Theorem evl1scvarpw 19727

Description: Univariate polynomial evaluation maps a multiple of an exponentiation of a variable to the multiple of an exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)

Hypotheses

Ref	Expression
evl1varpw.q	|- Q = (eval1 ` R )
evl1varpw.w	|- W = (Poly1 ` R )
evl1varpw.g	|- G = (mulGrp ` W )
evl1varpw.x	|- X = (var1 ` R )
evl1varpw.b	|- B = ( Base ` R )
evl1varpw.e	|- .^ = (.g ` G )
evl1varpw.r	|- ( ph -> R e. CRing )
evl1varpw.n	|- ( ph -> N e. NN0 )
evl1scvarpw.t1	|- .X. = ( .s ` W )
evl1scvarpw.a	|- ( ph -> A e. B )
evl1scvarpw.s	|- S = ( R ^s B )
evl1scvarpw.t2	|- .xb = ( .r ` S )
evl1scvarpw.m	|- M = (mulGrp ` S )
evl1scvarpw.f	|- F = (.g ` M )

Assertion

Ref	Expression
evl1scvarpw	|- ( ph -> ( Q ` ( A .X. ( N .^ X ) ) ) = ( ( B X. { A } ) .xb ( N F ( Q ` X ) ) ) )

Proof of Theorem evl1scvarpw

Step	Hyp	Ref	Expression
1		evl1varpw.r	. . . . . 6 |- ( ph -> R e. CRing )
2		evl1varpw.w	. . . . . . 7 |- W = (Poly1 ` R )
3	2	ply1assa 19569	. . . . . 6 |- ( R e. CRing -> W e. AssAlg )
4	1, 3	syl 17	. . . . 5 |- ( ph -> W e. AssAlg )
5		evl1scvarpw.a	. . . . . . 7 |- ( ph -> A e. B )
6		evl1varpw.b	. . . . . . 7 |- B = ( Base ` R )
7	5, 6	syl6eleq 2711	. . . . . 6 |- ( ph -> A e. ( Base ` R ) )
8	2	ply1sca 19623	. . . . . . . . 9 |- ( R e. CRing -> R = (Scalar ` W ) )
9	8	eqcomd 2628	. . . . . . . 8 |- ( R e. CRing -> (Scalar ` W ) = R )
10	1, 9	syl 17	. . . . . . 7 |- ( ph -> (Scalar ` W ) = R )
11	10	fveq2d 6195	. . . . . 6 |- ( ph -> ( Base ` (Scalar ` W ) ) = ( Base ` R ) )
12	7, 11	eleqtrrd 2704	. . . . 5 |- ( ph -> A e. ( Base ` (Scalar ` W ) ) )
13		crngring 18558	. . . . . . . . 9 |- ( R e. CRing -> R e. Ring )
14	1, 13	syl 17	. . . . . . . 8 |- ( ph -> R e. Ring )
15	2	ply1ring 19618	. . . . . . . 8 |- ( R e. Ring -> W e. Ring )
16	14, 15	syl 17	. . . . . . 7 |- ( ph -> W e. Ring )
17		evl1varpw.g	. . . . . . . 8 |- G = (mulGrp ` W )
18	17	ringmgp 18553	. . . . . . 7 |- ( W e. Ring -> G e. Mnd )
19	16, 18	syl 17	. . . . . 6 |- ( ph -> G e. Mnd )
20		evl1varpw.n	. . . . . 6 |- ( ph -> N e. NN0 )
21		evl1varpw.x	. . . . . . . 8 |- X = (var1 ` R )
22		eqid 2622	. . . . . . . 8 |- ( Base ` W ) = ( Base ` W )
23	21, 2, 22	vr1cl 19587	. . . . . . 7 |- ( R e. Ring -> X e. ( Base ` W ) )
24	14, 23	syl 17	. . . . . 6 |- ( ph -> X e. ( Base ` W ) )
25	17, 22	mgpbas 18495	. . . . . . 7 |- ( Base ` W ) = ( Base ` G )
26		evl1varpw.e	. . . . . . 7 |- .^ = (.g ` G )
27	25, 26	mulgnn0cl 17558	. . . . . 6 |- ( ( G e. Mnd /\ N e. NN0 /\ X e. ( Base ` W ) ) -> ( N .^ X ) e. ( Base ` W ) )
28	19, 20, 24, 27	syl3anc 1326	. . . . 5 |- ( ph -> ( N .^ X ) e. ( Base ` W ) )
29		eqid 2622	. . . . . 6 |- (algSc ` W ) = (algSc ` W )
30		eqid 2622	. . . . . 6 |- (Scalar ` W ) = (Scalar ` W )
31		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
32		eqid 2622	. . . . . 6 |- ( .r ` W ) = ( .r ` W )
33		evl1scvarpw.t1	. . . . . 6 |- .X. = ( .s ` W )
34	29, 30, 31, 22, 32, 33	asclmul1 19339	. . . . 5 |- ( ( W e. AssAlg /\ A e. ( Base ` (Scalar ` W ) ) /\ ( N .^ X ) e. ( Base ` W ) ) -> ( ( (algSc ` W ) ` A ) ( .r ` W ) ( N .^ X ) ) = ( A .X. ( N .^ X ) ) )
35	4, 12, 28, 34	syl3anc 1326	. . . 4 |- ( ph -> ( ( (algSc ` W ) ` A ) ( .r ` W ) ( N .^ X ) ) = ( A .X. ( N .^ X ) ) )
36	35	eqcomd 2628	. . 3 |- ( ph -> ( A .X. ( N .^ X ) ) = ( ( (algSc ` W ) ` A ) ( .r ` W ) ( N .^ X ) ) )
37	36	fveq2d 6195	. 2 |- ( ph -> ( Q ` ( A .X. ( N .^ X ) ) ) = ( Q ` ( ( (algSc ` W ) ` A ) ( .r ` W ) ( N .^ X ) ) ) )
38		evl1varpw.q	. . . . 5 |- Q = (eval1 ` R )
39		evl1scvarpw.s	. . . . 5 |- S = ( R ^s B )
40	38, 2, 39, 6	evl1rhm 19696	. . . 4 |- ( R e. CRing -> Q e. ( W RingHom S ) )
41	1, 40	syl 17	. . 3 |- ( ph -> Q e. ( W RingHom S ) )
42	2	ply1lmod 19622	. . . . . 6 |- ( R e. Ring -> W e. LMod )
43	14, 42	syl 17	. . . . 5 |- ( ph -> W e. LMod )
44	29, 30, 16, 43, 31, 22	asclf 19337	. . . 4 |- ( ph -> (algSc ` W ) : ( Base ` (Scalar ` W ) ) --> ( Base ` W ) )
45	44, 12	ffvelrnd 6360	. . 3 |- ( ph -> ( (algSc ` W ) ` A ) e. ( Base ` W ) )
46		evl1scvarpw.t2	. . . 4 |- .xb = ( .r ` S )
47	22, 32, 46	rhmmul 18727	. . 3 |- ( ( Q e. ( W RingHom S ) /\ ( (algSc ` W ) ` A ) e. ( Base ` W ) /\ ( N .^ X ) e. ( Base ` W ) ) -> ( Q ` ( ( (algSc ` W ) ` A ) ( .r ` W ) ( N .^ X ) ) ) = ( ( Q ` ( (algSc ` W ) ` A ) ) .xb ( Q ` ( N .^ X ) ) ) )
48	41, 45, 28, 47	syl3anc 1326	. 2 |- ( ph -> ( Q ` ( ( (algSc ` W ) ` A ) ( .r ` W ) ( N .^ X ) ) ) = ( ( Q ` ( (algSc ` W ) ` A ) ) .xb ( Q ` ( N .^ X ) ) ) )
49	38, 2, 6, 29	evl1sca 19698	. . . 4 |- ( ( R e. CRing /\ A e. B ) -> ( Q ` ( (algSc ` W ) ` A ) ) = ( B X. { A } ) )
50	1, 5, 49	syl2anc 693	. . 3 |- ( ph -> ( Q ` ( (algSc ` W ) ` A ) ) = ( B X. { A } ) )
51	38, 2, 17, 21, 6, 26, 1, 20	evl1varpw 19725	. . . 4 |- ( ph -> ( Q ` ( N .^ X ) ) = ( N (.g ` (mulGrp ` ( R ^s B ) ) ) ( Q ` X ) ) )
52		evl1scvarpw.f	. . . . . . . 8 |- F = (.g ` M )
53		evl1scvarpw.m	. . . . . . . . . 10 |- M = (mulGrp ` S )
54	39	fveq2i 6194	. . . . . . . . . 10 |- (mulGrp ` S ) = (mulGrp ` ( R ^s B ) )
55	53, 54	eqtri 2644	. . . . . . . . 9 |- M = (mulGrp ` ( R ^s B ) )
56	55	fveq2i 6194	. . . . . . . 8 |- (.g ` M ) = (.g ` (mulGrp ` ( R ^s B ) ) )
57	52, 56	eqtri 2644	. . . . . . 7 |- F = (.g ` (mulGrp ` ( R ^s B ) ) )
58	57	a1i 11	. . . . . 6 |- ( ph -> F = (.g ` (mulGrp ` ( R ^s B ) ) ) )
59	58	eqcomd 2628	. . . . 5 |- ( ph -> (.g ` (mulGrp ` ( R ^s B ) ) ) = F )
60	59	oveqd 6667	. . . 4 |- ( ph -> ( N (.g ` (mulGrp ` ( R ^s B ) ) ) ( Q ` X ) ) = ( N F ( Q ` X ) ) )
61	51, 60	eqtrd 2656	. . 3 |- ( ph -> ( Q ` ( N .^ X ) ) = ( N F ( Q ` X ) ) )
62	50, 61	oveq12d 6668	. 2 |- ( ph -> ( ( Q ` ( (algSc ` W ) ` A ) ) .xb ( Q ` ( N .^ X ) ) ) = ( ( B X. { A } ) .xb ( N F ( Q ` X ) ) ) )
63	37, 48, 62	3eqtrd 2660	1 |- ( ph -> ( Q ` ( A .X. ( N .^ X ) ) ) = ( ( B X. { A } ) .xb ( N F ( Q ` X ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650   NN0cn0 11292   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   ^_s cpws 16107   Mndcmnd 17294  ._gcmg 17540  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548   RingHom crh 18712   LModclmod 18863  AssAlgcasa 19309  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547  eval₁ce1 19679

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-int 4476  df-iun 4522  df-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-evl 19507  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-evls1 19680  df-evl1 19681

This theorem is referenced by: (None)