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Theorem assamulgscm 17763
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 6282 . . . . . . 7  |-  ( x  =  0  ->  (
x E ( A 
.x.  X ) )  =  ( 0 E ( A  .x.  X
) ) )
2 oveq1 6282 . . . . . . . 8  |-  ( x  =  0  ->  (
x  .^  A )  =  ( 0  .^  A ) )
3 oveq1 6282 . . . . . . . 8  |-  ( x  =  0  ->  (
x E X )  =  ( 0 E X ) )
42, 3oveq12d 6293 . . . . . . 7  |-  ( x  =  0  ->  (
( x  .^  A
)  .x.  ( x E X ) )  =  ( ( 0  .^  A )  .x.  (
0 E X ) ) )
51, 4eqeq12d 2482 . . . . . 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 6282 . . . . . . 7  |-  ( x  =  y  ->  (
x E ( A 
.x.  X ) )  =  ( y E ( A  .x.  X
) ) )
8 oveq1 6282 . . . . . . . 8  |-  ( x  =  y  ->  (
x  .^  A )  =  ( y  .^  A ) )
9 oveq1 6282 . . . . . . . 8  |-  ( x  =  y  ->  (
x E X )  =  ( y E X ) )
108, 9oveq12d 6293 . . . . . . 7  |-  ( x  =  y  ->  (
( x  .^  A
)  .x.  ( x E X ) )  =  ( ( y  .^  A )  .x.  (
y E X ) ) )
117, 10eqeq12d 2482 . . . . . 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 6282 . . . . . . 7  |-  ( x  =  ( y  +  1 )  ->  (
x E ( A 
.x.  X ) )  =  ( ( y  +  1 ) E ( A  .x.  X
) ) )
14 oveq1 6282 . . . . . . . 8  |-  ( x  =  ( y  +  1 )  ->  (
x  .^  A )  =  ( ( y  +  1 )  .^  A ) )
15 oveq1 6282 . . . . . . . 8  |-  ( x  =  ( y  +  1 )  ->  (
x E X )  =  ( ( y  +  1 ) E X ) )
1614, 15oveq12d 6293 . . . . . . 7  |-  ( x  =  ( y  +  1 )  ->  (
( x  .^  A
)  .x.  ( x E X ) )  =  ( ( ( y  +  1 )  .^  A )  .x.  (
( y  +  1 ) E X ) ) )
1713, 16eqeq12d 2482 . . . . . 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 6282 . . . . . . 7  |-  ( x  =  N  ->  (
x E ( A 
.x.  X ) )  =  ( N E ( A  .x.  X
) ) )
20 oveq1 6282 . . . . . . . 8  |-  ( x  =  N  ->  (
x  .^  A )  =  ( N  .^  A ) )
21 oveq1 6282 . . . . . . . 8  |-  ( x  =  N  ->  (
x E X )  =  ( N E X ) )
2220, 21oveq12d 6293 . . . . . . 7  |-  ( x  =  N  ->  (
( x  .^  A
)  .x.  ( x E X ) )  =  ( ( N  .^  A )  .x.  ( N E X ) ) )
2319, 22eqeq12d 2482 . . . . . 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 17761 . . . . 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 17762 . . . . . 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 10946 . . . 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 1185 . 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 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484   NN0cn0 10784   Basecbs 14479  Scalarcsca 14547   .scvsca 14548  .gcmg 15720  mulGrpcmgp 16924  AssAlgcasa 17722
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-rmo 2815  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-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-seq 12064  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-0g 14686  df-mnd 15721  df-mulg 15854  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-lmod 17290  df-assa 17725
This theorem is referenced by:  lply1binomsc  18113
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