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

Theorem assamulgscm 18126
Description: Exponentiation of a scalar multiplication in an associative algebra:  ( a  .x.  X ) ^ N  =  ( a ^ N )  .X.  ( X ^ N ). (Contributed by AV, 26-Aug-2019.)
Hypotheses
Ref Expression
assamulgscm.v  |-  V  =  ( Base `  W
)
assamulgscm.f  |-  F  =  (Scalar `  W )
assamulgscm.b  |-  B  =  ( Base `  F
)
assamulgscm.s  |-  .x.  =  ( .s `  W )
assamulgscm.g  |-  G  =  (mulGrp `  F )
assamulgscm.p  |-  .^  =  (.g
`  G )
assamulgscm.h  |-  H  =  (mulGrp `  W )
assamulgscm.e  |-  E  =  (.g `  H )
Assertion
Ref Expression
assamulgscm  |-  ( ( W  e. AssAlg  /\  ( N  e.  NN0  /\  A  e.  B  /\  X  e.  V ) )  -> 
( N E ( A  .x.  X ) )  =  ( ( N  .^  A )  .x.  ( N E X ) ) )

Proof of Theorem assamulgscm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6303 . . . . . . 7  |-  ( x  =  0  ->  (
x E ( A 
.x.  X ) )  =  ( 0 E ( A  .x.  X
) ) )
2 oveq1 6303 . . . . . . . 8  |-  ( x  =  0  ->  (
x  .^  A )  =  ( 0  .^  A ) )
3 oveq1 6303 . . . . . . . 8  |-  ( x  =  0  ->  (
x E X )  =  ( 0 E X ) )
42, 3oveq12d 6314 . . . . . . 7  |-  ( x  =  0  ->  (
( x  .^  A
)  .x.  ( x E X ) )  =  ( ( 0  .^  A )  .x.  (
0 E X ) ) )
51, 4eqeq12d 2479 . . . . . 6  |-  ( x  =  0  ->  (
( x E ( A  .x.  X ) )  =  ( ( x  .^  A )  .x.  ( x E X ) )  <->  ( 0 E ( A  .x.  X ) )  =  ( ( 0  .^  A )  .x.  (
0 E X ) ) ) )
65imbi2d 316 . . . . 5  |-  ( x  =  0  ->  (
( ( ( A  e.  B  /\  X  e.  V )  /\  W  e. AssAlg )  ->  ( x E ( A  .x.  X ) )  =  ( ( x  .^  A )  .x.  (
x E X ) ) )  <->  ( (
( A  e.  B  /\  X  e.  V
)  /\  W  e. AssAlg )  ->  ( 0 E ( A  .x.  X
) )  =  ( ( 0  .^  A
)  .x.  ( 0 E X ) ) ) ) )
7 oveq1 6303 . . . . . . 7  |-  ( x  =  y  ->  (
x E ( A 
.x.  X ) )  =  ( y E ( A  .x.  X
) ) )
8 oveq1 6303 . . . . . . . 8  |-  ( x  =  y  ->  (
x  .^  A )  =  ( y  .^  A ) )
9 oveq1 6303 . . . . . . . 8  |-  ( x  =  y  ->  (
x E X )  =  ( y E X ) )
108, 9oveq12d 6314 . . . . . . 7  |-  ( x  =  y  ->  (
( x  .^  A
)  .x.  ( x E X ) )  =  ( ( y  .^  A )  .x.  (
y E X ) ) )
117, 10eqeq12d 2479 . . . . . 6  |-  ( x  =  y  ->  (
( x E ( A  .x.  X ) )  =  ( ( x  .^  A )  .x.  ( x E X ) )  <->  ( y E ( A  .x.  X ) )  =  ( ( y  .^  A )  .x.  (
y E X ) ) ) )
1211imbi2d 316 . . . . 5  |-  ( x  =  y  ->  (
( ( ( A  e.  B  /\  X  e.  V )  /\  W  e. AssAlg )  ->  ( x E ( A  .x.  X ) )  =  ( ( x  .^  A )  .x.  (
x E X ) ) )  <->  ( (
( A  e.  B  /\  X  e.  V
)  /\  W  e. AssAlg )  ->  ( y E ( A  .x.  X
) )  =  ( ( y  .^  A
)  .x.  ( y E X ) ) ) ) )
13 oveq1 6303 . . . . . . 7  |-  ( x  =  ( y  +  1 )  ->  (
x E ( A 
.x.  X ) )  =  ( ( y  +  1 ) E ( A  .x.  X
) ) )
14 oveq1 6303 . . . . . . . 8  |-  ( x  =  ( y  +  1 )  ->  (
x  .^  A )  =  ( ( y  +  1 )  .^  A ) )
15 oveq1 6303 . . . . . . . 8  |-  ( x  =  ( y  +  1 )  ->  (
x E X )  =  ( ( y  +  1 ) E X ) )
1614, 15oveq12d 6314 . . . . . . 7  |-  ( x  =  ( y  +  1 )  ->  (
( x  .^  A
)  .x.  ( x E X ) )  =  ( ( ( y  +  1 )  .^  A )  .x.  (
( y  +  1 ) E X ) ) )
1713, 16eqeq12d 2479 . . . . . 6  |-  ( x  =  ( y  +  1 )  ->  (
( x E ( A  .x.  X ) )  =  ( ( x  .^  A )  .x.  ( x E X ) )  <->  ( (
y  +  1 ) E ( A  .x.  X ) )  =  ( ( ( y  +  1 )  .^  A )  .x.  (
( y  +  1 ) E X ) ) ) )
1817imbi2d 316 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
( ( ( A  e.  B  /\  X  e.  V )  /\  W  e. AssAlg )  ->  ( x E ( A  .x.  X ) )  =  ( ( x  .^  A )  .x.  (
x E X ) ) )  <->  ( (
( A  e.  B  /\  X  e.  V
)  /\  W  e. AssAlg )  ->  ( ( y  +  1 ) E ( A  .x.  X
) )  =  ( ( ( y  +  1 )  .^  A
)  .x.  ( (
y  +  1 ) E X ) ) ) ) )
19 oveq1 6303 . . . . . . 7  |-  ( x  =  N  ->  (
x E ( A 
.x.  X ) )  =  ( N E ( A  .x.  X
) ) )
20 oveq1 6303 . . . . . . . 8  |-  ( x  =  N  ->  (
x  .^  A )  =  ( N  .^  A ) )
21 oveq1 6303 . . . . . . . 8  |-  ( x  =  N  ->  (
x E X )  =  ( N E X ) )
2220, 21oveq12d 6314 . . . . . . 7  |-  ( x  =  N  ->  (
( x  .^  A
)  .x.  ( x E X ) )  =  ( ( N  .^  A )  .x.  ( N E X ) ) )
2319, 22eqeq12d 2479 . . . . . 6  |-  ( x  =  N  ->  (
( x E ( A  .x.  X ) )  =  ( ( x  .^  A )  .x.  ( x E X ) )  <->  ( N E ( A  .x.  X ) )  =  ( ( N  .^  A )  .x.  ( N E X ) ) ) )
2423imbi2d 316 . . . . 5  |-  ( x  =  N  ->  (
( ( ( A  e.  B  /\  X  e.  V )  /\  W  e. AssAlg )  ->  ( x E ( A  .x.  X ) )  =  ( ( x  .^  A )  .x.  (
x E X ) ) )  <->  ( (
( A  e.  B  /\  X  e.  V
)  /\  W  e. AssAlg )  ->  ( N E ( A  .x.  X
) )  =  ( ( N  .^  A
)  .x.  ( N E X ) ) ) ) )
25 assamulgscm.v . . . . . 6  |-  V  =  ( Base `  W
)
26 assamulgscm.f . . . . . 6  |-  F  =  (Scalar `  W )
27 assamulgscm.b . . . . . 6  |-  B  =  ( Base `  F
)
28 assamulgscm.s . . . . . 6  |-  .x.  =  ( .s `  W )
29 assamulgscm.g . . . . . 6  |-  G  =  (mulGrp `  F )
30 assamulgscm.p . . . . . 6  |-  .^  =  (.g
`  G )
31 assamulgscm.h . . . . . 6  |-  H  =  (mulGrp `  W )
32 assamulgscm.e . . . . . 6  |-  E  =  (.g `  H )
3325, 26, 27, 28, 29, 30, 31, 32assamulgscmlem1 18124 . . . . 5  |-  ( ( ( A  e.  B  /\  X  e.  V
)  /\  W  e. AssAlg )  ->  ( 0 E ( A  .x.  X
) )  =  ( ( 0  .^  A
)  .x.  ( 0 E X ) ) )
3425, 26, 27, 28, 29, 30, 31, 32assamulgscmlem2 18125 . . . . . 6  |-  ( y  e.  NN0  ->  ( ( ( A  e.  B  /\  X  e.  V
)  /\  W  e. AssAlg )  ->  ( ( y E ( A  .x.  X ) )  =  ( ( y  .^  A )  .x.  (
y E X ) )  ->  ( (
y  +  1 ) E ( A  .x.  X ) )  =  ( ( ( y  +  1 )  .^  A )  .x.  (
( y  +  1 ) E X ) ) ) ) )
3534a2d 26 . . . . 5  |-  ( y  e.  NN0  ->  ( ( ( ( A  e.  B  /\  X  e.  V )  /\  W  e. AssAlg )  ->  ( y E ( A  .x.  X ) )  =  ( ( y  .^  A )  .x.  (
y E X ) ) )  ->  (
( ( A  e.  B  /\  X  e.  V )  /\  W  e. AssAlg )  ->  ( (
y  +  1 ) E ( A  .x.  X ) )  =  ( ( ( y  +  1 )  .^  A )  .x.  (
( y  +  1 ) E X ) ) ) ) )
366, 12, 18, 24, 33, 35nn0ind 10980 . . . 4  |-  ( N  e.  NN0  ->  ( ( ( A  e.  B  /\  X  e.  V
)  /\  W  e. AssAlg )  ->  ( N E ( A  .x.  X
) )  =  ( ( N  .^  A
)  .x.  ( N E X ) ) ) )
3736exp4c 608 . . 3  |-  ( N  e.  NN0  ->  ( A  e.  B  ->  ( X  e.  V  ->  ( W  e. AssAlg  ->  ( N E ( A  .x.  X ) )  =  ( ( N  .^  A )  .x.  ( N E X ) ) ) ) ) )
38373imp 1190 . 2  |-  ( ( N  e.  NN0  /\  A  e.  B  /\  X  e.  V )  ->  ( W  e. AssAlg  ->  ( N E ( A 
.x.  X ) )  =  ( ( N 
.^  A )  .x.  ( N E X ) ) ) )
3938impcom 430 1  |-  ( ( W  e. AssAlg  /\  ( N  e.  NN0  /\  A  e.  B  /\  X  e.  V ) )  -> 
( N E ( A  .x.  X ) )  =  ( ( N  .^  A )  .x.  ( N E X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512   NN0cn0 10816   Basecbs 14644  Scalarcsca 14715   .scvsca 14716  .gcmg 16183  mulGrpcmgp 17268  AssAlgcasa 18085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12111  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-plusg 14725  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mulg 16187  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-lmod 17641  df-assa 18088
This theorem is referenced by:  lply1binomsc  18476
  Copyright terms: Public domain W3C validator