Theorem fta1lem 24062

Description: Lemma for fta1 24063. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypotheses

Ref	Expression
fta1.1	|- R = ( `' F " { 0 } )
fta1.2	|- ( ph -> D e. NN0 )
fta1.3	|- ( ph -> F e. ( (Poly ` CC ) \ { 0p } ) )
fta1.4	|- ( ph -> (deg ` F ) = ( D + 1 ) )
fta1.5	|- ( ph -> A e. ( `' F " { 0 } ) )
fta1.6	|- ( ph -> A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = D -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) )

Assertion

Ref	Expression
fta1lem	|- ( ph -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )

Distinct variable groups:   A, g   D, g   g, F

Allowed substitution hints:   ph( g)   R( g)

Proof of Theorem fta1lem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fta1.3	. . . . . . . . . 10 |- ( ph -> F e. ( (Poly ` CC ) \ { 0p } ) )
2		eldifsn 4317	. . . . . . . . . 10 |- ( F e. ( (Poly ` CC ) \ { 0p } ) <-> ( F e. (Poly ` CC ) /\ F =/= 0p ) )
3	1, 2	sylib 208	. . . . . . . . 9 |- ( ph -> ( F e. (Poly ` CC ) /\ F =/= 0p ) )
4	3	simpld 475	. . . . . . . 8 |- ( ph -> F e. (Poly ` CC ) )
5		fta1.5	. . . . . . . . . 10 |- ( ph -> A e. ( `' F " { 0 } ) )
6		plyf 23954	. . . . . . . . . . 11 |- ( F e. (Poly ` CC ) -> F : CC --> CC )
7		ffn 6045	. . . . . . . . . . 11 |- ( F : CC --> CC -> F Fn CC )
8		fniniseg 6338	. . . . . . . . . . 11 |- ( F Fn CC -> ( A e. ( `' F " { 0 } ) <-> ( A e. CC /\ ( F ` A ) = 0 ) ) )
9	4, 6, 7, 8	4syl 19	. . . . . . . . . 10 |- ( ph -> ( A e. ( `' F " { 0 } ) <-> ( A e. CC /\ ( F ` A ) = 0 ) ) )
10	5, 9	mpbid 222	. . . . . . . . 9 |- ( ph -> ( A e. CC /\ ( F ` A ) = 0 ) )
11	10	simpld 475	. . . . . . . 8 |- ( ph -> A e. CC )
12	10	simprd 479	. . . . . . . 8 |- ( ph -> ( F ` A ) = 0 )
13		eqid 2622	. . . . . . . . 9 |- ( Xp oF - ( CC X. { A } ) ) = ( Xp oF - ( CC X. { A } ) )
14	13	facth 24061	. . . . . . . 8 |- ( ( F e. (Poly ` CC ) /\ A e. CC /\ ( F ` A ) = 0 ) -> F = ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) )
15	4, 11, 12, 14	syl3anc 1326	. . . . . . 7 |- ( ph -> F = ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) )
16	15	cnveqd 5298	. . . . . 6 |- ( ph -> `' F = `' ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) )
17	16	imaeq1d 5465	. . . . 5 |- ( ph -> ( `' F " { 0 } ) = ( `' ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) " { 0 } ) )
18		cnex 10017	. . . . . . 7 |- CC e. _V
19	18	a1i 11	. . . . . 6 |- ( ph -> CC e. _V )
20		ssid 3624	. . . . . . . . 9 |- CC C_ CC
21		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
22		plyid 23965	. . . . . . . . 9 |- ( ( CC C_ CC /\ 1 e. CC ) -> Xp e. (Poly ` CC ) )
23	20, 21, 22	mp2an 708	. . . . . . . 8 |- Xp e. (Poly ` CC )
24		plyconst 23962	. . . . . . . . 9 |- ( ( CC C_ CC /\ A e. CC ) -> ( CC X. { A } ) e. (Poly ` CC ) )
25	20, 11, 24	sylancr 695	. . . . . . . 8 |- ( ph -> ( CC X. { A } ) e. (Poly ` CC ) )
26		plysubcl 23978	. . . . . . . 8 |- ( ( Xp e. (Poly ` CC ) /\ ( CC X. { A } ) e. (Poly ` CC ) ) -> ( Xp oF - ( CC X. { A } ) ) e. (Poly ` CC ) )
27	23, 25, 26	sylancr 695	. . . . . . 7 |- ( ph -> ( Xp oF - ( CC X. { A } ) ) e. (Poly ` CC ) )
28		plyf 23954	. . . . . . 7 |- ( ( Xp oF - ( CC X. { A } ) ) e. (Poly ` CC ) -> ( Xp oF - ( CC X. { A } ) ) : CC --> CC )
29	27, 28	syl 17	. . . . . 6 |- ( ph -> ( Xp oF - ( CC X. { A } ) ) : CC --> CC )
30	13	plyremlem 24059	. . . . . . . . . . . 12 |- ( A e. CC -> ( ( Xp oF - ( CC X. { A } ) ) e. (Poly ` CC ) /\ (deg ` ( Xp oF - ( CC X. { A } ) ) ) = 1 /\ ( `' ( Xp oF - ( CC X. { A } ) ) " { 0 } ) = { A } ) )
31	11, 30	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( Xp oF - ( CC X. { A } ) ) e. (Poly ` CC ) /\ (deg ` ( Xp oF - ( CC X. { A } ) ) ) = 1 /\ ( `' ( Xp oF - ( CC X. { A } ) ) " { 0 } ) = { A } ) )
32	31	simp2d 1074	. . . . . . . . . 10 |- ( ph -> (deg ` ( Xp oF - ( CC X. { A } ) ) ) = 1 )
33		ax-1ne0 10005	. . . . . . . . . . 11 |- 1 =/= 0
34	33	a1i 11	. . . . . . . . . 10 |- ( ph -> 1 =/= 0 )
35	32, 34	eqnetrd 2861	. . . . . . . . 9 |- ( ph -> (deg ` ( Xp oF - ( CC X. { A } ) ) ) =/= 0 )
36		fveq2 6191	. . . . . . . . . . 11 |- ( ( Xp oF - ( CC X. { A } ) ) = 0p -> (deg ` ( Xp oF - ( CC X. { A } ) ) ) = (deg ` 0p ) )
37		dgr0 24018	. . . . . . . . . . 11 |- (deg ` 0p ) = 0
38	36, 37	syl6eq 2672	. . . . . . . . . 10 |- ( ( Xp oF - ( CC X. { A } ) ) = 0p -> (deg ` ( Xp oF - ( CC X. { A } ) ) ) = 0 )
39	38	necon3i 2826	. . . . . . . . 9 |- ( (deg ` ( Xp oF - ( CC X. { A } ) ) ) =/= 0 -> ( Xp oF - ( CC X. { A } ) ) =/= 0p )
40	35, 39	syl 17	. . . . . . . 8 |- ( ph -> ( Xp oF - ( CC X. { A } ) ) =/= 0p )
41		quotcl2 24057	. . . . . . . 8 |- ( ( F e. (Poly ` CC ) /\ ( Xp oF - ( CC X. { A } ) ) e. (Poly ` CC ) /\ ( Xp oF - ( CC X. { A } ) ) =/= 0p ) -> ( F quot ( Xp oF - ( CC X. { A } ) ) ) e. (Poly ` CC ) )
42	4, 27, 40, 41	syl3anc 1326	. . . . . . 7 |- ( ph -> ( F quot ( Xp oF - ( CC X. { A } ) ) ) e. (Poly ` CC ) )
43		plyf 23954	. . . . . . 7 |- ( ( F quot ( Xp oF - ( CC X. { A } ) ) ) e. (Poly ` CC ) -> ( F quot ( Xp oF - ( CC X. { A } ) ) ) : CC --> CC )
44	42, 43	syl 17	. . . . . 6 |- ( ph -> ( F quot ( Xp oF - ( CC X. { A } ) ) ) : CC --> CC )
45		ofmulrt 24037	. . . . . 6 |- ( ( CC e. _V /\ ( Xp oF - ( CC X. { A } ) ) : CC --> CC /\ ( F quot ( Xp oF - ( CC X. { A } ) ) ) : CC --> CC ) -> ( `' ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) " { 0 } ) = ( ( `' ( Xp oF - ( CC X. { A } ) ) " { 0 } ) u. ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) )
46	19, 29, 44, 45	syl3anc 1326	. . . . 5 |- ( ph -> ( `' ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) " { 0 } ) = ( ( `' ( Xp oF - ( CC X. { A } ) ) " { 0 } ) u. ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) )
47	31	simp3d 1075	. . . . . 6 |- ( ph -> ( `' ( Xp oF - ( CC X. { A } ) ) " { 0 } ) = { A } )
48	47	uneq1d 3766	. . . . 5 |- ( ph -> ( ( `' ( Xp oF - ( CC X. { A } ) ) " { 0 } ) u. ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) = ( { A } u. ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) )
49	17, 46, 48	3eqtrd 2660	. . . 4 |- ( ph -> ( `' F " { 0 } ) = ( { A } u. ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) )
50		fta1.1	. . . 4 |- R = ( `' F " { 0 } )
51		uncom 3757	. . . 4 |- ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) = ( { A } u. ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) )
52	49, 50, 51	3eqtr4g 2681	. . 3 |- ( ph -> R = ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) )
53	3	simprd 479	. . . . . . . . 9 |- ( ph -> F =/= 0p )
54	15	eqcomd 2628	. . . . . . . . 9 |- ( ph -> ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) = F )
55		0cnd 10033	. . . . . . . . . . 11 |- ( ph -> 0 e. CC )
56		mul01 10215	. . . . . . . . . . . 12 |- ( x e. CC -> ( x x. 0 ) = 0 )
57	56	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> ( x x. 0 ) = 0 )
58	19, 29, 55, 55, 57	caofid1 6927	. . . . . . . . . 10 |- ( ph -> ( ( Xp oF - ( CC X. { A } ) ) oF x. ( CC X. { 0 } ) ) = ( CC X. { 0 } ) )
59		df-0p 23437	. . . . . . . . . . 11 |- 0p = ( CC X. { 0 } )
60	59	oveq2i 6661	. . . . . . . . . 10 |- ( ( Xp oF - ( CC X. { A } ) ) oF x. 0p ) = ( ( Xp oF - ( CC X. { A } ) ) oF x. ( CC X. { 0 } ) )
61	58, 60, 59	3eqtr4g 2681	. . . . . . . . 9 |- ( ph -> ( ( Xp oF - ( CC X. { A } ) ) oF x. 0p ) = 0p )
62	53, 54, 61	3netr4d 2871	. . . . . . . 8 |- ( ph -> ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) =/= ( ( Xp oF - ( CC X. { A } ) ) oF x. 0p ) )
63		oveq2 6658	. . . . . . . . 9 |- ( ( F quot ( Xp oF - ( CC X. { A } ) ) ) = 0p -> ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) = ( ( Xp oF - ( CC X. { A } ) ) oF x. 0p ) )
64	63	necon3i 2826	. . . . . . . 8 |- ( ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) =/= ( ( Xp oF - ( CC X. { A } ) ) oF x. 0p ) -> ( F quot ( Xp oF - ( CC X. { A } ) ) ) =/= 0p )
65	62, 64	syl 17	. . . . . . 7 |- ( ph -> ( F quot ( Xp oF - ( CC X. { A } ) ) ) =/= 0p )
66		eldifsn 4317	. . . . . . 7 |- ( ( F quot ( Xp oF - ( CC X. { A } ) ) ) e. ( (Poly ` CC ) \ { 0p } ) <-> ( ( F quot ( Xp oF - ( CC X. { A } ) ) ) e. (Poly ` CC ) /\ ( F quot ( Xp oF - ( CC X. { A } ) ) ) =/= 0p ) )
67	42, 65, 66	sylanbrc 698	. . . . . 6 |- ( ph -> ( F quot ( Xp oF - ( CC X. { A } ) ) ) e. ( (Poly ` CC ) \ { 0p } ) )
68		fta1.6	. . . . . 6 |- ( ph -> A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = D -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) )
69	21	a1i 11	. . . . . . 7 |- ( ph -> 1 e. CC )
70		dgrcl 23989	. . . . . . . . 9 |- ( ( F quot ( Xp oF - ( CC X. { A } ) ) ) e. (Poly ` CC ) -> (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) e. NN0 )
71	42, 70	syl 17	. . . . . . . 8 |- ( ph -> (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) e. NN0 )
72	71	nn0cnd 11353	. . . . . . 7 |- ( ph -> (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) e. CC )
73		fta1.2	. . . . . . . 8 |- ( ph -> D e. NN0 )
74	73	nn0cnd 11353	. . . . . . 7 |- ( ph -> D e. CC )
75		addcom 10222	. . . . . . . . 9 |- ( ( 1 e. CC /\ D e. CC ) -> ( 1 + D ) = ( D + 1 ) )
76	21, 74, 75	sylancr 695	. . . . . . . 8 |- ( ph -> ( 1 + D ) = ( D + 1 ) )
77	15	fveq2d 6195	. . . . . . . . 9 |- ( ph -> (deg ` F ) = (deg ` ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) )
78		fta1.4	. . . . . . . . 9 |- ( ph -> (deg ` F ) = ( D + 1 ) )
79		eqid 2622	. . . . . . . . . . 11 |- (deg ` ( Xp oF - ( CC X. { A } ) ) ) = (deg ` ( Xp oF - ( CC X. { A } ) ) )
80		eqid 2622	. . . . . . . . . . 11 |- (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) = (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) )
81	79, 80	dgrmul 24026	. . . . . . . . . 10 |- ( ( ( ( Xp oF - ( CC X. { A } ) ) e. (Poly ` CC ) /\ ( Xp oF - ( CC X. { A } ) ) =/= 0p ) /\ ( ( F quot ( Xp oF - ( CC X. { A } ) ) ) e. (Poly ` CC ) /\ ( F quot ( Xp oF - ( CC X. { A } ) ) ) =/= 0p ) ) -> (deg ` ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) = ( (deg ` ( Xp oF - ( CC X. { A } ) ) ) + (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) )
82	27, 40, 42, 65, 81	syl22anc 1327	. . . . . . . . 9 |- ( ph -> (deg ` ( ( Xp oF - ( CC X. { A } ) ) oF x. ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) = ( (deg ` ( Xp oF - ( CC X. { A } ) ) ) + (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) )
83	77, 78, 82	3eqtr3d 2664	. . . . . . . 8 |- ( ph -> ( D + 1 ) = ( (deg ` ( Xp oF - ( CC X. { A } ) ) ) + (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) )
84	32	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( (deg ` ( Xp oF - ( CC X. { A } ) ) ) + (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) = ( 1 + (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) )
85	76, 83, 84	3eqtrrd 2661	. . . . . . 7 |- ( ph -> ( 1 + (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) = ( 1 + D ) )
86	69, 72, 74, 85	addcanad 10241	. . . . . 6 |- ( ph -> (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) = D )
87		fveq2 6191	. . . . . . . . 9 |- ( g = ( F quot ( Xp oF - ( CC X. { A } ) ) ) -> (deg ` g ) = (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) )
88	87	eqeq1d 2624	. . . . . . . 8 |- ( g = ( F quot ( Xp oF - ( CC X. { A } ) ) ) -> ( (deg ` g ) = D <-> (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) = D ) )
89		cnveq 5296	. . . . . . . . . . 11 |- ( g = ( F quot ( Xp oF - ( CC X. { A } ) ) ) -> `' g = `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) )
90	89	imaeq1d 5465	. . . . . . . . . 10 |- ( g = ( F quot ( Xp oF - ( CC X. { A } ) ) ) -> ( `' g " { 0 } ) = ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) )
91	90	eleq1d 2686	. . . . . . . . 9 |- ( g = ( F quot ( Xp oF - ( CC X. { A } ) ) ) -> ( ( `' g " { 0 } ) e. Fin <-> ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) e. Fin ) )
92	90	fveq2d 6195	. . . . . . . . . 10 |- ( g = ( F quot ( Xp oF - ( CC X. { A } ) ) ) -> ( # ` ( `' g " { 0 } ) ) = ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) )
93	92, 87	breq12d 4666	. . . . . . . . 9 |- ( g = ( F quot ( Xp oF - ( CC X. { A } ) ) ) -> ( ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) <-> ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) <_ (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) )
94	91, 93	anbi12d 747	. . . . . . . 8 |- ( g = ( F quot ( Xp oF - ( CC X. { A } ) ) ) -> ( ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) <-> ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) e. Fin /\ ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) <_ (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) ) )
95	88, 94	imbi12d 334	. . . . . . 7 |- ( g = ( F quot ( Xp oF - ( CC X. { A } ) ) ) -> ( ( (deg ` g ) = D -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) <-> ( (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) = D -> ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) e. Fin /\ ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) <_ (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) ) ) )
96	95	rspcv 3305	. . . . . 6 |- ( ( F quot ( Xp oF - ( CC X. { A } ) ) ) e. ( (Poly ` CC ) \ { 0p } ) -> ( A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = D -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) -> ( (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) = D -> ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) e. Fin /\ ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) <_ (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) ) ) )
97	67, 68, 86, 96	syl3c 66	. . . . 5 |- ( ph -> ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) e. Fin /\ ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) <_ (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) ) )
98	97	simpld 475	. . . 4 |- ( ph -> ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) e. Fin )
99		snfi 8038	. . . 4 |- { A } e. Fin
100		unfi 8227	. . . 4 |- ( ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) e. Fin /\ { A } e. Fin ) -> ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) e. Fin )
101	98, 99, 100	sylancl 694	. . 3 |- ( ph -> ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) e. Fin )
102	52, 101	eqeltrd 2701	. 2 |- ( ph -> R e. Fin )
103	52	fveq2d 6195	. . 3 |- ( ph -> ( # ` R ) = ( # ` ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) ) )
104		hashcl 13147	. . . . . 6 |- ( ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) e. Fin -> ( # ` ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) ) e. NN0 )
105	101, 104	syl 17	. . . . 5 |- ( ph -> ( # ` ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) ) e. NN0 )
106	105	nn0red 11352	. . . 4 |- ( ph -> ( # ` ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) ) e. RR )
107		hashcl 13147	. . . . . . 7 |- ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) e. Fin -> ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) e. NN0 )
108	98, 107	syl 17	. . . . . 6 |- ( ph -> ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) e. NN0 )
109	108	nn0red 11352	. . . . 5 |- ( ph -> ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) e. RR )
110		peano2re 10209	. . . . 5 |- ( ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) e. RR -> ( ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) + 1 ) e. RR )
111	109, 110	syl 17	. . . 4 |- ( ph -> ( ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) + 1 ) e. RR )
112		dgrcl 23989	. . . . . 6 |- ( F e. (Poly ` CC ) -> (deg ` F ) e. NN0 )
113	4, 112	syl 17	. . . . 5 |- ( ph -> (deg ` F ) e. NN0 )
114	113	nn0red 11352	. . . 4 |- ( ph -> (deg ` F ) e. RR )
115		hashun2 13172	. . . . . 6 |- ( ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) e. Fin /\ { A } e. Fin ) -> ( # ` ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) ) <_ ( ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) + ( # ` { A } ) ) )
116	98, 99, 115	sylancl 694	. . . . 5 |- ( ph -> ( # ` ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) ) <_ ( ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) + ( # ` { A } ) ) )
117		hashsng 13159	. . . . . . 7 |- ( A e. CC -> ( # ` { A } ) = 1 )
118	11, 117	syl 17	. . . . . 6 |- ( ph -> ( # ` { A } ) = 1 )
119	118	oveq2d 6666	. . . . 5 |- ( ph -> ( ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) + ( # ` { A } ) ) = ( ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) + 1 ) )
120	116, 119	breqtrd 4679	. . . 4 |- ( ph -> ( # ` ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) ) <_ ( ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) + 1 ) )
121	73	nn0red 11352	. . . . . 6 |- ( ph -> D e. RR )
122		1red 10055	. . . . . 6 |- ( ph -> 1 e. RR )
123	97	simprd 479	. . . . . . 7 |- ( ph -> ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) <_ (deg ` ( F quot ( Xp oF - ( CC X. { A } ) ) ) ) )
124	123, 86	breqtrd 4679	. . . . . 6 |- ( ph -> ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) <_ D )
125	109, 121, 122, 124	leadd1dd 10641	. . . . 5 |- ( ph -> ( ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) + 1 ) <_ ( D + 1 ) )
126	125, 78	breqtrrd 4681	. . . 4 |- ( ph -> ( ( # ` ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) ) + 1 ) <_ (deg ` F ) )
127	106, 111, 114, 120, 126	letrd 10194	. . 3 |- ( ph -> ( # ` ( ( `' ( F quot ( Xp oF - ( CC X. { A } ) ) ) " { 0 } ) u. { A } ) ) <_ (deg ` F ) )
128	103, 127	eqbrtrd 4675	. 2 |- ( ph -> ( # ` R ) <_ (deg ` F ) )
129	102, 128	jca 554	1 |- ( ph -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   {csn 4177   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   NN0cn0 11292   #chash 13117   0pc0p 23436  Polycply 23940   Xpcidp 23941  degcdgr 23943   quot cquot 24045

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046

This theorem is referenced by:  fta1  24063