Theorem assamulgscm 15026

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 6086	. . . . . . 7 |- ( x = 0 -> ( x E ( A .x. X ) ) = ( 0 E ( A .x. X ) ) )
2		oveq1 6086	. . . . . . . 8 |- ( x = 0 -> ( x .^ A ) = ( 0 .^ A ) )
3		oveq1 6086	. . . . . . . 8 |- ( x = 0 -> ( x E X ) = ( 0 E X ) )
4	2, 3	oveq12d 6097	. . . . . . 7 |- ( x = 0 -> ( ( x .^ A ) .x. ( x E X ) ) = ( ( 0 .^ A ) .x. ( 0 E X ) ) )
5	1, 4	eqeq12d 2253	. . . . . 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 230	. . . . 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 6086	. . . . . . 7 |- ( x = y -> ( x E ( A .x. X ) ) = ( y E ( A .x. X ) ) )
8		oveq1 6086	. . . . . . . 8 |- ( x = y -> ( x .^ A ) = ( y .^ A ) )
9		oveq1 6086	. . . . . . . 8 |- ( x = y -> ( x E X ) = ( y E X ) )
10	8, 9	oveq12d 6097	. . . . . . 7 |- ( x = y -> ( ( x .^ A ) .x. ( x E X ) ) = ( ( y .^ A ) .x. ( y E X ) ) )
11	7, 10	eqeq12d 2253	. . . . . 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 230	. . . . 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 6086	. . . . . . 7 |- ( x = ( y + 1 ) -> ( x E ( A .x. X ) ) = ( ( y + 1 ) E ( A .x. X ) ) )
14		oveq1 6086	. . . . . . . 8 |- ( x = ( y + 1 ) -> ( x .^ A ) = ( ( y + 1 ) .^ A ) )
15		oveq1 6086	. . . . . . . 8 |- ( x = ( y + 1 ) -> ( x E X ) = ( ( y + 1 ) E X ) )
16	14, 15	oveq12d 6097	. . . . . . 7 |- ( x = ( y + 1 ) -> ( ( x .^ A ) .x. ( x E X ) ) = ( ( ( y + 1 ) .^ A ) .x. ( ( y + 1 ) E X ) ) )
17	13, 16	eqeq12d 2253	. . . . . 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 230	. . . . 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 6086	. . . . . . 7 |- ( x = N -> ( x E ( A .x. X ) ) = ( N E ( A .x. X ) ) )
20		oveq1 6086	. . . . . . . 8 |- ( x = N -> ( x .^ A ) = ( N .^ A ) )
21		oveq1 6086	. . . . . . . 8 |- ( x = N -> ( x E X ) = ( N E X ) )
22	20, 21	oveq12d 6097	. . . . . . 7 |- ( x = N -> ( ( x .^ A ) .x. ( x E X ) ) = ( ( N .^ A ) .x. ( N E X ) ) )
23	19, 22	eqeq12d 2253	. . . . . 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 230	. . . . 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 15024	. . . . 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 15025	. . . . . 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 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 ) ) ) ) )
36	6, 12, 18, 24, 33, 35	nn0ind 9743	. . . 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 368	. . 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 1224	. 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 125	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 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5375  (class class class)co 6079   0cc0 8173   1c1 8174   + caddc 8176   NN0cn0 9546   Basecbs 13335  Scalarcsca 13417   .scvsca 13418  ._gcmg 13905  mulGrpcmgp 14200  AssAlgcasa 14979

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-n0 9547  df-z 9628  df-uz 9905  df-seqfrec 10868  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-minusg 13792  df-mulg 13906  df-mgp 14201  df-ur 14246  df-ring 14285  df-lmod 14608  df-assa 14982

This theorem is referenced by: (None)