Theorem assamulgscm 19350

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.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . . . 7 |- ( x = 0 -> ( x E ( A .x. X ) ) = ( 0 E ( A .x. X ) ) )
2		oveq1 6657	. . . . . . . 8 |- ( x = 0 -> ( x .^ A ) = ( 0 .^ A ) )
3		oveq1 6657	. . . . . . . 8 |- ( x = 0 -> ( x E X ) = ( 0 E X ) )
4	2, 3	oveq12d 6668	. . . . . . 7 |- ( x = 0 -> ( ( x .^ A ) .x. ( x E X ) ) = ( ( 0 .^ A ) .x. ( 0 E X ) ) )
5	1, 4	eqeq12d 2637	. . . . . 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 ) ) ) )
6	5	imbi2d 330	. . . . 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 6657	. . . . . . 7 |- ( x = y -> ( x E ( A .x. X ) ) = ( y E ( A .x. X ) ) )
8		oveq1 6657	. . . . . . . 8 |- ( x = y -> ( x .^ A ) = ( y .^ A ) )
9		oveq1 6657	. . . . . . . 8 |- ( x = y -> ( x E X ) = ( y E X ) )
10	8, 9	oveq12d 6668	. . . . . . 7 |- ( x = y -> ( ( x .^ A ) .x. ( x E X ) ) = ( ( y .^ A ) .x. ( y E X ) ) )
11	7, 10	eqeq12d 2637	. . . . . 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 ) ) ) )
12	11	imbi2d 330	. . . . 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 6657	. . . . . . 7 |- ( x = ( y + 1 ) -> ( x E ( A .x. X ) ) = ( ( y + 1 ) E ( A .x. X ) ) )
14		oveq1 6657	. . . . . . . 8 |- ( x = ( y + 1 ) -> ( x .^ A ) = ( ( y + 1 ) .^ A ) )
15		oveq1 6657	. . . . . . . 8 |- ( x = ( y + 1 ) -> ( x E X ) = ( ( y + 1 ) E X ) )
16	14, 15	oveq12d 6668	. . . . . . 7 |- ( x = ( y + 1 ) -> ( ( x .^ A ) .x. ( x E X ) ) = ( ( ( y + 1 ) .^ A ) .x. ( ( y + 1 ) E X ) ) )
17	13, 16	eqeq12d 2637	. . . . . 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 ) ) ) )
18	17	imbi2d 330	. . . . 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 6657	. . . . . . 7 |- ( x = N -> ( x E ( A .x. X ) ) = ( N E ( A .x. X ) ) )
20		oveq1 6657	. . . . . . . 8 |- ( x = N -> ( x .^ A ) = ( N .^ A ) )
21		oveq1 6657	. . . . . . . 8 |- ( x = N -> ( x E X ) = ( N E X ) )
22	20, 21	oveq12d 6668	. . . . . . 7 |- ( x = N -> ( ( x .^ A ) .x. ( x E X ) ) = ( ( N .^ A ) .x. ( N E X ) ) )
23	19, 22	eqeq12d 2637	. . . . . 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 ) ) ) )
24	23	imbi2d 330	. . . . 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 )
33	25, 26, 27, 28, 29, 30, 31, 32	assamulgscmlem1 19348	. . . . 5 |- ( ( ( A e. B /\ X e. V ) /\ W e. AssAlg ) -> ( 0 E ( A .x. X ) ) = ( ( 0 .^ A ) .x. ( 0 E X ) ) )
34	25, 26, 27, 28, 29, 30, 31, 32	assamulgscmlem2 19349	. . . . . 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 ) ) ) ) )
35	34	a2d 29	. . . . 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 ) ) ) ) )
36	6, 12, 18, 24, 33, 35	nn0ind 11472	. . . 4 |- ( N e. NN0 -> ( ( ( A e. B /\ X e. V ) /\ W e. AssAlg ) -> ( N E ( A .x. X ) ) = ( ( N .^ A ) .x. ( N E X ) ) ) )
37	36	exp4c 636	. . 3 |- ( N e. NN0 -> ( A e. B -> ( X e. V -> ( W e. AssAlg -> ( N E ( A .x. X ) ) = ( ( N .^ A ) .x. ( N E X ) ) ) ) ) )
38	37	3imp 1256	. 2 |- ( ( N e. NN0 /\ A e. B /\ X e. V ) -> ( W e. AssAlg -> ( N E ( A .x. X ) ) = ( ( N .^ A ) .x. ( N E X ) ) ) )
39	38	impcom 446	1 |- ( ( W e. AssAlg /\ ( N e. NN0 /\ A e. B /\ X e. V ) ) -> ( N E ( A .x. X ) ) = ( ( N .^ A ) .x. ( N E X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   Basecbs 15857  Scalarcsca 15944   .scvsca 15945  ._gcmg 17540  mulGrpcmgp 18489  AssAlgcasa 19309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mulg 17541  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-lmod 18865  df-assa 19312

This theorem is referenced by:  lply1binomsc  19677