Theorem rngunsnply 37743

Description: Adjoining one element to a ring results in a set of polynomial evaluations. (Contributed by Stefan O'Rear, 30-Nov-2014.)

Hypotheses

Ref	Expression
rngunsnply.b	|- ( ph -> B e. (SubRing ` ℂfld ) )
rngunsnply.x	|- ( ph -> X e. CC )
rngunsnply.s	|- ( ph -> S = ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) )

Assertion

Ref	Expression
rngunsnply	|- ( ph -> ( V e. S <-> E. p e. (Poly ` B ) V = ( p ` X ) ) )

Distinct variable groups:   ph, p   B, p   X, p   V, p

Allowed substitution hint:   S( p)

Proof of Theorem rngunsnply

Dummy variables a b c d e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngunsnply.s	. . 3 |- ( ph -> S = ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) )
2	1	eleq2d 2687	. 2 |- ( ph -> ( V e. S <-> V e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) ) )
3		cnring 19768	. . . . . . 7 |- ℂfld e. Ring
4	3	a1i 11	. . . . . 6 |- ( ph -> ℂfld e. Ring )
5		cnfldbas 19750	. . . . . . 7 |- CC = ( Base ` ℂfld )
6	5	a1i 11	. . . . . 6 |- ( ph -> CC = ( Base ` ℂfld ) )
7		rngunsnply.b	. . . . . . . 8 |- ( ph -> B e. (SubRing ` ℂfld ) )
8	5	subrgss 18781	. . . . . . . 8 |- ( B e. (SubRing ` ℂfld ) -> B C_ CC )
9	7, 8	syl 17	. . . . . . 7 |- ( ph -> B C_ CC )
10		rngunsnply.x	. . . . . . . 8 |- ( ph -> X e. CC )
11	10	snssd 4340	. . . . . . 7 |- ( ph -> { X } C_ CC )
12	9, 11	unssd 3789	. . . . . 6 |- ( ph -> ( B u. { X } ) C_ CC )
13		eqidd 2623	. . . . . 6 |- ( ph -> (RingSpan ` ℂfld ) = (RingSpan ` ℂfld ) )
14		eqidd 2623	. . . . . 6 |- ( ph -> ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) = ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) )
15		eqidd 2623	. . . . . . 7 |- ( ph -> (ℂfld ↾s { a | E. p e. (Poly ` B ) a = ( p ` X ) } ) = (ℂfld ↾s { a | E. p e. (Poly ` B ) a = ( p ` X ) } ) )
16		cnfld0 19770	. . . . . . . 8 |- 0 = ( 0g ` ℂfld )
17	16	a1i 11	. . . . . . 7 |- ( ph -> 0 = ( 0g ` ℂfld ) )
18		cnfldadd 19751	. . . . . . . 8 |- + = ( +g ` ℂfld )
19	18	a1i 11	. . . . . . 7 |- ( ph -> + = ( +g ` ℂfld ) )
20		plyf 23954	. . . . . . . . . . . 12 |- ( p e. (Poly ` B ) -> p : CC --> CC )
21		ffvelrn 6357	. . . . . . . . . . . 12 |- ( ( p : CC --> CC /\ X e. CC ) -> ( p ` X ) e. CC )
22	20, 10, 21	syl2anr 495	. . . . . . . . . . 11 |- ( ( ph /\ p e. (Poly ` B ) ) -> ( p ` X ) e. CC )
23		eleq1 2689	. . . . . . . . . . 11 |- ( a = ( p ` X ) -> ( a e. CC <-> ( p ` X ) e. CC ) )
24	22, 23	syl5ibrcom 237	. . . . . . . . . 10 |- ( ( ph /\ p e. (Poly ` B ) ) -> ( a = ( p ` X ) -> a e. CC ) )
25	24	rexlimdva 3031	. . . . . . . . 9 |- ( ph -> ( E. p e. (Poly ` B ) a = ( p ` X ) -> a e. CC ) )
26	25	ss2abdv 3675	. . . . . . . 8 |- ( ph -> { a | E. p e. (Poly ` B ) a = ( p ` X ) } C_ { a | a e. CC } )
27		abid2 2745	. . . . . . . . 9 |- { a | a e. CC } = CC
28	27, 5	eqtri 2644	. . . . . . . 8 |- { a | a e. CC } = ( Base ` ℂfld )
29	26, 28	syl6sseq 3651	. . . . . . 7 |- ( ph -> { a | E. p e. (Poly ` B ) a = ( p ` X ) } C_ ( Base ` ℂfld ) )
30		abid2 2745	. . . . . . . . 9 |- { a | a e. B } = B
31		plyconst 23962	. . . . . . . . . . . . 13 |- ( ( B C_ CC /\ a e. B ) -> ( CC X. { a } ) e. (Poly ` B ) )
32	9, 31	sylan 488	. . . . . . . . . . . 12 |- ( ( ph /\ a e. B ) -> ( CC X. { a } ) e. (Poly ` B ) )
33	10	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ a e. B ) -> X e. CC )
34		vex 3203	. . . . . . . . . . . . . . 15 |- a e. _V
35	34	fvconst2 6469	. . . . . . . . . . . . . 14 |- ( X e. CC -> ( ( CC X. { a } ) ` X ) = a )
36	33, 35	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. B ) -> ( ( CC X. { a } ) ` X ) = a )
37	36	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ph /\ a e. B ) -> a = ( ( CC X. { a } ) ` X ) )
38		fveq1 6190	. . . . . . . . . . . . . 14 |- ( p = ( CC X. { a } ) -> ( p ` X ) = ( ( CC X. { a } ) ` X ) )
39	38	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( p = ( CC X. { a } ) -> ( a = ( p ` X ) <-> a = ( ( CC X. { a } ) ` X ) ) )
40	39	rspcev 3309	. . . . . . . . . . . 12 |- ( ( ( CC X. { a } ) e. (Poly ` B ) /\ a = ( ( CC X. { a } ) ` X ) ) -> E. p e. (Poly ` B ) a = ( p ` X ) )
41	32, 37, 40	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ a e. B ) -> E. p e. (Poly ` B ) a = ( p ` X ) )
42	41	ex 450	. . . . . . . . . 10 |- ( ph -> ( a e. B -> E. p e. (Poly ` B ) a = ( p ` X ) ) )
43	42	ss2abdv 3675	. . . . . . . . 9 |- ( ph -> { a | a e. B } C_ { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
44	30, 43	syl5eqssr 3650	. . . . . . . 8 |- ( ph -> B C_ { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
45		subrgsubg 18786	. . . . . . . . . 10 |- ( B e. (SubRing ` ℂfld ) -> B e. (SubGrp ` ℂfld ) )
46	7, 45	syl 17	. . . . . . . . 9 |- ( ph -> B e. (SubGrp ` ℂfld ) )
47	16	subg0cl 17602	. . . . . . . . 9 |- ( B e. (SubGrp ` ℂfld ) -> 0 e. B )
48	46, 47	syl 17	. . . . . . . 8 |- ( ph -> 0 e. B )
49	44, 48	sseldd 3604	. . . . . . 7 |- ( ph -> 0 e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
50		biid 251	. . . . . . . . 9 |- ( ph <-> ph )
51		vex 3203	. . . . . . . . . 10 |- b e. _V
52		eqeq1 2626	. . . . . . . . . . . 12 |- ( a = b -> ( a = ( p ` X ) <-> b = ( p ` X ) ) )
53	52	rexbidv 3052	. . . . . . . . . . 11 |- ( a = b -> ( E. p e. (Poly ` B ) a = ( p ` X ) <-> E. p e. (Poly ` B ) b = ( p ` X ) ) )
54		fveq1 6190	. . . . . . . . . . . . 13 |- ( p = e -> ( p ` X ) = ( e ` X ) )
55	54	eqeq2d 2632	. . . . . . . . . . . 12 |- ( p = e -> ( b = ( p ` X ) <-> b = ( e ` X ) ) )
56	55	cbvrexv 3172	. . . . . . . . . . 11 |- ( E. p e. (Poly ` B ) b = ( p ` X ) <-> E. e e. (Poly ` B ) b = ( e ` X ) )
57	53, 56	syl6bb 276	. . . . . . . . . 10 |- ( a = b -> ( E. p e. (Poly ` B ) a = ( p ` X ) <-> E. e e. (Poly ` B ) b = ( e ` X ) ) )
58	51, 57	elab 3350	. . . . . . . . 9 |- ( b e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } <-> E. e e. (Poly ` B ) b = ( e ` X ) )
59		vex 3203	. . . . . . . . . 10 |- c e. _V
60		eqeq1 2626	. . . . . . . . . . . 12 |- ( a = c -> ( a = ( p ` X ) <-> c = ( p ` X ) ) )
61	60	rexbidv 3052	. . . . . . . . . . 11 |- ( a = c -> ( E. p e. (Poly ` B ) a = ( p ` X ) <-> E. p e. (Poly ` B ) c = ( p ` X ) ) )
62		fveq1 6190	. . . . . . . . . . . . 13 |- ( p = d -> ( p ` X ) = ( d ` X ) )
63	62	eqeq2d 2632	. . . . . . . . . . . 12 |- ( p = d -> ( c = ( p ` X ) <-> c = ( d ` X ) ) )
64	63	cbvrexv 3172	. . . . . . . . . . 11 |- ( E. p e. (Poly ` B ) c = ( p ` X ) <-> E. d e. (Poly ` B ) c = ( d ` X ) )
65	61, 64	syl6bb 276	. . . . . . . . . 10 |- ( a = c -> ( E. p e. (Poly ` B ) a = ( p ` X ) <-> E. d e. (Poly ` B ) c = ( d ` X ) ) )
66	59, 65	elab 3350	. . . . . . . . 9 |- ( c e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } <-> E. d e. (Poly ` B ) c = ( d ` X ) )
67		simplr 792	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> e e. (Poly ` B ) )
68		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> d e. (Poly ` B ) )
69	18	subrgacl 18791	. . . . . . . . . . . . . . . . . . . 20 |- ( ( B e. (SubRing ` ℂfld ) /\ a e. B /\ b e. B ) -> ( a + b ) e. B )
70	69	3expb 1266	. . . . . . . . . . . . . . . . . . 19 |- ( ( B e. (SubRing ` ℂfld ) /\ ( a e. B /\ b e. B ) ) -> ( a + b ) e. B )
71	7, 70	sylan 488	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a + b ) e. B )
72	71	adantlr 751	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ ( a e. B /\ b e. B ) ) -> ( a + b ) e. B )
73	72	adantlr 751	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) /\ ( a e. B /\ b e. B ) ) -> ( a + b ) e. B )
74	67, 68, 73	plyadd 23973	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> ( e oF + d ) e. (Poly ` B ) )
75		plyf 23954	. . . . . . . . . . . . . . . . . . 19 |- ( e e. (Poly ` B ) -> e : CC --> CC )
76		ffn 6045	. . . . . . . . . . . . . . . . . . 19 |- ( e : CC --> CC -> e Fn CC )
77	75, 76	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( e e. (Poly ` B ) -> e Fn CC )
78	77	ad2antlr 763	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> e Fn CC )
79		plyf 23954	. . . . . . . . . . . . . . . . . . 19 |- ( d e. (Poly ` B ) -> d : CC --> CC )
80		ffn 6045	. . . . . . . . . . . . . . . . . . 19 |- ( d : CC --> CC -> d Fn CC )
81	79, 80	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( d e. (Poly ` B ) -> d Fn CC )
82	81	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> d Fn CC )
83		cnex 10017	. . . . . . . . . . . . . . . . . 18 |- CC e. _V
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> CC e. _V )
85	10	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> X e. CC )
86		fnfvof 6911	. . . . . . . . . . . . . . . . 17 |- ( ( ( e Fn CC /\ d Fn CC ) /\ ( CC e. _V /\ X e. CC ) ) -> ( ( e oF + d ) ` X ) = ( ( e ` X ) + ( d ` X ) ) )
87	78, 82, 84, 85, 86	syl22anc 1327	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> ( ( e oF + d ) ` X ) = ( ( e ` X ) + ( d ` X ) ) )
88	87	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> ( ( e ` X ) + ( d ` X ) ) = ( ( e oF + d ) ` X ) )
89		fveq1 6190	. . . . . . . . . . . . . . . . 17 |- ( p = ( e oF + d ) -> ( p ` X ) = ( ( e oF + d ) ` X ) )
90	89	eqeq2d 2632	. . . . . . . . . . . . . . . 16 |- ( p = ( e oF + d ) -> ( ( ( e ` X ) + ( d ` X ) ) = ( p ` X ) <-> ( ( e ` X ) + ( d ` X ) ) = ( ( e oF + d ) ` X ) ) )
91	90	rspcev 3309	. . . . . . . . . . . . . . 15 |- ( ( ( e oF + d ) e. (Poly ` B ) /\ ( ( e ` X ) + ( d ` X ) ) = ( ( e oF + d ) ` X ) ) -> E. p e. (Poly ` B ) ( ( e ` X ) + ( d ` X ) ) = ( p ` X ) )
92	74, 88, 91	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> E. p e. (Poly ` B ) ( ( e ` X ) + ( d ` X ) ) = ( p ` X ) )
93		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( c = ( d ` X ) -> ( ( e ` X ) + c ) = ( ( e ` X ) + ( d ` X ) ) )
94	93	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( c = ( d ` X ) -> ( ( ( e ` X ) + c ) = ( p ` X ) <-> ( ( e ` X ) + ( d ` X ) ) = ( p ` X ) ) )
95	94	rexbidv 3052	. . . . . . . . . . . . . 14 |- ( c = ( d ` X ) -> ( E. p e. (Poly ` B ) ( ( e ` X ) + c ) = ( p ` X ) <-> E. p e. (Poly ` B ) ( ( e ` X ) + ( d ` X ) ) = ( p ` X ) ) )
96	92, 95	syl5ibrcom 237	. . . . . . . . . . . . 13 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> ( c = ( d ` X ) -> E. p e. (Poly ` B ) ( ( e ` X ) + c ) = ( p ` X ) ) )
97	96	rexlimdva 3031	. . . . . . . . . . . 12 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( E. d e. (Poly ` B ) c = ( d ` X ) -> E. p e. (Poly ` B ) ( ( e ` X ) + c ) = ( p ` X ) ) )
98		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( b = ( e ` X ) -> ( b + c ) = ( ( e ` X ) + c ) )
99	98	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( b = ( e ` X ) -> ( ( b + c ) = ( p ` X ) <-> ( ( e ` X ) + c ) = ( p ` X ) ) )
100	99	rexbidv 3052	. . . . . . . . . . . . 13 |- ( b = ( e ` X ) -> ( E. p e. (Poly ` B ) ( b + c ) = ( p ` X ) <-> E. p e. (Poly ` B ) ( ( e ` X ) + c ) = ( p ` X ) ) )
101	100	imbi2d 330	. . . . . . . . . . . 12 |- ( b = ( e ` X ) -> ( ( E. d e. (Poly ` B ) c = ( d ` X ) -> E. p e. (Poly ` B ) ( b + c ) = ( p ` X ) ) <-> ( E. d e. (Poly ` B ) c = ( d ` X ) -> E. p e. (Poly ` B ) ( ( e ` X ) + c ) = ( p ` X ) ) ) )
102	97, 101	syl5ibrcom 237	. . . . . . . . . . 11 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( b = ( e ` X ) -> ( E. d e. (Poly ` B ) c = ( d ` X ) -> E. p e. (Poly ` B ) ( b + c ) = ( p ` X ) ) ) )
103	102	rexlimdva 3031	. . . . . . . . . 10 |- ( ph -> ( E. e e. (Poly ` B ) b = ( e ` X ) -> ( E. d e. (Poly ` B ) c = ( d ` X ) -> E. p e. (Poly ` B ) ( b + c ) = ( p ` X ) ) ) )
104	103	3imp 1256	. . . . . . . . 9 |- ( ( ph /\ E. e e. (Poly ` B ) b = ( e ` X ) /\ E. d e. (Poly ` B ) c = ( d ` X ) ) -> E. p e. (Poly ` B ) ( b + c ) = ( p ` X ) )
105	50, 58, 66, 104	syl3anb 1369	. . . . . . . 8 |- ( ( ph /\ b e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } /\ c e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } ) -> E. p e. (Poly ` B ) ( b + c ) = ( p ` X ) )
106		ovex 6678	. . . . . . . . 9 |- ( b + c ) e. _V
107		eqeq1 2626	. . . . . . . . . 10 |- ( a = ( b + c ) -> ( a = ( p ` X ) <-> ( b + c ) = ( p ` X ) ) )
108	107	rexbidv 3052	. . . . . . . . 9 |- ( a = ( b + c ) -> ( E. p e. (Poly ` B ) a = ( p ` X ) <-> E. p e. (Poly ` B ) ( b + c ) = ( p ` X ) ) )
109	106, 108	elab 3350	. . . . . . . 8 |- ( ( b + c ) e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } <-> E. p e. (Poly ` B ) ( b + c ) = ( p ` X ) )
110	105, 109	sylibr 224	. . . . . . 7 |- ( ( ph /\ b e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } /\ c e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } ) -> ( b + c ) e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
111		ax-1cn 9994	. . . . . . . . . . . . . . . . . 18 |- 1 e. CC
112		cnfldneg 19772	. . . . . . . . . . . . . . . . . 18 |- ( 1 e. CC -> ( ( invg ` ℂfld ) ` 1 ) = -u 1 )
113	111, 112	mp1i 13	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( invg ` ℂfld ) ` 1 ) = -u 1 )
114		cnfld1 19771	. . . . . . . . . . . . . . . . . . . 20 |- 1 = ( 1r ` ℂfld )
115	114	subrg1cl 18788	. . . . . . . . . . . . . . . . . . 19 |- ( B e. (SubRing ` ℂfld ) -> 1 e. B )
116	7, 115	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ph -> 1 e. B )
117		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- ( invg ` ℂfld ) = ( invg ` ℂfld )
118	117	subginvcl 17603	. . . . . . . . . . . . . . . . . 18 |- ( ( B e. (SubGrp ` ℂfld ) /\ 1 e. B ) -> ( ( invg ` ℂfld ) ` 1 ) e. B )
119	46, 116, 118	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( invg ` ℂfld ) ` 1 ) e. B )
120	113, 119	eqeltrrd 2702	. . . . . . . . . . . . . . . 16 |- ( ph -> -u 1 e. B )
121		plyconst 23962	. . . . . . . . . . . . . . . 16 |- ( ( B C_ CC /\ -u 1 e. B ) -> ( CC X. { -u 1 } ) e. (Poly ` B ) )
122	9, 120, 121	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> ( CC X. { -u 1 } ) e. (Poly ` B ) )
123	122	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( CC X. { -u 1 } ) e. (Poly ` B ) )
124		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ph /\ e e. (Poly ` B ) ) -> e e. (Poly ` B ) )
125		cnfldmul 19752	. . . . . . . . . . . . . . . . . 18 |- x. = ( .r ` ℂfld )
126	125	subrgmcl 18792	. . . . . . . . . . . . . . . . 17 |- ( ( B e. (SubRing ` ℂfld ) /\ a e. B /\ b e. B ) -> ( a x. b ) e. B )
127	126	3expb 1266	. . . . . . . . . . . . . . . 16 |- ( ( B e. (SubRing ` ℂfld ) /\ ( a e. B /\ b e. B ) ) -> ( a x. b ) e. B )
128	7, 127	sylan 488	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a x. b ) e. B )
129	128	adantlr 751	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ ( a e. B /\ b e. B ) ) -> ( a x. b ) e. B )
130	123, 124, 72, 129	plymul 23974	. . . . . . . . . . . . 13 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( ( CC X. { -u 1 } ) oF x. e ) e. (Poly ` B ) )
131		ffvelrn 6357	. . . . . . . . . . . . . . . 16 |- ( ( e : CC --> CC /\ X e. CC ) -> ( e ` X ) e. CC )
132	75, 10, 131	syl2anr 495	. . . . . . . . . . . . . . 15 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( e ` X ) e. CC )
133		cnfldneg 19772	. . . . . . . . . . . . . . 15 |- ( ( e ` X ) e. CC -> ( ( invg ` ℂfld ) ` ( e ` X ) ) = -u ( e ` X ) )
134	132, 133	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( ( invg ` ℂfld ) ` ( e ` X ) ) = -u ( e ` X ) )
135		negex 10279	. . . . . . . . . . . . . . . . 17 |- -u 1 e. _V
136		fnconstg 6093	. . . . . . . . . . . . . . . . 17 |- ( -u 1 e. _V -> ( CC X. { -u 1 } ) Fn CC )
137	135, 136	mp1i 13	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( CC X. { -u 1 } ) Fn CC )
138	77	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ e e. (Poly ` B ) ) -> e Fn CC )
139	83	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ e e. (Poly ` B ) ) -> CC e. _V )
140	10	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ e e. (Poly ` B ) ) -> X e. CC )
141		fnfvof 6911	. . . . . . . . . . . . . . . 16 |- ( ( ( ( CC X. { -u 1 } ) Fn CC /\ e Fn CC ) /\ ( CC e. _V /\ X e. CC ) ) -> ( ( ( CC X. { -u 1 } ) oF x. e ) ` X ) = ( ( ( CC X. { -u 1 } ) ` X ) x. ( e ` X ) ) )
142	137, 138, 139, 140, 141	syl22anc 1327	. . . . . . . . . . . . . . 15 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( ( ( CC X. { -u 1 } ) oF x. e ) ` X ) = ( ( ( CC X. { -u 1 } ) ` X ) x. ( e ` X ) ) )
143	135	fvconst2 6469	. . . . . . . . . . . . . . . . 17 |- ( X e. CC -> ( ( CC X. { -u 1 } ) ` X ) = -u 1 )
144	140, 143	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( ( CC X. { -u 1 } ) ` X ) = -u 1 )
145	144	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( ( ( CC X. { -u 1 } ) ` X ) x. ( e ` X ) ) = ( -u 1 x. ( e ` X ) ) )
146	132	mulm1d 10482	. . . . . . . . . . . . . . 15 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( -u 1 x. ( e ` X ) ) = -u ( e ` X ) )
147	142, 145, 146	3eqtrd 2660	. . . . . . . . . . . . . 14 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( ( ( CC X. { -u 1 } ) oF x. e ) ` X ) = -u ( e ` X ) )
148	134, 147	eqtr4d 2659	. . . . . . . . . . . . 13 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( ( invg ` ℂfld ) ` ( e ` X ) ) = ( ( ( CC X. { -u 1 } ) oF x. e ) ` X ) )
149		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( p = ( ( CC X. { -u 1 } ) oF x. e ) -> ( p ` X ) = ( ( ( CC X. { -u 1 } ) oF x. e ) ` X ) )
150	149	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( p = ( ( CC X. { -u 1 } ) oF x. e ) -> ( ( ( invg ` ℂfld ) ` ( e ` X ) ) = ( p ` X ) <-> ( ( invg ` ℂfld ) ` ( e ` X ) ) = ( ( ( CC X. { -u 1 } ) oF x. e ) ` X ) ) )
151	150	rspcev 3309	. . . . . . . . . . . . 13 |- ( ( ( ( CC X. { -u 1 } ) oF x. e ) e. (Poly ` B ) /\ ( ( invg ` ℂfld ) ` ( e ` X ) ) = ( ( ( CC X. { -u 1 } ) oF x. e ) ` X ) ) -> E. p e. (Poly ` B ) ( ( invg ` ℂfld ) ` ( e ` X ) ) = ( p ` X ) )
152	130, 148, 151	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ e e. (Poly ` B ) ) -> E. p e. (Poly ` B ) ( ( invg ` ℂfld ) ` ( e ` X ) ) = ( p ` X ) )
153		fveq2 6191	. . . . . . . . . . . . . 14 |- ( b = ( e ` X ) -> ( ( invg ` ℂfld ) ` b ) = ( ( invg ` ℂfld ) ` ( e ` X ) ) )
154	153	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( b = ( e ` X ) -> ( ( ( invg ` ℂfld ) ` b ) = ( p ` X ) <-> ( ( invg ` ℂfld ) ` ( e ` X ) ) = ( p ` X ) ) )
155	154	rexbidv 3052	. . . . . . . . . . . 12 |- ( b = ( e ` X ) -> ( E. p e. (Poly ` B ) ( ( invg ` ℂfld ) ` b ) = ( p ` X ) <-> E. p e. (Poly ` B ) ( ( invg ` ℂfld ) ` ( e ` X ) ) = ( p ` X ) ) )
156	152, 155	syl5ibrcom 237	. . . . . . . . . . 11 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( b = ( e ` X ) -> E. p e. (Poly ` B ) ( ( invg ` ℂfld ) ` b ) = ( p ` X ) ) )
157	156	rexlimdva 3031	. . . . . . . . . 10 |- ( ph -> ( E. e e. (Poly ` B ) b = ( e ` X ) -> E. p e. (Poly ` B ) ( ( invg ` ℂfld ) ` b ) = ( p ` X ) ) )
158	157	imp 445	. . . . . . . . 9 |- ( ( ph /\ E. e e. (Poly ` B ) b = ( e ` X ) ) -> E. p e. (Poly ` B ) ( ( invg ` ℂfld ) ` b ) = ( p ` X ) )
159	58, 158	sylan2b 492	. . . . . . . 8 |- ( ( ph /\ b e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } ) -> E. p e. (Poly ` B ) ( ( invg ` ℂfld ) ` b ) = ( p ` X ) )
160		fvex 6201	. . . . . . . . 9 |- ( ( invg ` ℂfld ) ` b ) e. _V
161		eqeq1 2626	. . . . . . . . . 10 |- ( a = ( ( invg ` ℂfld ) ` b ) -> ( a = ( p ` X ) <-> ( ( invg ` ℂfld ) ` b ) = ( p ` X ) ) )
162	161	rexbidv 3052	. . . . . . . . 9 |- ( a = ( ( invg ` ℂfld ) ` b ) -> ( E. p e. (Poly ` B ) a = ( p ` X ) <-> E. p e. (Poly ` B ) ( ( invg ` ℂfld ) ` b ) = ( p ` X ) ) )
163	160, 162	elab 3350	. . . . . . . 8 |- ( ( ( invg ` ℂfld ) ` b ) e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } <-> E. p e. (Poly ` B ) ( ( invg ` ℂfld ) ` b ) = ( p ` X ) )
164	159, 163	sylibr 224	. . . . . . 7 |- ( ( ph /\ b e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } ) -> ( ( invg ` ℂfld ) ` b ) e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
165	114	a1i 11	. . . . . . 7 |- ( ph -> 1 = ( 1r ` ℂfld ) )
166	125	a1i 11	. . . . . . 7 |- ( ph -> x. = ( .r ` ℂfld ) )
167	44, 116	sseldd 3604	. . . . . . 7 |- ( ph -> 1 e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
168	129	adantlr 751	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) /\ ( a e. B /\ b e. B ) ) -> ( a x. b ) e. B )
169	67, 68, 73, 168	plymul 23974	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> ( e oF x. d ) e. (Poly ` B ) )
170		fnfvof 6911	. . . . . . . . . . . . . . . . 17 |- ( ( ( e Fn CC /\ d Fn CC ) /\ ( CC e. _V /\ X e. CC ) ) -> ( ( e oF x. d ) ` X ) = ( ( e ` X ) x. ( d ` X ) ) )
171	78, 82, 84, 85, 170	syl22anc 1327	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> ( ( e oF x. d ) ` X ) = ( ( e ` X ) x. ( d ` X ) ) )
172	171	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> ( ( e ` X ) x. ( d ` X ) ) = ( ( e oF x. d ) ` X ) )
173		fveq1 6190	. . . . . . . . . . . . . . . . 17 |- ( p = ( e oF x. d ) -> ( p ` X ) = ( ( e oF x. d ) ` X ) )
174	173	eqeq2d 2632	. . . . . . . . . . . . . . . 16 |- ( p = ( e oF x. d ) -> ( ( ( e ` X ) x. ( d ` X ) ) = ( p ` X ) <-> ( ( e ` X ) x. ( d ` X ) ) = ( ( e oF x. d ) ` X ) ) )
175	174	rspcev 3309	. . . . . . . . . . . . . . 15 |- ( ( ( e oF x. d ) e. (Poly ` B ) /\ ( ( e ` X ) x. ( d ` X ) ) = ( ( e oF x. d ) ` X ) ) -> E. p e. (Poly ` B ) ( ( e ` X ) x. ( d ` X ) ) = ( p ` X ) )
176	169, 172, 175	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> E. p e. (Poly ` B ) ( ( e ` X ) x. ( d ` X ) ) = ( p ` X ) )
177		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( c = ( d ` X ) -> ( ( e ` X ) x. c ) = ( ( e ` X ) x. ( d ` X ) ) )
178	177	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( c = ( d ` X ) -> ( ( ( e ` X ) x. c ) = ( p ` X ) <-> ( ( e ` X ) x. ( d ` X ) ) = ( p ` X ) ) )
179	178	rexbidv 3052	. . . . . . . . . . . . . 14 |- ( c = ( d ` X ) -> ( E. p e. (Poly ` B ) ( ( e ` X ) x. c ) = ( p ` X ) <-> E. p e. (Poly ` B ) ( ( e ` X ) x. ( d ` X ) ) = ( p ` X ) ) )
180	176, 179	syl5ibrcom 237	. . . . . . . . . . . . 13 |- ( ( ( ph /\ e e. (Poly ` B ) ) /\ d e. (Poly ` B ) ) -> ( c = ( d ` X ) -> E. p e. (Poly ` B ) ( ( e ` X ) x. c ) = ( p ` X ) ) )
181	180	rexlimdva 3031	. . . . . . . . . . . 12 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( E. d e. (Poly ` B ) c = ( d ` X ) -> E. p e. (Poly ` B ) ( ( e ` X ) x. c ) = ( p ` X ) ) )
182		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( b = ( e ` X ) -> ( b x. c ) = ( ( e ` X ) x. c ) )
183	182	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( b = ( e ` X ) -> ( ( b x. c ) = ( p ` X ) <-> ( ( e ` X ) x. c ) = ( p ` X ) ) )
184	183	rexbidv 3052	. . . . . . . . . . . . 13 |- ( b = ( e ` X ) -> ( E. p e. (Poly ` B ) ( b x. c ) = ( p ` X ) <-> E. p e. (Poly ` B ) ( ( e ` X ) x. c ) = ( p ` X ) ) )
185	184	imbi2d 330	. . . . . . . . . . . 12 |- ( b = ( e ` X ) -> ( ( E. d e. (Poly ` B ) c = ( d ` X ) -> E. p e. (Poly ` B ) ( b x. c ) = ( p ` X ) ) <-> ( E. d e. (Poly ` B ) c = ( d ` X ) -> E. p e. (Poly ` B ) ( ( e ` X ) x. c ) = ( p ` X ) ) ) )
186	181, 185	syl5ibrcom 237	. . . . . . . . . . 11 |- ( ( ph /\ e e. (Poly ` B ) ) -> ( b = ( e ` X ) -> ( E. d e. (Poly ` B ) c = ( d ` X ) -> E. p e. (Poly ` B ) ( b x. c ) = ( p ` X ) ) ) )
187	186	rexlimdva 3031	. . . . . . . . . 10 |- ( ph -> ( E. e e. (Poly ` B ) b = ( e ` X ) -> ( E. d e. (Poly ` B ) c = ( d ` X ) -> E. p e. (Poly ` B ) ( b x. c ) = ( p ` X ) ) ) )
188	187	3imp 1256	. . . . . . . . 9 |- ( ( ph /\ E. e e. (Poly ` B ) b = ( e ` X ) /\ E. d e. (Poly ` B ) c = ( d ` X ) ) -> E. p e. (Poly ` B ) ( b x. c ) = ( p ` X ) )
189	50, 58, 66, 188	syl3anb 1369	. . . . . . . 8 |- ( ( ph /\ b e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } /\ c e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } ) -> E. p e. (Poly ` B ) ( b x. c ) = ( p ` X ) )
190		ovex 6678	. . . . . . . . 9 |- ( b x. c ) e. _V
191		eqeq1 2626	. . . . . . . . . 10 |- ( a = ( b x. c ) -> ( a = ( p ` X ) <-> ( b x. c ) = ( p ` X ) ) )
192	191	rexbidv 3052	. . . . . . . . 9 |- ( a = ( b x. c ) -> ( E. p e. (Poly ` B ) a = ( p ` X ) <-> E. p e. (Poly ` B ) ( b x. c ) = ( p ` X ) ) )
193	190, 192	elab 3350	. . . . . . . 8 |- ( ( b x. c ) e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } <-> E. p e. (Poly ` B ) ( b x. c ) = ( p ` X ) )
194	189, 193	sylibr 224	. . . . . . 7 |- ( ( ph /\ b e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } /\ c e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } ) -> ( b x. c ) e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
195	15, 17, 19, 29, 49, 110, 164, 165, 166, 167, 194, 4	issubrngd2 19189	. . . . . 6 |- ( ph -> { a | E. p e. (Poly ` B ) a = ( p ` X ) } e. (SubRing ` ℂfld ) )
196		plyid 23965	. . . . . . . . . . 11 |- ( ( B C_ CC /\ 1 e. B ) -> Xp e. (Poly ` B ) )
197	9, 116, 196	syl2anc 693	. . . . . . . . . 10 |- ( ph -> Xp e. (Poly ` B ) )
198		df-idp 23945	. . . . . . . . . . . 12 |- Xp = ( _I |` CC )
199	198	fveq1i 6192	. . . . . . . . . . 11 |- ( Xp ` X ) = ( ( _I |` CC ) ` X )
200		fvresi 6439	. . . . . . . . . . . 12 |- ( X e. CC -> ( ( _I |` CC ) ` X ) = X )
201	10, 200	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( _I |` CC ) ` X ) = X )
202	199, 201	syl5req 2669	. . . . . . . . . 10 |- ( ph -> X = ( Xp ` X ) )
203		fveq1 6190	. . . . . . . . . . . 12 |- ( p = Xp -> ( p ` X ) = ( Xp ` X ) )
204	203	eqeq2d 2632	. . . . . . . . . . 11 |- ( p = Xp -> ( X = ( p ` X ) <-> X = ( Xp ` X ) ) )
205	204	rspcev 3309	. . . . . . . . . 10 |- ( ( Xp e. (Poly ` B ) /\ X = ( Xp ` X ) ) -> E. p e. (Poly ` B ) X = ( p ` X ) )
206	197, 202, 205	syl2anc 693	. . . . . . . . 9 |- ( ph -> E. p e. (Poly ` B ) X = ( p ` X ) )
207		eqeq1 2626	. . . . . . . . . . . 12 |- ( a = X -> ( a = ( p ` X ) <-> X = ( p ` X ) ) )
208	207	rexbidv 3052	. . . . . . . . . . 11 |- ( a = X -> ( E. p e. (Poly ` B ) a = ( p ` X ) <-> E. p e. (Poly ` B ) X = ( p ` X ) ) )
209	208	elabg 3351	. . . . . . . . . 10 |- ( X e. CC -> ( X e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } <-> E. p e. (Poly ` B ) X = ( p ` X ) ) )
210	10, 209	syl 17	. . . . . . . . 9 |- ( ph -> ( X e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } <-> E. p e. (Poly ` B ) X = ( p ` X ) ) )
211	206, 210	mpbird 247	. . . . . . . 8 |- ( ph -> X e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
212	211	snssd 4340	. . . . . . 7 |- ( ph -> { X } C_ { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
213	44, 212	unssd 3789	. . . . . 6 |- ( ph -> ( B u. { X } ) C_ { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
214	4, 6, 12, 13, 14, 195, 213	rgspnmin 37741	. . . . 5 |- ( ph -> ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) C_ { a | E. p e. (Poly ` B ) a = ( p ` X ) } )
215	214	sseld 3602	. . . 4 |- ( ph -> ( V e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) -> V e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } ) )
216		fvex 6201	. . . . . . 7 |- ( p ` X ) e. _V
217		eleq1 2689	. . . . . . 7 |- ( V = ( p ` X ) -> ( V e. _V <-> ( p ` X ) e. _V ) )
218	216, 217	mpbiri 248	. . . . . 6 |- ( V = ( p ` X ) -> V e. _V )
219	218	rexlimivw 3029	. . . . 5 |- ( E. p e. (Poly ` B ) V = ( p ` X ) -> V e. _V )
220		eqeq1 2626	. . . . . 6 |- ( a = V -> ( a = ( p ` X ) <-> V = ( p ` X ) ) )
221	220	rexbidv 3052	. . . . 5 |- ( a = V -> ( E. p e. (Poly ` B ) a = ( p ` X ) <-> E. p e. (Poly ` B ) V = ( p ` X ) ) )
222	219, 221	elab3 3358	. . . 4 |- ( V e. { a | E. p e. (Poly ` B ) a = ( p ` X ) } <-> E. p e. (Poly ` B ) V = ( p ` X ) )
223	215, 222	syl6ib 241	. . 3 |- ( ph -> ( V e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) -> E. p e. (Poly ` B ) V = ( p ` X ) ) )
224	4, 6, 12, 13, 14	rgspncl 37739	. . . . . . 7 |- ( ph -> ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) e. (SubRing ` ℂfld ) )
225	224	adantr 481	. . . . . 6 |- ( ( ph /\ p e. (Poly ` B ) ) -> ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) e. (SubRing ` ℂfld ) )
226		simpr 477	. . . . . 6 |- ( ( ph /\ p e. (Poly ` B ) ) -> p e. (Poly ` B ) )
227	4, 6, 12, 13, 14	rgspnssid 37740	. . . . . . . . 9 |- ( ph -> ( B u. { X } ) C_ ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) )
228	227	unssbd 3791	. . . . . . . 8 |- ( ph -> { X } C_ ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) )
229		snidg 4206	. . . . . . . . 9 |- ( X e. CC -> X e. { X } )
230	10, 229	syl 17	. . . . . . . 8 |- ( ph -> X e. { X } )
231	228, 230	sseldd 3604	. . . . . . 7 |- ( ph -> X e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) )
232	231	adantr 481	. . . . . 6 |- ( ( ph /\ p e. (Poly ` B ) ) -> X e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) )
233	227	unssad 3790	. . . . . . 7 |- ( ph -> B C_ ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) )
234	233	adantr 481	. . . . . 6 |- ( ( ph /\ p e. (Poly ` B ) ) -> B C_ ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) )
235	225, 226, 232, 234	cnsrplycl 37737	. . . . 5 |- ( ( ph /\ p e. (Poly ` B ) ) -> ( p ` X ) e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) )
236		eleq1 2689	. . . . 5 |- ( V = ( p ` X ) -> ( V e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) <-> ( p ` X ) e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) ) )
237	235, 236	syl5ibrcom 237	. . . 4 |- ( ( ph /\ p e. (Poly ` B ) ) -> ( V = ( p ` X ) -> V e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) ) )
238	237	rexlimdva 3031	. . 3 |- ( ph -> ( E. p e. (Poly ` B ) V = ( p ` X ) -> V e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) ) )
239	223, 238	impbid 202	. 2 |- ( ph -> ( V e. ( (RingSpan ` ℂfld ) ` ( B u. { X } ) ) <-> E. p e. (Poly ` B ) V = ( p ` X ) ) )
240	2, 239	bitrd 268	1 |- ( ph -> ( V e. S <-> E. p e. (Poly ` B ) V = ( p ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   _I cid 5023   X. cxp 5112   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   -ucneg 10267   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   invgcminusg 17423  SubGrpcsubg 17588   1rcur 18501   Ringcrg 18547  SubRingcsubrg 18776  RingSpancrgspn 18777  ℂ_fldccnfld 19746  Polycply 23940   Xpcidp 23941

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-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-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-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-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-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-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-rgspn 18779  df-cnfld 19747  df-0p 23437  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947

This theorem is referenced by: (None)