( x
( +g  `  G ) y )  =  ( x ( +g  `  H
) y ) )
2415, 20, 22, 23seqfeq3 10485 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { b } ) )  =  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) )
2524fveq1d 5512 . . . . . 6  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { b } ) ) `  a )  =  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { b } ) ) `  a ) )
261, 2, 3, 4, 12grpinvpropdg 12821 . . . . . . . 8  |-  ( ph  ->  ( invg `  G )  =  ( invg `  H
) )
27263ad2ant1 1018 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ  /\  b  e.  B
)  ->  ( invg `  G )  =  ( invg `  H ) )
2824fveq1d 5512 . . . . . . 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
) )
2927, 28fveq12d 5517 . . . . . 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 ) ) )
3025, 29ifeq12d 3553 . . . . 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 ) ) ) )
3114, 30ifeq12d 3553 . . . 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 ) ) ) ) )
3231mpoeq3dva 5932 . . 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 ) ) ) ) ) )
33 eqidd 2178 . . . 4  |-  ( ph  ->  ZZ  =  ZZ )
34 eqidd 2178 . . . 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 ) ) ) ) )
3533, 1, 34mpoeq123dv 5930 . . 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 ) ) ) ) ) )
36 eqidd 2178 . . . 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 ) ) ) ) )
3733, 2, 36mpoeq123dv 5930 . . 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 ) ) ) ) ) )
3832, 35, 373eqtr3d 2218 . 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 ) ) ) ) ) )
39 mulgpropdg.m . . 3  |-  ( ph  ->  .x.  =  (.g `  G
) )
40 eqid 2177 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
41 eqid 2177 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
42 eqid 2177 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
43 eqid 2177 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
44 eqid 2177 . . . . 5  |-  (.g `  G
)  =  (.g `  G
)
4540, 41, 42, 43, 44mulgfvalg 12861 . . . 4  |-  ( G  e.  V  ->  (.g `  G )  =  ( 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 ) ) ) ) ) )
463, 45syl 14 . . 3  |-  ( ph  ->  (.g `  G )  =  ( 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 ) ) ) ) ) )
4739, 46eqtrd 2210 . 2  |-  ( ph  ->  .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
) ) ) ) ) )
48 mulgpropdg.n . . 3  |-  ( ph  ->  .X.  =  (.g `  H
) )
49 eqid 2177 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
50 eqid 2177 . . . . 5  |-  ( +g  `  H )  =  ( +g  `  H )
51 eqid 2177 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
52 eqid 2177 . . . . 5  |-  ( invg `  H )  =  ( invg `  H )
53 eqid 2177 . . . . 5  |-  (.g `  H
)  =  (.g `  H
)
5449, 50, 51, 52, 53mulgfvalg 12861 . . . 4  |-  ( H  e.  W  ->  (.g `  H )  =  ( 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 ) ) ) ) ) )
554, 54syl 14 . . 3  |-  ( ph  ->  (.g `  H )  =  ( 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 ) ) ) ) ) )
5648, 55eqtrd 2210 . 2  |-  ( ph  ->  .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
) ) ) ) ) )
5738, 47, 563eqtr4d 2220 1  |-  ( ph  ->  .x.  =  .X.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148    C_ wss 3129   ifcif 3534   {csn 3591   class class class wbr 4000    X. cxp 4620   ` cfv 5211  (class class class)co 5868    e. cmpo 5870   0cc0 7789   1c1 7790    < clt 7969   -ucneg 8106   NNcn 8895   ZZcz 9229    seqcseq 10418   Basecbs 12432   +g cplusg 12505   0gc0g 12640   invgcminusg 12755  .gcmg 12859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-n0 9153  df-z 9230  df-uz 9505  df-seqfrec 10419  df-ndx 12435  df-slot 12436  df-base 12438  df-0g 12642  df-minusg 12758  df-mulg 12860
This theorem is referenced by:  mulgass3  13066
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