Theorem fta1glem1 23925

Description: Lemma for fta1g 23927. (Contributed by Mario Carneiro, 7-Jun-2016.)

Hypotheses

Ref	Expression
fta1g.p	|- P = (Poly1 ` R )
fta1g.b	|- B = ( Base ` P )
fta1g.d	|- D = ( deg1 ` R )
fta1g.o	|- O = (eval1 ` R )
fta1g.w	|- W = ( 0g ` R )
fta1g.z	|- .0. = ( 0g ` P )
fta1g.1	|- ( ph -> R e. IDomn )
fta1g.2	|- ( ph -> F e. B )
fta1glem.k	|- K = ( Base ` R )
fta1glem.x	|- X = (var1 ` R )
fta1glem.m	|- .- = ( -g ` P )
fta1glem.a	|- A = (algSc ` P )
fta1glem.g	|- G = ( X .- ( A ` T ) )
fta1glem.3	|- ( ph -> N e. NN0 )
fta1glem.4	|- ( ph -> ( D ` F ) = ( N + 1 ) )
fta1glem.5	|- ( ph -> T e. ( `' ( O ` F ) " { W } ) )

Assertion

Ref	Expression
fta1glem1	|- ( ph -> ( D ` ( F (quot1p ` R ) G ) ) = N )

Proof of Theorem fta1glem1

Step	Hyp	Ref	Expression
1		1cnd 10056	. 2 |- ( ph -> 1 e. CC )
2		fta1g.1	. . . . . 6 |- ( ph -> R e. IDomn )
3		isidom 19304	. . . . . . . 8 |- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) )
4	3	simprbi 480	. . . . . . 7 |- ( R e. IDomn -> R e. Domn )
5		domnnzr 19295	. . . . . . 7 |- ( R e. Domn -> R e. NzRing )
6	4, 5	syl 17	. . . . . 6 |- ( R e. IDomn -> R e. NzRing )
7	2, 6	syl 17	. . . . 5 |- ( ph -> R e. NzRing )
8		nzrring 19261	. . . . 5 |- ( R e. NzRing -> R e. Ring )
9	7, 8	syl 17	. . . 4 |- ( ph -> R e. Ring )
10		fta1g.2	. . . . 5 |- ( ph -> F e. B )
11		fta1g.p	. . . . . . . 8 |- P = (Poly1 ` R )
12		fta1g.b	. . . . . . . 8 |- B = ( Base ` P )
13		fta1glem.k	. . . . . . . 8 |- K = ( Base ` R )
14		fta1glem.x	. . . . . . . 8 |- X = (var1 ` R )
15		fta1glem.m	. . . . . . . 8 |- .- = ( -g ` P )
16		fta1glem.a	. . . . . . . 8 |- A = (algSc ` P )
17		fta1glem.g	. . . . . . . 8 |- G = ( X .- ( A ` T ) )
18		fta1g.o	. . . . . . . 8 |- O = (eval1 ` R )
19	3	simplbi 476	. . . . . . . . 9 |- ( R e. IDomn -> R e. CRing )
20	2, 19	syl 17	. . . . . . . 8 |- ( ph -> R e. CRing )
21		fta1glem.5	. . . . . . . . . 10 |- ( ph -> T e. ( `' ( O ` F ) " { W } ) )
22		eqid 2622	. . . . . . . . . . . . 13 |- ( R ^s K ) = ( R ^s K )
23		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` ( R ^s K ) ) = ( Base ` ( R ^s K ) )
24		fvex 6201	. . . . . . . . . . . . . . 15 |- ( Base ` R ) e. _V
25	13, 24	eqeltri 2697	. . . . . . . . . . . . . 14 |- K e. _V
26	25	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> K e. _V )
27	18, 11, 22, 13	evl1rhm 19696	. . . . . . . . . . . . . . . 16 |- ( R e. CRing -> O e. ( P RingHom ( R ^s K ) ) )
28	20, 27	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> O e. ( P RingHom ( R ^s K ) ) )
29	12, 23	rhmf 18726	. . . . . . . . . . . . . . 15 |- ( O e. ( P RingHom ( R ^s K ) ) -> O : B --> ( Base ` ( R ^s K ) ) )
30	28, 29	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> O : B --> ( Base ` ( R ^s K ) ) )
31	30, 10	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ph -> ( O ` F ) e. ( Base ` ( R ^s K ) ) )
32	22, 13, 23, 2, 26, 31	pwselbas 16149	. . . . . . . . . . . 12 |- ( ph -> ( O ` F ) : K --> K )
33		ffn 6045	. . . . . . . . . . . 12 |- ( ( O ` F ) : K --> K -> ( O ` F ) Fn K )
34	32, 33	syl 17	. . . . . . . . . . 11 |- ( ph -> ( O ` F ) Fn K )
35		fniniseg 6338	. . . . . . . . . . 11 |- ( ( O ` F ) Fn K -> ( T e. ( `' ( O ` F ) " { W } ) <-> ( T e. K /\ ( ( O ` F ) ` T ) = W ) ) )
36	34, 35	syl 17	. . . . . . . . . 10 |- ( ph -> ( T e. ( `' ( O ` F ) " { W } ) <-> ( T e. K /\ ( ( O ` F ) ` T ) = W ) ) )
37	21, 36	mpbid 222	. . . . . . . . 9 |- ( ph -> ( T e. K /\ ( ( O ` F ) ` T ) = W ) )
38	37	simpld 475	. . . . . . . 8 |- ( ph -> T e. K )
39		eqid 2622	. . . . . . . 8 |- (Monic1p ` R ) = (Monic1p ` R )
40		fta1g.d	. . . . . . . 8 |- D = ( deg1 ` R )
41		fta1g.w	. . . . . . . 8 |- W = ( 0g ` R )
42	11, 12, 13, 14, 15, 16, 17, 18, 7, 20, 38, 39, 40, 41	ply1remlem 23922	. . . . . . 7 |- ( ph -> ( G e. (Monic1p ` R ) /\ ( D ` G ) = 1 /\ ( `' ( O ` G ) " { W } ) = { T } ) )
43	42	simp1d 1073	. . . . . 6 |- ( ph -> G e. (Monic1p ` R ) )
44		eqid 2622	. . . . . . 7 |- (Unic1p ` R ) = (Unic1p ` R )
45	44, 39	mon1puc1p 23910	. . . . . 6 |- ( ( R e. Ring /\ G e. (Monic1p ` R ) ) -> G e. (Unic1p ` R ) )
46	9, 43, 45	syl2anc 693	. . . . 5 |- ( ph -> G e. (Unic1p ` R ) )
47		eqid 2622	. . . . . 6 |- (quot1p ` R ) = (quot1p ` R )
48	47, 11, 12, 44	q1pcl 23915	. . . . 5 |- ( ( R e. Ring /\ F e. B /\ G e. (Unic1p ` R ) ) -> ( F (quot1p ` R ) G ) e. B )
49	9, 10, 46, 48	syl3anc 1326	. . . 4 |- ( ph -> ( F (quot1p ` R ) G ) e. B )
50		fta1glem.4	. . . . . . . 8 |- ( ph -> ( D ` F ) = ( N + 1 ) )
51		fta1glem.3	. . . . . . . . 9 |- ( ph -> N e. NN0 )
52		peano2nn0 11333	. . . . . . . . 9 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
53	51, 52	syl 17	. . . . . . . 8 |- ( ph -> ( N + 1 ) e. NN0 )
54	50, 53	eqeltrd 2701	. . . . . . 7 |- ( ph -> ( D ` F ) e. NN0 )
55		fta1g.z	. . . . . . . . 9 |- .0. = ( 0g ` P )
56	40, 11, 55, 12	deg1nn0clb 23850	. . . . . . . 8 |- ( ( R e. Ring /\ F e. B ) -> ( F =/= .0. <-> ( D ` F ) e. NN0 ) )
57	9, 10, 56	syl2anc 693	. . . . . . 7 |- ( ph -> ( F =/= .0. <-> ( D ` F ) e. NN0 ) )
58	54, 57	mpbird 247	. . . . . 6 |- ( ph -> F =/= .0. )
59	37	simprd 479	. . . . . . . . 9 |- ( ph -> ( ( O ` F ) ` T ) = W )
60		eqid 2622	. . . . . . . . . 10 |- ( ||r ` P ) = ( ||r ` P )
61	11, 12, 13, 14, 15, 16, 17, 18, 7, 20, 38, 10, 41, 60	facth1 23924	. . . . . . . . 9 |- ( ph -> ( G ( ||r ` P ) F <-> ( ( O ` F ) ` T ) = W ) )
62	59, 61	mpbird 247	. . . . . . . 8 |- ( ph -> G ( ||r ` P ) F )
63		eqid 2622	. . . . . . . . . 10 |- ( .r ` P ) = ( .r ` P )
64	11, 60, 12, 44, 63, 47	dvdsq1p 23920	. . . . . . . . 9 |- ( ( R e. Ring /\ F e. B /\ G e. (Unic1p ` R ) ) -> ( G ( ||r ` P ) F <-> F = ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) )
65	9, 10, 46, 64	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( G ( ||r ` P ) F <-> F = ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) )
66	62, 65	mpbid 222	. . . . . . 7 |- ( ph -> F = ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) )
67	66	eqcomd 2628	. . . . . 6 |- ( ph -> ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) = F )
68	11	ply1crng 19568	. . . . . . . . 9 |- ( R e. CRing -> P e. CRing )
69	20, 68	syl 17	. . . . . . . 8 |- ( ph -> P e. CRing )
70		crngring 18558	. . . . . . . 8 |- ( P e. CRing -> P e. Ring )
71	69, 70	syl 17	. . . . . . 7 |- ( ph -> P e. Ring )
72	11, 12, 39	mon1pcl 23904	. . . . . . . 8 |- ( G e. (Monic1p ` R ) -> G e. B )
73	43, 72	syl 17	. . . . . . 7 |- ( ph -> G e. B )
74	12, 63, 55	ringlz 18587	. . . . . . 7 |- ( ( P e. Ring /\ G e. B ) -> ( .0. ( .r ` P ) G ) = .0. )
75	71, 73, 74	syl2anc 693	. . . . . 6 |- ( ph -> ( .0. ( .r ` P ) G ) = .0. )
76	58, 67, 75	3netr4d 2871	. . . . 5 |- ( ph -> ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) =/= ( .0. ( .r ` P ) G ) )
77		oveq1 6657	. . . . . 6 |- ( ( F (quot1p ` R ) G ) = .0. -> ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) = ( .0. ( .r ` P ) G ) )
78	77	necon3i 2826	. . . . 5 |- ( ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) =/= ( .0. ( .r ` P ) G ) -> ( F (quot1p ` R ) G ) =/= .0. )
79	76, 78	syl 17	. . . 4 |- ( ph -> ( F (quot1p ` R ) G ) =/= .0. )
80	40, 11, 55, 12	deg1nn0cl 23848	. . . 4 |- ( ( R e. Ring /\ ( F (quot1p ` R ) G ) e. B /\ ( F (quot1p ` R ) G ) =/= .0. ) -> ( D ` ( F (quot1p ` R ) G ) ) e. NN0 )
81	9, 49, 79, 80	syl3anc 1326	. . 3 |- ( ph -> ( D ` ( F (quot1p ` R ) G ) ) e. NN0 )
82	81	nn0cnd 11353	. 2 |- ( ph -> ( D ` ( F (quot1p ` R ) G ) ) e. CC )
83	51	nn0cnd 11353	. 2 |- ( ph -> N e. CC )
84	12, 63	crngcom 18562	. . . . . . 7 |- ( ( P e. CRing /\ ( F (quot1p ` R ) G ) e. B /\ G e. B ) -> ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) = ( G ( .r ` P ) ( F (quot1p ` R ) G ) ) )
85	69, 49, 73, 84	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) = ( G ( .r ` P ) ( F (quot1p ` R ) G ) ) )
86	66, 85	eqtrd 2656	. . . . 5 |- ( ph -> F = ( G ( .r ` P ) ( F (quot1p ` R ) G ) ) )
87	86	fveq2d 6195	. . . 4 |- ( ph -> ( D ` F ) = ( D ` ( G ( .r ` P ) ( F (quot1p ` R ) G ) ) ) )
88		eqid 2622	. . . . 5 |- (RLReg ` R ) = (RLReg ` R )
89	42	simp2d 1074	. . . . . . 7 |- ( ph -> ( D ` G ) = 1 )
90		1nn0 11308	. . . . . . 7 |- 1 e. NN0
91	89, 90	syl6eqel 2709	. . . . . 6 |- ( ph -> ( D ` G ) e. NN0 )
92	40, 11, 55, 12	deg1nn0clb 23850	. . . . . . 7 |- ( ( R e. Ring /\ G e. B ) -> ( G =/= .0. <-> ( D ` G ) e. NN0 ) )
93	9, 73, 92	syl2anc 693	. . . . . 6 |- ( ph -> ( G =/= .0. <-> ( D ` G ) e. NN0 ) )
94	91, 93	mpbird 247	. . . . 5 |- ( ph -> G =/= .0. )
95		eqid 2622	. . . . . . . 8 |- (Unit ` R ) = (Unit ` R )
96	88, 95	unitrrg 19293	. . . . . . 7 |- ( R e. Ring -> (Unit ` R ) C_ (RLReg ` R ) )
97	9, 96	syl 17	. . . . . 6 |- ( ph -> (Unit ` R ) C_ (RLReg ` R ) )
98	40, 95, 44	uc1pldg 23908	. . . . . . 7 |- ( G e. (Unic1p ` R ) -> ( (coe1 ` G ) ` ( D ` G ) ) e. (Unit ` R ) )
99	46, 98	syl 17	. . . . . 6 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) e. (Unit ` R ) )
100	97, 99	sseldd 3604	. . . . 5 |- ( ph -> ( (coe1 ` G ) ` ( D ` G ) ) e. (RLReg ` R ) )
101	40, 11, 88, 12, 63, 55, 9, 73, 94, 100, 49, 79	deg1mul2 23874	. . . 4 |- ( ph -> ( D ` ( G ( .r ` P ) ( F (quot1p ` R ) G ) ) ) = ( ( D ` G ) + ( D ` ( F (quot1p ` R ) G ) ) ) )
102	87, 50, 101	3eqtr3d 2664	. . 3 |- ( ph -> ( N + 1 ) = ( ( D ` G ) + ( D ` ( F (quot1p ` R ) G ) ) ) )
103		ax-1cn 9994	. . . 4 |- 1 e. CC
104		addcom 10222	. . . 4 |- ( ( N e. CC /\ 1 e. CC ) -> ( N + 1 ) = ( 1 + N ) )
105	83, 103, 104	sylancl 694	. . 3 |- ( ph -> ( N + 1 ) = ( 1 + N ) )
106	89	oveq1d 6665	. . 3 |- ( ph -> ( ( D ` G ) + ( D ` ( F (quot1p ` R ) G ) ) ) = ( 1 + ( D ` ( F (quot1p ` R ) G ) ) ) )
107	102, 105, 106	3eqtr3rd 2665	. 2 |- ( ph -> ( 1 + ( D ` ( F (quot1p ` R ) G ) ) ) = ( 1 + N ) )
108	1, 82, 83, 107	addcanad 10241	1 |- ( ph -> ( D ` ( F (quot1p ` R ) G ) ) = N )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   {csn 4177   class class class wbr 4653   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   NN0cn0 11292   Basecbs 15857   .rcmulr 15942   0gc0g 16100   ^_s cpws 16107   -gcsg 17424   Ringcrg 18547   CRingccrg 18548   ||_rcdsr 18638  Unitcui 18639   RingHom crh 18712  NzRingcnzr 19257  RLRegcrlreg 19279  Domncdomn 19280  IDomncidom 19281  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548  eval₁ce1 19679   deg₁ cdg1 23814  Monic_1pcmn1 23885  Unic_1pcuc1p 23886  quot_1pcq1p 23887

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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

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-tpos 7352  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-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  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-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-nzr 19258  df-rlreg 19283  df-domn 19284  df-idom 19285  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-coe1 19553  df-evl1 19681  df-cnfld 19747  df-mdeg 23815  df-deg1 23816  df-mon1 23890  df-uc1p 23891  df-q1p 23892  df-r1p 23893

This theorem is referenced by:  fta1glem2  23926