Theorem coe1subfv 19636

Description: A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Hypotheses

Ref	Expression
coe1sub.y	|- Y = (Poly1 ` R )
coe1sub.b	|- B = ( Base ` Y )
coe1sub.p	|- .- = ( -g ` Y )
coe1sub.q	|- N = ( -g ` R )

Assertion

Ref	Expression
coe1subfv	|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( (coe1 ` ( F .- G ) ) ` X ) = ( ( (coe1 ` F ) ` X ) N ( (coe1 ` G ) ` X ) ) )

Proof of Theorem coe1subfv

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> R e. Ring )
2		coe1sub.y	. . . . . . . . 9 |- Y = (Poly1 ` R )
3	2	ply1ring 19618	. . . . . . . 8 |- ( R e. Ring -> Y e. Ring )
4		ringgrp 18552	. . . . . . . 8 |- ( Y e. Ring -> Y e. Grp )
5	3, 4	syl 17	. . . . . . 7 |- ( R e. Ring -> Y e. Grp )
6		coe1sub.b	. . . . . . . 8 |- B = ( Base ` Y )
7		coe1sub.p	. . . . . . . 8 |- .- = ( -g ` Y )
8	6, 7	grpsubcl 17495	. . . . . . 7 |- ( ( Y e. Grp /\ F e. B /\ G e. B ) -> ( F .- G ) e. B )
9	5, 8	syl3an1 1359	. . . . . 6 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( F .- G ) e. B )
10	9	adantr 481	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( F .- G ) e. B )
11		simpl3 1066	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> G e. B )
12		simpr 477	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> X e. NN0 )
13		eqid 2622	. . . . . 6 |- ( +g ` Y ) = ( +g ` Y )
14		eqid 2622	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
15	2, 6, 13, 14	coe1addfv 19635	. . . . 5 |- ( ( ( R e. Ring /\ ( F .- G ) e. B /\ G e. B ) /\ X e. NN0 ) -> ( (coe1 ` ( ( F .- G ) ( +g ` Y ) G ) ) ` X ) = ( ( (coe1 ` ( F .- G ) ) ` X ) ( +g ` R ) ( (coe1 ` G ) ` X ) ) )
16	1, 10, 11, 12, 15	syl31anc 1329	. . . 4 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( (coe1 ` ( ( F .- G ) ( +g ` Y ) G ) ) ` X ) = ( ( (coe1 ` ( F .- G ) ) ` X ) ( +g ` R ) ( (coe1 ` G ) ` X ) ) )
17	5	3ad2ant1 1082	. . . . . . . 8 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> Y e. Grp )
18	17	adantr 481	. . . . . . 7 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> Y e. Grp )
19		simpl2 1065	. . . . . . 7 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> F e. B )
20	6, 13, 7	grpnpcan 17507	. . . . . . 7 |- ( ( Y e. Grp /\ F e. B /\ G e. B ) -> ( ( F .- G ) ( +g ` Y ) G ) = F )
21	18, 19, 11, 20	syl3anc 1326	. . . . . 6 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( F .- G ) ( +g ` Y ) G ) = F )
22	21	fveq2d 6195	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> (coe1 ` ( ( F .- G ) ( +g ` Y ) G ) ) = (coe1 ` F ) )
23	22	fveq1d 6193	. . . 4 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( (coe1 ` ( ( F .- G ) ( +g ` Y ) G ) ) ` X ) = ( (coe1 ` F ) ` X ) )
24	16, 23	eqtr3d 2658	. . 3 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( (coe1 ` ( F .- G ) ) ` X ) ( +g ` R ) ( (coe1 ` G ) ` X ) ) = ( (coe1 ` F ) ` X ) )
25		ringgrp 18552	. . . . . 6 |- ( R e. Ring -> R e. Grp )
26	25	3ad2ant1 1082	. . . . 5 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> R e. Grp )
27	26	adantr 481	. . . 4 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> R e. Grp )
28		eqid 2622	. . . . . . 7 |- (coe1 ` F ) = (coe1 ` F )
29		eqid 2622	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
30	28, 6, 2, 29	coe1f 19581	. . . . . 6 |- ( F e. B -> (coe1 ` F ) : NN0 --> ( Base ` R ) )
31	30	3ad2ant2 1083	. . . . 5 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> (coe1 ` F ) : NN0 --> ( Base ` R ) )
32	31	ffvelrnda 6359	. . . 4 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( (coe1 ` F ) ` X ) e. ( Base ` R ) )
33		eqid 2622	. . . . . . 7 |- (coe1 ` G ) = (coe1 ` G )
34	33, 6, 2, 29	coe1f 19581	. . . . . 6 |- ( G e. B -> (coe1 ` G ) : NN0 --> ( Base ` R ) )
35	34	3ad2ant3 1084	. . . . 5 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> (coe1 ` G ) : NN0 --> ( Base ` R ) )
36	35	ffvelrnda 6359	. . . 4 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( (coe1 ` G ) ` X ) e. ( Base ` R ) )
37		eqid 2622	. . . . . . 7 |- (coe1 ` ( F .- G ) ) = (coe1 ` ( F .- G ) )
38	37, 6, 2, 29	coe1f 19581	. . . . . 6 |- ( ( F .- G ) e. B -> (coe1 ` ( F .- G ) ) : NN0 --> ( Base ` R ) )
39	9, 38	syl 17	. . . . 5 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> (coe1 ` ( F .- G ) ) : NN0 --> ( Base ` R ) )
40	39	ffvelrnda 6359	. . . 4 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( (coe1 ` ( F .- G ) ) ` X ) e. ( Base ` R ) )
41		coe1sub.q	. . . . 5 |- N = ( -g ` R )
42	29, 14, 41	grpsubadd 17503	. . . 4 |- ( ( R e. Grp /\ ( ( (coe1 ` F ) ` X ) e. ( Base ` R ) /\ ( (coe1 ` G ) ` X ) e. ( Base ` R ) /\ ( (coe1 ` ( F .- G ) ) ` X ) e. ( Base ` R ) ) ) -> ( ( ( (coe1 ` F ) ` X ) N ( (coe1 ` G ) ` X ) ) = ( (coe1 ` ( F .- G ) ) ` X ) <-> ( ( (coe1 ` ( F .- G ) ) ` X ) ( +g ` R ) ( (coe1 ` G ) ` X ) ) = ( (coe1 ` F ) ` X ) ) )
43	27, 32, 36, 40, 42	syl13anc 1328	. . 3 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( ( (coe1 ` F ) ` X ) N ( (coe1 ` G ) ` X ) ) = ( (coe1 ` ( F .- G ) ) ` X ) <-> ( ( (coe1 ` ( F .- G ) ) ` X ) ( +g ` R ) ( (coe1 ` G ) ` X ) ) = ( (coe1 ` F ) ` X ) ) )
44	24, 43	mpbird 247	. 2 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( (coe1 ` F ) ` X ) N ( (coe1 ` G ) ` X ) ) = ( (coe1 ` ( F .- G ) ) ` X ) )
45	44	eqcomd 2628	1 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( (coe1 ` ( F .- G ) ) ` X ) = ( ( (coe1 ` F ) ` X ) N ( (coe1 ` G ) ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   -->wf 5884   ` cfv 5888  (class class class)co 6650   NN0cn0 11292   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   -gcsg 17424   Ringcrg 18547  Poly₁cpl1 19547  coe₁cco1 19548

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-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-tset 15960  df-ple 15961  df-0g 16102  df-gsum 16103  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-ring 18549  df-subrg 18778  df-psr 19356  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-ply1 19552  df-coe1 19553

This theorem is referenced by:  deg1sublt  23870  ply1remlem  23922