Theorem fta1glem2 23926

Description: Lemma for fta1g 23927. (Contributed by Mario Carneiro, 12-Jun-2015.)

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 } ) )
fta1glem.6	|- ( ph -> A. g e. B ( ( D ` g ) = N -> ( # ` ( `' ( O ` g ) " { W } ) ) <_ ( D ` g ) ) )

Assertion

Ref	Expression
fta1glem2	|- ( ph -> ( # ` ( `' ( O ` F ) " { W } ) ) <_ ( D ` F ) )

Distinct variable groups:   B, g   D, g   g, F   g, N   g, O   g, G   P, g   R, g   g, W

Allowed substitution hints:   ph( g)   A( g)   T( g)   K( g)   .- ( g)   X( g)   .0. ( g)

Proof of Theorem fta1glem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fta1glem.5	. . . . . . . . . . . . . . . . . 18 |- ( ph -> T e. ( `' ( O ` F ) " { W } ) )
2		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 |- ( R ^s K ) = ( R ^s K )
3		fta1glem.k	. . . . . . . . . . . . . . . . . . . . 21 |- K = ( Base ` R )
4		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 |- ( Base ` ( R ^s K ) ) = ( Base ` ( R ^s K ) )
5		fta1g.1	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> R e. IDomn )
6		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( Base ` R ) e. _V
7	3, 6	eqeltri 2697	. . . . . . . . . . . . . . . . . . . . . 22 |- K e. _V
8	7	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> K e. _V )
9		isidom 19304	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) )
10	9	simplbi 476	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( R e. IDomn -> R e. CRing )
11	5, 10	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ph -> R e. CRing )
12		fta1g.o	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- O = (eval1 ` R )
13		fta1g.p	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- P = (Poly1 ` R )
14	12, 13, 2, 3	evl1rhm 19696	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( R e. CRing -> O e. ( P RingHom ( R ^s K ) ) )
15	11, 14	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> O e. ( P RingHom ( R ^s K ) ) )
16		fta1g.b	. . . . . . . . . . . . . . . . . . . . . . . 24 |- B = ( Base ` P )
17	16, 4	rhmf 18726	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( O e. ( P RingHom ( R ^s K ) ) -> O : B --> ( Base ` ( R ^s K ) ) )
18	15, 17	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> O : B --> ( Base ` ( R ^s K ) ) )
19		fta1g.2	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> F e. B )
20	18, 19	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> ( O ` F ) e. ( Base ` ( R ^s K ) ) )
21	2, 3, 4, 5, 8, 20	pwselbas 16149	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( O ` F ) : K --> K )
22	21	ffnd 6046	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( O ` F ) Fn K )
23		fniniseg 6338	. . . . . . . . . . . . . . . . . . 19 |- ( ( O ` F ) Fn K -> ( T e. ( `' ( O ` F ) " { W } ) <-> ( T e. K /\ ( ( O ` F ) ` T ) = W ) ) )
24	22, 23	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( T e. ( `' ( O ` F ) " { W } ) <-> ( T e. K /\ ( ( O ` F ) ` T ) = W ) ) )
25	1, 24	mpbid 222	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( T e. K /\ ( ( O ` F ) ` T ) = W ) )
26	25	simprd 479	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( O ` F ) ` T ) = W )
27		fta1glem.x	. . . . . . . . . . . . . . . . 17 |- X = (var1 ` R )
28		fta1glem.m	. . . . . . . . . . . . . . . . 17 |- .- = ( -g ` P )
29		fta1glem.a	. . . . . . . . . . . . . . . . 17 |- A = (algSc ` P )
30		fta1glem.g	. . . . . . . . . . . . . . . . 17 |- G = ( X .- ( A ` T ) )
31	9	simprbi 480	. . . . . . . . . . . . . . . . . . 19 |- ( R e. IDomn -> R e. Domn )
32		domnnzr 19295	. . . . . . . . . . . . . . . . . . 19 |- ( R e. Domn -> R e. NzRing )
33	31, 32	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( R e. IDomn -> R e. NzRing )
34	5, 33	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> R e. NzRing )
35	25	simpld 475	. . . . . . . . . . . . . . . . 17 |- ( ph -> T e. K )
36		fta1g.w	. . . . . . . . . . . . . . . . 17 |- W = ( 0g ` R )
37		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( ||r ` P ) = ( ||r ` P )
38	13, 16, 3, 27, 28, 29, 30, 12, 34, 11, 35, 19, 36, 37	facth1 23924	. . . . . . . . . . . . . . . 16 |- ( ph -> ( G ( ||r ` P ) F <-> ( ( O ` F ) ` T ) = W ) )
39	26, 38	mpbird 247	. . . . . . . . . . . . . . 15 |- ( ph -> G ( ||r ` P ) F )
40		nzrring 19261	. . . . . . . . . . . . . . . . 17 |- ( R e. NzRing -> R e. Ring )
41	34, 40	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> R e. Ring )
42		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- (Monic1p ` R ) = (Monic1p ` R )
43		fta1g.d	. . . . . . . . . . . . . . . . . . 19 |- D = ( deg1 ` R )
44	13, 16, 3, 27, 28, 29, 30, 12, 34, 11, 35, 42, 43, 36	ply1remlem 23922	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( G e. (Monic1p ` R ) /\ ( D ` G ) = 1 /\ ( `' ( O ` G ) " { W } ) = { T } ) )
45	44	simp1d 1073	. . . . . . . . . . . . . . . . 17 |- ( ph -> G e. (Monic1p ` R ) )
46		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- (Unic1p ` R ) = (Unic1p ` R )
47	46, 42	mon1puc1p 23910	. . . . . . . . . . . . . . . . 17 |- ( ( R e. Ring /\ G e. (Monic1p ` R ) ) -> G e. (Unic1p ` R ) )
48	41, 45, 47	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ph -> G e. (Unic1p ` R ) )
49		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( .r ` P ) = ( .r ` P )
50		eqid 2622	. . . . . . . . . . . . . . . . 17 |- (quot1p ` R ) = (quot1p ` R )
51	13, 37, 16, 46, 49, 50	dvdsq1p 23920	. . . . . . . . . . . . . . . 16 |- ( ( R e. Ring /\ F e. B /\ G e. (Unic1p ` R ) ) -> ( G ( ||r ` P ) F <-> F = ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) )
52	41, 19, 48, 51	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ph -> ( G ( ||r ` P ) F <-> F = ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) )
53	39, 52	mpbid 222	. . . . . . . . . . . . . 14 |- ( ph -> F = ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) )
54	53	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ph -> ( O ` F ) = ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) )
55	50, 13, 16, 46	q1pcl 23915	. . . . . . . . . . . . . . 15 |- ( ( R e. Ring /\ F e. B /\ G e. (Unic1p ` R ) ) -> ( F (quot1p ` R ) G ) e. B )
56	41, 19, 48, 55	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ph -> ( F (quot1p ` R ) G ) e. B )
57	13, 16, 42	mon1pcl 23904	. . . . . . . . . . . . . . 15 |- ( G e. (Monic1p ` R ) -> G e. B )
58	45, 57	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> G e. B )
59		eqid 2622	. . . . . . . . . . . . . . 15 |- ( .r ` ( R ^s K ) ) = ( .r ` ( R ^s K ) )
60	16, 49, 59	rhmmul 18727	. . . . . . . . . . . . . 14 |- ( ( O e. ( P RingHom ( R ^s K ) ) /\ ( F (quot1p ` R ) G ) e. B /\ G e. B ) -> ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) = ( ( O ` ( F (quot1p ` R ) G ) ) ( .r ` ( R ^s K ) ) ( O ` G ) ) )
61	15, 56, 58, 60	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ph -> ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) = ( ( O ` ( F (quot1p ` R ) G ) ) ( .r ` ( R ^s K ) ) ( O ` G ) ) )
62	18, 56	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ph -> ( O ` ( F (quot1p ` R ) G ) ) e. ( Base ` ( R ^s K ) ) )
63	18, 58	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ph -> ( O ` G ) e. ( Base ` ( R ^s K ) ) )
64		eqid 2622	. . . . . . . . . . . . . 14 |- ( .r ` R ) = ( .r ` R )
65	2, 4, 5, 8, 62, 63, 64, 59	pwsmulrval 16151	. . . . . . . . . . . . 13 |- ( ph -> ( ( O ` ( F (quot1p ` R ) G ) ) ( .r ` ( R ^s K ) ) ( O ` G ) ) = ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) )
66	54, 61, 65	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ph -> ( O ` F ) = ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) )
67	66	fveq1d 6193	. . . . . . . . . . 11 |- ( ph -> ( ( O ` F ) ` x ) = ( ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) ` x ) )
68	67	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. K ) -> ( ( O ` F ) ` x ) = ( ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) ` x ) )
69	2, 3, 4, 5, 8, 62	pwselbas 16149	. . . . . . . . . . . . 13 |- ( ph -> ( O ` ( F (quot1p ` R ) G ) ) : K --> K )
70	69	ffnd 6046	. . . . . . . . . . . 12 |- ( ph -> ( O ` ( F (quot1p ` R ) G ) ) Fn K )
71	70	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> ( O ` ( F (quot1p ` R ) G ) ) Fn K )
72	2, 3, 4, 5, 8, 63	pwselbas 16149	. . . . . . . . . . . . 13 |- ( ph -> ( O ` G ) : K --> K )
73	72	ffnd 6046	. . . . . . . . . . . 12 |- ( ph -> ( O ` G ) Fn K )
74	73	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> ( O ` G ) Fn K )
75	7	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> K e. _V )
76		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ x e. K ) -> x e. K )
77		fnfvof 6911	. . . . . . . . . . 11 |- ( ( ( ( O ` ( F (quot1p ` R ) G ) ) Fn K /\ ( O ` G ) Fn K ) /\ ( K e. _V /\ x e. K ) ) -> ( ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) ` x ) = ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) ( .r ` R ) ( ( O ` G ) ` x ) ) )
78	71, 74, 75, 76, 77	syl22anc 1327	. . . . . . . . . 10 |- ( ( ph /\ x e. K ) -> ( ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) ` x ) = ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) ( .r ` R ) ( ( O ` G ) ` x ) ) )
79	68, 78	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ x e. K ) -> ( ( O ` F ) ` x ) = ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) ( .r ` R ) ( ( O ` G ) ` x ) ) )
80	79	eqeq1d 2624	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( ( ( O ` F ) ` x ) = W <-> ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) ( .r ` R ) ( ( O ` G ) ` x ) ) = W ) )
81	5, 31	syl 17	. . . . . . . . . 10 |- ( ph -> R e. Domn )
82	81	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. K ) -> R e. Domn )
83	69	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ x e. K ) -> ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) e. K )
84	72	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ x e. K ) -> ( ( O ` G ) ` x ) e. K )
85	3, 64, 36	domneq0 19297	. . . . . . . . 9 |- ( ( R e. Domn /\ ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) e. K /\ ( ( O ` G ) ` x ) e. K ) -> ( ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) ( .r ` R ) ( ( O ` G ) ` x ) ) = W <-> ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W \/ ( ( O ` G ) ` x ) = W ) ) )
86	82, 83, 84, 85	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ x e. K ) -> ( ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) ( .r ` R ) ( ( O ` G ) ` x ) ) = W <-> ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W \/ ( ( O ` G ) ` x ) = W ) ) )
87	80, 86	bitrd 268	. . . . . . 7 |- ( ( ph /\ x e. K ) -> ( ( ( O ` F ) ` x ) = W <-> ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W \/ ( ( O ` G ) ` x ) = W ) ) )
88	87	pm5.32da 673	. . . . . 6 |- ( ph -> ( ( x e. K /\ ( ( O ` F ) ` x ) = W ) <-> ( x e. K /\ ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W \/ ( ( O ` G ) ` x ) = W ) ) ) )
89		andi 911	. . . . . 6 |- ( ( x e. K /\ ( ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W \/ ( ( O ` G ) ` x ) = W ) ) <-> ( ( x e. K /\ ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W ) \/ ( x e. K /\ ( ( O ` G ) ` x ) = W ) ) )
90	88, 89	syl6bb 276	. . . . 5 |- ( ph -> ( ( x e. K /\ ( ( O ` F ) ` x ) = W ) <-> ( ( x e. K /\ ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W ) \/ ( x e. K /\ ( ( O ` G ) ` x ) = W ) ) ) )
91		fniniseg 6338	. . . . . 6 |- ( ( O ` F ) Fn K -> ( x e. ( `' ( O ` F ) " { W } ) <-> ( x e. K /\ ( ( O ` F ) ` x ) = W ) ) )
92	22, 91	syl 17	. . . . 5 |- ( ph -> ( x e. ( `' ( O ` F ) " { W } ) <-> ( x e. K /\ ( ( O ` F ) ` x ) = W ) ) )
93		elun 3753	. . . . . 6 |- ( x e. ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) <-> ( x e. ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) \/ x e. { T } ) )
94		fniniseg 6338	. . . . . . . 8 |- ( ( O ` ( F (quot1p ` R ) G ) ) Fn K -> ( x e. ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) <-> ( x e. K /\ ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W ) ) )
95	70, 94	syl 17	. . . . . . 7 |- ( ph -> ( x e. ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) <-> ( x e. K /\ ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W ) ) )
96	44	simp3d 1075	. . . . . . . . 9 |- ( ph -> ( `' ( O ` G ) " { W } ) = { T } )
97	96	eleq2d 2687	. . . . . . . 8 |- ( ph -> ( x e. ( `' ( O ` G ) " { W } ) <-> x e. { T } ) )
98		fniniseg 6338	. . . . . . . . 9 |- ( ( O ` G ) Fn K -> ( x e. ( `' ( O ` G ) " { W } ) <-> ( x e. K /\ ( ( O ` G ) ` x ) = W ) ) )
99	73, 98	syl 17	. . . . . . . 8 |- ( ph -> ( x e. ( `' ( O ` G ) " { W } ) <-> ( x e. K /\ ( ( O ` G ) ` x ) = W ) ) )
100	97, 99	bitr3d 270	. . . . . . 7 |- ( ph -> ( x e. { T } <-> ( x e. K /\ ( ( O ` G ) ` x ) = W ) ) )
101	95, 100	orbi12d 746	. . . . . 6 |- ( ph -> ( ( x e. ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) \/ x e. { T } ) <-> ( ( x e. K /\ ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W ) \/ ( x e. K /\ ( ( O ` G ) ` x ) = W ) ) ) )
102	93, 101	syl5bb 272	. . . . 5 |- ( ph -> ( x e. ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) <-> ( ( x e. K /\ ( ( O ` ( F (quot1p ` R ) G ) ) ` x ) = W ) \/ ( x e. K /\ ( ( O ` G ) ` x ) = W ) ) ) )
103	90, 92, 102	3bitr4d 300	. . . 4 |- ( ph -> ( x e. ( `' ( O ` F ) " { W } ) <-> x e. ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) ) )
104	103	eqrdv 2620	. . 3 |- ( ph -> ( `' ( O ` F ) " { W } ) = ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) )
105	104	fveq2d 6195	. 2 |- ( ph -> ( # ` ( `' ( O ` F ) " { W } ) ) = ( # ` ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) ) )
106		fvex 6201	. . . . . . . . . 10 |- ( O ` ( F (quot1p ` R ) G ) ) e. _V
107	106	cnvex 7113	. . . . . . . . 9 |- `' ( O ` ( F (quot1p ` R ) G ) ) e. _V
108	107	imaex 7104	. . . . . . . 8 |- ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) e. _V
109	108	a1i 11	. . . . . . 7 |- ( ph -> ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) e. _V )
110		fta1glem.3	. . . . . . 7 |- ( ph -> N e. NN0 )
111		fta1glem.6	. . . . . . . . 9 |- ( ph -> A. g e. B ( ( D ` g ) = N -> ( # ` ( `' ( O ` g ) " { W } ) ) <_ ( D ` g ) ) )
112		fta1g.z	. . . . . . . . . 10 |- .0. = ( 0g ` P )
113		fta1glem.4	. . . . . . . . . 10 |- ( ph -> ( D ` F ) = ( N + 1 ) )
114	13, 16, 43, 12, 36, 112, 5, 19, 3, 27, 28, 29, 30, 110, 113, 1	fta1glem1 23925	. . . . . . . . 9 |- ( ph -> ( D ` ( F (quot1p ` R ) G ) ) = N )
115		fveq2 6191	. . . . . . . . . . . 12 |- ( g = ( F (quot1p ` R ) G ) -> ( D ` g ) = ( D ` ( F (quot1p ` R ) G ) ) )
116	115	eqeq1d 2624	. . . . . . . . . . 11 |- ( g = ( F (quot1p ` R ) G ) -> ( ( D ` g ) = N <-> ( D ` ( F (quot1p ` R ) G ) ) = N ) )
117		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( g = ( F (quot1p ` R ) G ) -> ( O ` g ) = ( O ` ( F (quot1p ` R ) G ) ) )
118	117	cnveqd 5298	. . . . . . . . . . . . . 14 |- ( g = ( F (quot1p ` R ) G ) -> `' ( O ` g ) = `' ( O ` ( F (quot1p ` R ) G ) ) )
119	118	imaeq1d 5465	. . . . . . . . . . . . 13 |- ( g = ( F (quot1p ` R ) G ) -> ( `' ( O ` g ) " { W } ) = ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) )
120	119	fveq2d 6195	. . . . . . . . . . . 12 |- ( g = ( F (quot1p ` R ) G ) -> ( # ` ( `' ( O ` g ) " { W } ) ) = ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) )
121	120, 115	breq12d 4666	. . . . . . . . . . 11 |- ( g = ( F (quot1p ` R ) G ) -> ( ( # ` ( `' ( O ` g ) " { W } ) ) <_ ( D ` g ) <-> ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) <_ ( D ` ( F (quot1p ` R ) G ) ) ) )
122	116, 121	imbi12d 334	. . . . . . . . . 10 |- ( g = ( F (quot1p ` R ) G ) -> ( ( ( D ` g ) = N -> ( # ` ( `' ( O ` g ) " { W } ) ) <_ ( D ` g ) ) <-> ( ( D ` ( F (quot1p ` R ) G ) ) = N -> ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) <_ ( D ` ( F (quot1p ` R ) G ) ) ) ) )
123	122	rspcv 3305	. . . . . . . . 9 |- ( ( F (quot1p ` R ) G ) e. B -> ( A. g e. B ( ( D ` g ) = N -> ( # ` ( `' ( O ` g ) " { W } ) ) <_ ( D ` g ) ) -> ( ( D ` ( F (quot1p ` R ) G ) ) = N -> ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) <_ ( D ` ( F (quot1p ` R ) G ) ) ) ) )
124	56, 111, 114, 123	syl3c 66	. . . . . . . 8 |- ( ph -> ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) <_ ( D ` ( F (quot1p ` R ) G ) ) )
125	124, 114	breqtrd 4679	. . . . . . 7 |- ( ph -> ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) <_ N )
126		hashbnd 13123	. . . . . . 7 |- ( ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) e. _V /\ N e. NN0 /\ ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) <_ N ) -> ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) e. Fin )
127	109, 110, 125, 126	syl3anc 1326	. . . . . 6 |- ( ph -> ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) e. Fin )
128		snfi 8038	. . . . . 6 |- { T } e. Fin
129		unfi 8227	. . . . . 6 |- ( ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) e. Fin /\ { T } e. Fin ) -> ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) e. Fin )
130	127, 128, 129	sylancl 694	. . . . 5 |- ( ph -> ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) e. Fin )
131		hashcl 13147	. . . . 5 |- ( ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) e. Fin -> ( # ` ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) ) e. NN0 )
132	130, 131	syl 17	. . . 4 |- ( ph -> ( # ` ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) ) e. NN0 )
133	132	nn0red 11352	. . 3 |- ( ph -> ( # ` ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) ) e. RR )
134		hashcl 13147	. . . . . 6 |- ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) e. Fin -> ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) e. NN0 )
135	127, 134	syl 17	. . . . 5 |- ( ph -> ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) e. NN0 )
136	135	nn0red 11352	. . . 4 |- ( ph -> ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) e. RR )
137		peano2re 10209	. . . 4 |- ( ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) e. RR -> ( ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) + 1 ) e. RR )
138	136, 137	syl 17	. . 3 |- ( ph -> ( ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) + 1 ) e. RR )
139		peano2nn0 11333	. . . . . 6 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
140	110, 139	syl 17	. . . . 5 |- ( ph -> ( N + 1 ) e. NN0 )
141	113, 140	eqeltrd 2701	. . . 4 |- ( ph -> ( D ` F ) e. NN0 )
142	141	nn0red 11352	. . 3 |- ( ph -> ( D ` F ) e. RR )
143		hashun2 13172	. . . . 5 |- ( ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) e. Fin /\ { T } e. Fin ) -> ( # ` ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) ) <_ ( ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) + ( # ` { T } ) ) )
144	127, 128, 143	sylancl 694	. . . 4 |- ( ph -> ( # ` ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) ) <_ ( ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) + ( # ` { T } ) ) )
145		hashsng 13159	. . . . . 6 |- ( T e. ( `' ( O ` F ) " { W } ) -> ( # ` { T } ) = 1 )
146	1, 145	syl 17	. . . . 5 |- ( ph -> ( # ` { T } ) = 1 )
147	146	oveq2d 6666	. . . 4 |- ( ph -> ( ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) + ( # ` { T } ) ) = ( ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) + 1 ) )
148	144, 147	breqtrd 4679	. . 3 |- ( ph -> ( # ` ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) ) <_ ( ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) + 1 ) )
149	110	nn0red 11352	. . . . 5 |- ( ph -> N e. RR )
150		1red 10055	. . . . 5 |- ( ph -> 1 e. RR )
151	136, 149, 150, 125	leadd1dd 10641	. . . 4 |- ( ph -> ( ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) + 1 ) <_ ( N + 1 ) )
152	151, 113	breqtrrd 4681	. . 3 |- ( ph -> ( ( # ` ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) ) + 1 ) <_ ( D ` F ) )
153	133, 138, 142, 148, 152	letrd 10194	. 2 |- ( ph -> ( # ` ( ( `' ( O ` ( F (quot1p ` R ) G ) ) " { W } ) u. { T } ) ) <_ ( D ` F ) )
154	105, 153	eqbrtrd 4675	1 |- ( ph -> ( # ` ( `' ( O ` F ) " { W } ) ) <_ ( D ` F ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   {csn 4177   class class class wbr 4653   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   RRcr 9935   1c1 9937   + caddc 9939   <_ cle 10075   NN0cn0 11292   #chash 13117   Basecbs 15857   .rcmulr 15942   0gc0g 16100   ^_s cpws 16107   -gcsg 17424   Ringcrg 18547   CRingccrg 18548   ||_rcdsr 18638   RingHom crh 18712  NzRingcnzr 19257  Domncdomn 19280  IDomncidom 19281  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547  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-cda 8990  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-xnn0 11364  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:  fta1g  23927