Theorem evl1expd 19709

Description: Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
evl1addd.q	|- O = (eval1 ` R )
evl1addd.p	|- P = (Poly1 ` R )
evl1addd.b	|- B = ( Base ` R )
evl1addd.u	|- U = ( Base ` P )
evl1addd.1	|- ( ph -> R e. CRing )
evl1addd.2	|- ( ph -> Y e. B )
evl1addd.3	|- ( ph -> ( M e. U /\ ( ( O ` M ) ` Y ) = V ) )
evl1expd.f	|- .xb = (.g ` (mulGrp ` P ) )
evl1expd.e	|- .^ = (.g ` (mulGrp ` R ) )
evl1expd.4	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
evl1expd	|- ( ph -> ( ( N .xb M ) e. U /\ ( ( O ` ( N .xb M ) ) ` Y ) = ( N .^ V ) ) )

Proof of Theorem evl1expd

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evl1addd.1	. . . . 5 |- ( ph -> R e. CRing )
2		crngring 18558	. . . . 5 |- ( R e. CRing -> R e. Ring )
3	1, 2	syl 17	. . . 4 |- ( ph -> R e. Ring )
4		evl1addd.p	. . . . 5 |- P = (Poly1 ` R )
5	4	ply1ring 19618	. . . 4 |- ( R e. Ring -> P e. Ring )
6		eqid 2622	. . . . 5 |- (mulGrp ` P ) = (mulGrp ` P )
7	6	ringmgp 18553	. . . 4 |- ( P e. Ring -> (mulGrp ` P ) e. Mnd )
8	3, 5, 7	3syl 18	. . 3 |- ( ph -> (mulGrp ` P ) e. Mnd )
9		evl1expd.4	. . 3 |- ( ph -> N e. NN0 )
10		evl1addd.3	. . . 4 |- ( ph -> ( M e. U /\ ( ( O ` M ) ` Y ) = V ) )
11	10	simpld 475	. . 3 |- ( ph -> M e. U )
12		evl1addd.u	. . . . 5 |- U = ( Base ` P )
13	6, 12	mgpbas 18495	. . . 4 |- U = ( Base ` (mulGrp ` P ) )
14		evl1expd.f	. . . 4 |- .xb = (.g ` (mulGrp ` P ) )
15	13, 14	mulgnn0cl 17558	. . 3 |- ( ( (mulGrp ` P ) e. Mnd /\ N e. NN0 /\ M e. U ) -> ( N .xb M ) e. U )
16	8, 9, 11, 15	syl3anc 1326	. 2 |- ( ph -> ( N .xb M ) e. U )
17		evl1addd.q	. . . . . . . . 9 |- O = (eval1 ` R )
18		eqid 2622	. . . . . . . . 9 |- ( R ^s B ) = ( R ^s B )
19		evl1addd.b	. . . . . . . . 9 |- B = ( Base ` R )
20	17, 4, 18, 19	evl1rhm 19696	. . . . . . . 8 |- ( R e. CRing -> O e. ( P RingHom ( R ^s B ) ) )
21	1, 20	syl 17	. . . . . . 7 |- ( ph -> O e. ( P RingHom ( R ^s B ) ) )
22		eqid 2622	. . . . . . . 8 |- (mulGrp ` ( R ^s B ) ) = (mulGrp ` ( R ^s B ) )
23	6, 22	rhmmhm 18722	. . . . . . 7 |- ( O e. ( P RingHom ( R ^s B ) ) -> O e. ( (mulGrp ` P ) MndHom (mulGrp ` ( R ^s B ) ) ) )
24	21, 23	syl 17	. . . . . 6 |- ( ph -> O e. ( (mulGrp ` P ) MndHom (mulGrp ` ( R ^s B ) ) ) )
25		eqid 2622	. . . . . . 7 |- (.g ` (mulGrp ` ( R ^s B ) ) ) = (.g ` (mulGrp ` ( R ^s B ) ) )
26	13, 14, 25	mhmmulg 17583	. . . . . 6 |- ( ( O e. ( (mulGrp ` P ) MndHom (mulGrp ` ( R ^s B ) ) ) /\ N e. NN0 /\ M e. U ) -> ( O ` ( N .xb M ) ) = ( N (.g ` (mulGrp ` ( R ^s B ) ) ) ( O ` M ) ) )
27	24, 9, 11, 26	syl3anc 1326	. . . . 5 |- ( ph -> ( O ` ( N .xb M ) ) = ( N (.g ` (mulGrp ` ( R ^s B ) ) ) ( O ` M ) ) )
28		eqid 2622	. . . . . . 7 |- (.g ` ( (mulGrp ` R ) ^s B ) ) = (.g ` ( (mulGrp ` R ) ^s B ) )
29		eqidd 2623	. . . . . . 7 |- ( ph -> ( Base ` (mulGrp ` ( R ^s B ) ) ) = ( Base ` (mulGrp ` ( R ^s B ) ) ) )
30		fvex 6201	. . . . . . . . . 10 |- ( Base ` R ) e. _V
31	19, 30	eqeltri 2697	. . . . . . . . 9 |- B e. _V
32		eqid 2622	. . . . . . . . . 10 |- (mulGrp ` R ) = (mulGrp ` R )
33		eqid 2622	. . . . . . . . . 10 |- ( (mulGrp ` R ) ^s B ) = ( (mulGrp ` R ) ^s B )
34		eqid 2622	. . . . . . . . . 10 |- ( Base ` (mulGrp ` ( R ^s B ) ) ) = ( Base ` (mulGrp ` ( R ^s B ) ) )
35		eqid 2622	. . . . . . . . . 10 |- ( Base ` ( (mulGrp ` R ) ^s B ) ) = ( Base ` ( (mulGrp ` R ) ^s B ) )
36		eqid 2622	. . . . . . . . . 10 |- ( +g ` (mulGrp ` ( R ^s B ) ) ) = ( +g ` (mulGrp ` ( R ^s B ) ) )
37		eqid 2622	. . . . . . . . . 10 |- ( +g ` ( (mulGrp ` R ) ^s B ) ) = ( +g ` ( (mulGrp ` R ) ^s B ) )
38	18, 32, 33, 22, 34, 35, 36, 37	pwsmgp 18618	. . . . . . . . 9 |- ( ( R e. CRing /\ B e. _V ) -> ( ( Base ` (mulGrp ` ( R ^s B ) ) ) = ( Base ` ( (mulGrp ` R ) ^s B ) ) /\ ( +g ` (mulGrp ` ( R ^s B ) ) ) = ( +g ` ( (mulGrp ` R ) ^s B ) ) ) )
39	1, 31, 38	sylancl 694	. . . . . . . 8 |- ( ph -> ( ( Base ` (mulGrp ` ( R ^s B ) ) ) = ( Base ` ( (mulGrp ` R ) ^s B ) ) /\ ( +g ` (mulGrp ` ( R ^s B ) ) ) = ( +g ` ( (mulGrp ` R ) ^s B ) ) ) )
40	39	simpld 475	. . . . . . 7 |- ( ph -> ( Base ` (mulGrp ` ( R ^s B ) ) ) = ( Base ` ( (mulGrp ` R ) ^s B ) ) )
41		ssv 3625	. . . . . . . 8 |- ( Base ` (mulGrp ` ( R ^s B ) ) ) C_ _V
42	41	a1i 11	. . . . . . 7 |- ( ph -> ( Base ` (mulGrp ` ( R ^s B ) ) ) C_ _V )
43		ovexd 6680	. . . . . . 7 |- ( ( ph /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` (mulGrp ` ( R ^s B ) ) ) y ) e. _V )
44	39	simprd 479	. . . . . . . 8 |- ( ph -> ( +g ` (mulGrp ` ( R ^s B ) ) ) = ( +g ` ( (mulGrp ` R ) ^s B ) ) )
45	44	oveqdr 6674	. . . . . . 7 |- ( ( ph /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` (mulGrp ` ( R ^s B ) ) ) y ) = ( x ( +g ` ( (mulGrp ` R ) ^s B ) ) y ) )
46	25, 28, 29, 40, 42, 43, 45	mulgpropd 17584	. . . . . 6 |- ( ph -> (.g ` (mulGrp ` ( R ^s B ) ) ) = (.g ` ( (mulGrp ` R ) ^s B ) ) )
47	46	oveqd 6667	. . . . 5 |- ( ph -> ( N (.g ` (mulGrp ` ( R ^s B ) ) ) ( O ` M ) ) = ( N (.g ` ( (mulGrp ` R ) ^s B ) ) ( O ` M ) ) )
48	27, 47	eqtrd 2656	. . . 4 |- ( ph -> ( O ` ( N .xb M ) ) = ( N (.g ` ( (mulGrp ` R ) ^s B ) ) ( O ` M ) ) )
49	48	fveq1d 6193	. . 3 |- ( ph -> ( ( O ` ( N .xb M ) ) ` Y ) = ( ( N (.g ` ( (mulGrp ` R ) ^s B ) ) ( O ` M ) ) ` Y ) )
50	32	ringmgp 18553	. . . . . 6 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
51	3, 50	syl 17	. . . . 5 |- ( ph -> (mulGrp ` R ) e. Mnd )
52	31	a1i 11	. . . . 5 |- ( ph -> B e. _V )
53		eqid 2622	. . . . . . . . 9 |- ( Base ` ( R ^s B ) ) = ( Base ` ( R ^s B ) )
54	12, 53	rhmf 18726	. . . . . . . 8 |- ( O e. ( P RingHom ( R ^s B ) ) -> O : U --> ( Base ` ( R ^s B ) ) )
55	21, 54	syl 17	. . . . . . 7 |- ( ph -> O : U --> ( Base ` ( R ^s B ) ) )
56	55, 11	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( O ` M ) e. ( Base ` ( R ^s B ) ) )
57	22, 53	mgpbas 18495	. . . . . . 7 |- ( Base ` ( R ^s B ) ) = ( Base ` (mulGrp ` ( R ^s B ) ) )
58	57, 40	syl5eq 2668	. . . . . 6 |- ( ph -> ( Base ` ( R ^s B ) ) = ( Base ` ( (mulGrp ` R ) ^s B ) ) )
59	56, 58	eleqtrd 2703	. . . . 5 |- ( ph -> ( O ` M ) e. ( Base ` ( (mulGrp ` R ) ^s B ) ) )
60		evl1addd.2	. . . . 5 |- ( ph -> Y e. B )
61		evl1expd.e	. . . . . 6 |- .^ = (.g ` (mulGrp ` R ) )
62	33, 35, 28, 61	pwsmulg 17587	. . . . 5 |- ( ( ( (mulGrp ` R ) e. Mnd /\ B e. _V ) /\ ( N e. NN0 /\ ( O ` M ) e. ( Base ` ( (mulGrp ` R ) ^s B ) ) /\ Y e. B ) ) -> ( ( N (.g ` ( (mulGrp ` R ) ^s B ) ) ( O ` M ) ) ` Y ) = ( N .^ ( ( O ` M ) ` Y ) ) )
63	51, 52, 9, 59, 60, 62	syl23anc 1333	. . . 4 |- ( ph -> ( ( N (.g ` ( (mulGrp ` R ) ^s B ) ) ( O ` M ) ) ` Y ) = ( N .^ ( ( O ` M ) ` Y ) ) )
64	10	simprd 479	. . . . 5 |- ( ph -> ( ( O ` M ) ` Y ) = V )
65	64	oveq2d 6666	. . . 4 |- ( ph -> ( N .^ ( ( O ` M ) ` Y ) ) = ( N .^ V ) )
66	63, 65	eqtrd 2656	. . 3 |- ( ph -> ( ( N (.g ` ( (mulGrp ` R ) ^s B ) ) ( O ` M ) ) ` Y ) = ( N .^ V ) )
67	49, 66	eqtrd 2656	. 2 |- ( ph -> ( ( O ` ( N .xb M ) ) ` Y ) = ( N .^ V ) )
68	16, 67	jca 554	1 |- ( ph -> ( ( N .xb M ) e. U /\ ( ( O ` ( N .xb M ) ) ` Y ) = ( N .^ V ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   -->wf 5884   ` cfv 5888  (class class class)co 6650   NN0cn0 11292   Basecbs 15857   +g cplusg 15941   ^_s cpws 16107   Mndcmnd 17294   MndHom cmhm 17333  ._gcmg 17540  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548   RingHom crh 18712  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-ply1 19552  df-evl1 19681

This theorem is referenced by:  evl1varpwval  19726  plypf1  23968  lgsqrlem1  25071  idomrootle  37773