Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngunsnply Structured version   Unicode version

Theorem rngunsnply 30755
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 ` fld ) )
rngunsnply.x  |-  ( ph  ->  X  e.  CC )
rngunsnply.s  |-  ( ph  ->  S  =  ( (RingSpan ` fld ) `  ( 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.
StepHypRef Expression
1 rngunsnply.s . . 3  |-  ( ph  ->  S  =  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) )
21eleq2d 2537 . 2  |-  ( ph  ->  ( V  e.  S  <->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
3 cnrng 18239 . . . . . . 7  |-fld  e.  Ring
43a1i 11 . . . . . 6  |-  ( ph  ->fld  e. 
Ring )
5 cnfldbas 18223 . . . . . . 7  |-  CC  =  ( Base ` fld )
65a1i 11 . . . . . 6  |-  ( ph  ->  CC  =  ( Base ` fld ) )
7 rngunsnply.b . . . . . . . 8  |-  ( ph  ->  B  e.  (SubRing ` fld ) )
85subrgss 17230 . . . . . . . 8  |-  ( B  e.  (SubRing ` fld )  ->  B  C_  CC )
97, 8syl 16 . . . . . . 7  |-  ( ph  ->  B  C_  CC )
10 rngunsnply.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
1110snssd 4172 . . . . . . 7  |-  ( ph  ->  { X }  C_  CC )
129, 11unssd 3680 . . . . . 6  |-  ( ph  ->  ( B  u.  { X } )  C_  CC )
13 eqidd 2468 . . . . . 6  |-  ( ph  ->  (RingSpan ` fld )  =  (RingSpan ` fld ) )
14 eqidd 2468 . . . . . 6  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  =  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
15 eqidd 2468 . . . . . . 7  |-  ( ph  ->  (flds  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  =  (flds  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } ) )
16 cnfld0 18241 . . . . . . . 8  |-  0  =  ( 0g ` fld )
1716a1i 11 . . . . . . 7  |-  ( ph  ->  0  =  ( 0g
` fld
) )
18 cnfldadd 18224 . . . . . . . 8  |-  +  =  ( +g  ` fld )
1918a1i 11 . . . . . . 7  |-  ( ph  ->  +  =  ( +g  ` fld ) )
20 plyf 22358 . . . . . . . . . . . 12  |-  ( p  e.  (Poly `  B
)  ->  p : CC
--> CC )
21 ffvelrn 6019 . . . . . . . . . . . 12  |-  ( ( p : CC --> CC  /\  X  e.  CC )  ->  ( p `  X
)  e.  CC )
2220, 10, 21syl2anr 478 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( p `  X )  e.  CC )
23 eleq1 2539 . . . . . . . . . . 11  |-  ( a  =  ( p `  X )  ->  (
a  e.  CC  <->  ( p `  X )  e.  CC ) )
2422, 23syl5ibrcom 222 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( a  =  ( p `  X
)  ->  a  e.  CC ) )
2524rexlimdva 2955 . . . . . . . . 9  |-  ( ph  ->  ( E. p  e.  (Poly `  B )
a  =  ( p `
 X )  -> 
a  e.  CC ) )
2625ss2abdv 3573 . . . . . . . 8  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  C_  { a  |  a  e.  CC } )
27 abid2 2607 . . . . . . . . 9  |-  { a  |  a  e.  CC }  =  CC
2827, 5eqtri 2496 . . . . . . . 8  |-  { a  |  a  e.  CC }  =  ( Base ` fld )
2926, 28syl6sseq 3550 . . . . . . 7  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  C_  ( Base ` fld ) )
30 abid2 2607 . . . . . . . . 9  |-  { a  |  a  e.  B }  =  B
31 plyconst 22366 . . . . . . . . . . . . 13  |-  ( ( B  C_  CC  /\  a  e.  B )  ->  ( CC  X.  { a } )  e.  (Poly `  B ) )
329, 31sylan 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  ( CC  X.  { a } )  e.  (Poly `  B ) )
3310adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  B )  ->  X  e.  CC )
34 vex 3116 . . . . . . . . . . . . . . 15  |-  a  e. 
_V
3534fvconst2 6116 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
( CC  X.  {
a } ) `  X )  =  a )
3633, 35syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  B )  ->  (
( CC  X.  {
a } ) `  X )  =  a )
3736eqcomd 2475 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  a  =  ( ( CC 
X.  { a } ) `  X ) )
38 fveq1 5865 . . . . . . . . . . . . . 14  |-  ( p  =  ( CC  X.  { a } )  ->  ( p `  X )  =  ( ( CC  X.  {
a } ) `  X ) )
3938eqeq2d 2481 . . . . . . . . . . . . 13  |-  ( p  =  ( CC  X.  { a } )  ->  ( a  =  ( p `  X
)  <->  a  =  ( ( CC  X.  {
a } ) `  X ) ) )
4039rspcev 3214 . . . . . . . . . . . 12  |-  ( ( ( CC  X.  {
a } )  e.  (Poly `  B )  /\  a  =  (
( CC  X.  {
a } ) `  X ) )  ->  E. p  e.  (Poly `  B ) a  =  ( p `  X
) )
4132, 37, 40syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  B )  ->  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) )
4241ex 434 . . . . . . . . . 10  |-  ( ph  ->  ( a  e.  B  ->  E. p  e.  (Poly `  B ) a  =  ( p `  X
) ) )
4342ss2abdv 3573 . . . . . . . . 9  |-  ( ph  ->  { a  |  a  e.  B }  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
4430, 43syl5eqssr 3549 . . . . . . . 8  |-  ( ph  ->  B  C_  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) } )
45 subrgsubg 17235 . . . . . . . . . 10  |-  ( B  e.  (SubRing ` fld )  ->  B  e.  (SubGrp ` fld ) )
467, 45syl 16 . . . . . . . . 9  |-  ( ph  ->  B  e.  (SubGrp ` fld )
)
4716subg0cl 16014 . . . . . . . . 9  |-  ( B  e.  (SubGrp ` fld )  ->  0  e.  B )
4846, 47syl 16 . . . . . . . 8  |-  ( ph  ->  0  e.  B )
4944, 48sseldd 3505 . . . . . . 7  |-  ( ph  ->  0  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
50 biid 236 . . . . . . . . 9  |-  ( ph  <->  ph )
51 vex 3116 . . . . . . . . . 10  |-  b  e. 
_V
52 eqeq1 2471 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
a  =  ( p `
 X )  <->  b  =  ( p `  X
) ) )
5352rexbidv 2973 . . . . . . . . . . 11  |-  ( a  =  b  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) b  =  ( p `  X
) ) )
54 fveq1 5865 . . . . . . . . . . . . 13  |-  ( p  =  e  ->  (
p `  X )  =  ( e `  X ) )
5554eqeq2d 2481 . . . . . . . . . . . 12  |-  ( p  =  e  ->  (
b  =  ( p `
 X )  <->  b  =  ( e `  X
) ) )
5655cbvrexv 3089 . . . . . . . . . . 11  |-  ( E. p  e.  (Poly `  B ) b  =  ( p `  X
)  <->  E. e  e.  (Poly `  B ) b  =  ( e `  X
) )
5753, 56syl6bb 261 . . . . . . . . . 10  |-  ( a  =  b  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. e  e.  (Poly `  B ) b  =  ( e `  X
) ) )
5851, 57elab 3250 . . . . . . . . 9  |-  ( b  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. e  e.  (Poly `  B )
b  =  ( e `
 X ) )
59 vex 3116 . . . . . . . . . 10  |-  c  e. 
_V
60 eqeq1 2471 . . . . . . . . . . . 12  |-  ( a  =  c  ->  (
a  =  ( p `
 X )  <->  c  =  ( p `  X
) ) )
6160rexbidv 2973 . . . . . . . . . . 11  |-  ( a  =  c  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) c  =  ( p `  X
) ) )
62 fveq1 5865 . . . . . . . . . . . . 13  |-  ( p  =  d  ->  (
p `  X )  =  ( d `  X ) )
6362eqeq2d 2481 . . . . . . . . . . . 12  |-  ( p  =  d  ->  (
c  =  ( p `
 X )  <->  c  =  ( d `  X
) ) )
6463cbvrexv 3089 . . . . . . . . . . 11  |-  ( E. p  e.  (Poly `  B ) c  =  ( p `  X
)  <->  E. d  e.  (Poly `  B ) c  =  ( d `  X
) )
6561, 64syl6bb 261 . . . . . . . . . 10  |-  ( a  =  c  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. d  e.  (Poly `  B ) c  =  ( d `  X
) ) )
6659, 65elab 3250 . . . . . . . . 9  |-  ( c  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. d  e.  (Poly `  B )
c  =  ( d `
 X ) )
67 simplr 754 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  e  e.  (Poly `  B ) )
68 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  d  e.  (Poly `  B ) )
6918subrgacl 17240 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  e.  (SubRing ` fld )  /\  a  e.  B  /\  b  e.  B )  ->  (
a  +  b )  e.  B )
70693expb 1197 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  e.  (SubRing ` fld )  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  +  b )  e.  B )
717, 70sylan 471 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7271adantlr 714 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7372adantlr 714 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  e  e.  (Poly `  B
) )  /\  d  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7467, 68, 73plyadd 22377 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( e  oF  +  d )  e.  (Poly `  B
) )
75 plyf 22358 . . . . . . . . . . . . . . . . . . 19  |-  ( e  e.  (Poly `  B
)  ->  e : CC
--> CC )
76 ffn 5731 . . . . . . . . . . . . . . . . . . 19  |-  ( e : CC --> CC  ->  e  Fn  CC )
7775, 76syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( e  e.  (Poly `  B
)  ->  e  Fn  CC )
7877ad2antlr 726 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  e  Fn  CC )
79 plyf 22358 . . . . . . . . . . . . . . . . . . 19  |-  ( d  e.  (Poly `  B
)  ->  d : CC
--> CC )
80 ffn 5731 . . . . . . . . . . . . . . . . . . 19  |-  ( d : CC --> CC  ->  d  Fn  CC )
8179, 80syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( d  e.  (Poly `  B
)  ->  d  Fn  CC )
8281adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  d  Fn  CC )
83 cnex 9573 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  _V
8483a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  CC  e.  _V )
8510ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  X  e.  CC )
86 fnfvof 6537 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  Fn  CC  /\  d  Fn  CC )  /\  ( CC  e.  _V  /\  X  e.  CC ) )  ->  (
( e  oF  +  d ) `  X )  =  ( ( e `  X
)  +  ( d `
 X ) ) )
8778, 82, 84, 85, 86syl22anc 1229 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e  oF  +  d ) `  X )  =  ( ( e `
 X )  +  ( d `  X
) ) )
8887eqcomd 2475 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e `
 X )  +  ( d `  X
) )  =  ( ( e  oF  +  d ) `  X ) )
89 fveq1 5865 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( e  oF  +  d )  ->  ( p `  X )  =  ( ( e  oF  +  d ) `  X ) )
9089eqeq2d 2481 . . . . . . . . . . . . . . . 16  |-  ( p  =  ( e  oF  +  d )  ->  ( ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
)  <->  ( ( e `
 X )  +  ( d `  X
) )  =  ( ( e  oF  +  d ) `  X ) ) )
9190rspcev 3214 . . . . . . . . . . . . . . 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 ) )
9274, 88, 91syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
) )
93 oveq2 6292 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( d `  X )  ->  (
( e `  X
)  +  c )  =  ( ( e `
 X )  +  ( d `  X
) ) )
9493eqeq1d 2469 . . . . . . . . . . . . . . 15  |-  ( c  =  ( d `  X )  ->  (
( ( e `  X )  +  c )  =  ( p `
 X )  <->  ( (
e `  X )  +  ( d `  X ) )  =  ( p `  X
) ) )
9594rexbidv 2973 . . . . . . . . . . . . . 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
) ) )
9692, 95syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( c  =  ( d `  X
)  ->  E. p  e.  (Poly `  B )
( ( e `  X )  +  c )  =  ( p `
 X ) ) )
9796rexlimdva 2955 . . . . . . . . . . . 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 6291 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e `  X )  ->  (
b  +  c )  =  ( ( e `
 X )  +  c ) )
9998eqeq1d 2469 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( b  +  c )  =  ( p `
 X )  <->  ( (
e `  X )  +  c )  =  ( p `  X
) ) )
10099rexbidv 2973 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  ( E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  c )  =  ( p `  X
) ) )
101100imbi2d 316 . . . . . . . . . . . 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
) ) ) )
10297, 101syl5ibrcom 222 . . . . . . . . . . 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
) ) ) )
103102rexlimdva 2955 . . . . . . . . . 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
) ) ) )
1041033imp 1190 . . . . . . . . 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 ) )
10550, 58, 66, 104syl3anb 1271 . . . . . . . 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 6309 . . . . . . . . 9  |-  ( b  +  c )  e. 
_V
107 eqeq1 2471 . . . . . . . . . 10  |-  ( a  =  ( b  +  c )  ->  (
a  =  ( p `
 X )  <->  ( b  +  c )  =  ( p `  X
) ) )
108107rexbidv 2973 . . . . . . . . 9  |-  ( a  =  ( b  +  c )  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
) ) )
109106, 108elab 3250 . . . . . . . 8  |-  ( ( b  +  c )  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B )
( b  +  c )  =  ( p `
 X ) )
110105, 109sylibr 212 . . . . . . 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 9550 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
112 cnfldneg 18243 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  CC  ->  (
( invg ` fld ) `  1 )  = 
-u 1 )
113111, 112mp1i 12 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( invg ` fld ) `  1 )  =  -u 1 )
114 cnfld1 18242 . . . . . . . . . . . . . . . . . . . 20  |-  1  =  ( 1r ` fld )
115114subrg1cl 17237 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  (SubRing ` fld )  ->  1  e.  B )
1167, 115syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  1  e.  B )
117 eqid 2467 . . . . . . . . . . . . . . . . . . 19  |-  ( invg ` fld )  =  ( invg ` fld )
118117subginvcl 16015 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  e.  (SubGrp ` fld )  /\  1  e.  B
)  ->  ( ( invg ` fld ) `  1 )  e.  B )
11946, 116, 118syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( invg ` fld ) `  1 )  e.  B )
120113, 119eqeltrrd 2556 . . . . . . . . . . . . . . . 16  |-  ( ph  -> 
-u 1  e.  B
)
121 plyconst 22366 . . . . . . . . . . . . . . . 16  |-  ( ( B  C_  CC  /\  -u 1  e.  B )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B ) )
1229, 120, 121syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B )
)
123122adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B
) )
124 simpr 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  e  e.  (Poly `  B ) )
125 cnfldmul 18225 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
126125subrgmcl 17241 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  (SubRing ` fld )  /\  a  e.  B  /\  b  e.  B )  ->  (
a  x.  b )  e.  B )
1271263expb 1197 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  (SubRing ` fld )  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  x.  b )  e.  B
)
1287, 127sylan 471 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
129128adantlr 714 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
130123, 124, 72, 129plymul 22378 . . . . . . . . . . . . 13  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( CC 
X.  { -u 1 } )  oF  x.  e )  e.  (Poly `  B )
)
131 ffvelrn 6019 . . . . . . . . . . . . . . . 16  |-  ( ( e : CC --> CC  /\  X  e.  CC )  ->  ( e `  X
)  e.  CC )
13275, 10, 131syl2anr 478 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( e `  X )  e.  CC )
133 cnfldneg 18243 . . . . . . . . . . . . . . 15  |-  ( ( e `  X )  e.  CC  ->  (
( invg ` fld ) `  ( e `  X
) )  =  -u ( e `  X
) )
134132, 133syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( invg ` fld ) `  ( e `
 X ) )  =  -u ( e `  X ) )
135 negex 9818 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  _V
136 fnconstg 5773 . . . . . . . . . . . . . . . . 17  |-  ( -u
1  e.  _V  ->  ( CC  X.  { -u
1 } )  Fn  CC )
137135, 136mp1i 12 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( CC  X.  { -u 1 } )  Fn  CC )
13877adantl 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  e  Fn  CC )
13983a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  CC  e.  _V )
14010adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  X  e.  CC )
141 fnfvof 6537 . . . . . . . . . . . . . . . 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
) ) )
142137, 138, 139, 140, 141syl22anc 1229 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } )  oF  x.  e ) `
 X )  =  ( ( ( CC 
X.  { -u 1 } ) `  X
)  x.  ( e `
 X ) ) )
143135fvconst2 6116 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  CC  ->  (
( CC  X.  { -u 1 } ) `  X )  =  -u
1 )
144140, 143syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( CC 
X.  { -u 1 } ) `  X
)  =  -u 1
)
145144oveq1d 6299 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } ) `  X )  x.  (
e `  X )
)  =  ( -u
1  x.  ( e `
 X ) ) )
146132mulm1d 10008 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( -u 1  x.  ( e `  X
) )  =  -u ( e `  X
) )
147142, 145, 1463eqtrd 2512 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } )  oF  x.  e ) `
 X )  = 
-u ( e `  X ) )
148134, 147eqtr4d 2511 . . . . . . . . . . . . 13  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( invg ` fld ) `  ( e `
 X ) )  =  ( ( ( CC  X.  { -u
1 } )  oF  x.  e ) `
 X ) )
149 fveq1 5865 . . . . . . . . . . . . . . 15  |-  ( p  =  ( ( CC 
X.  { -u 1 } )  oF  x.  e )  -> 
( p `  X
)  =  ( ( ( CC  X.  { -u 1 } )  oF  x.  e ) `
 X ) )
150149eqeq2d 2481 . . . . . . . . . . . . . 14  |-  ( p  =  ( ( CC 
X.  { -u 1 } )  oF  x.  e )  -> 
( ( ( invg ` fld ) `  ( e `
 X ) )  =  ( p `  X )  <->  ( ( invg ` fld ) `  ( e `
 X ) )  =  ( ( ( CC  X.  { -u
1 } )  oF  x.  e ) `
 X ) ) )
151150rspcev 3214 . . . . . . . . . . . . 13  |-  ( ( ( ( CC  X.  { -u 1 } )  oF  x.  e
)  e.  (Poly `  B )  /\  (
( invg ` fld ) `  ( e `  X
) )  =  ( ( ( CC  X.  { -u 1 } )  oF  x.  e
) `  X )
)  ->  E. p  e.  (Poly `  B )
( ( invg ` fld ) `  ( e `  X ) )  =  ( p `  X
) )
152130, 148, 151syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  ( e `
 X ) )  =  ( p `  X ) )
153 fveq2 5866 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( invg ` fld ) `  b )  =  ( ( invg ` fld ) `  ( e `  X
) ) )
154153eqeq1d 2469 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  (
( ( invg ` fld ) `  b )  =  ( p `  X )  <->  ( ( invg ` fld ) `  ( e `
 X ) )  =  ( p `  X ) ) )
155154rexbidv 2973 . . . . . . . . . . . 12  |-  ( b  =  ( e `  X )  ->  ( E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  b )  =  ( p `  X )  <->  E. p  e.  (Poly `  B )
( ( invg ` fld ) `  ( e `  X ) )  =  ( p `  X
) ) )
156152, 155syl5ibrcom 222 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( b  =  ( e `  X
)  ->  E. p  e.  (Poly `  B )
( ( invg ` fld ) `  b )  =  ( p `  X ) ) )
157156rexlimdva 2955 . . . . . . . . . 10  |-  ( ph  ->  ( E. e  e.  (Poly `  B )
b  =  ( e `
 X )  ->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  b )  =  ( p `  X ) ) )
158157imp 429 . . . . . . . . 9  |-  ( (
ph  /\  E. e  e.  (Poly `  B )
b  =  ( e `
 X ) )  ->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  b )  =  ( p `  X ) )
15958, 158sylan2b 475 . . . . . . . 8  |-  ( (
ph  /\  b  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  ->  E. p  e.  (Poly `  B )
( ( invg ` fld ) `  b )  =  ( p `  X ) )
160 fvex 5876 . . . . . . . . 9  |-  ( ( invg ` fld ) `  b )  e.  _V
161 eqeq1 2471 . . . . . . . . . 10  |-  ( a  =  ( ( invg ` fld ) `  b )  ->  ( a  =  ( p `  X
)  <->  ( ( invg ` fld ) `  b )  =  ( p `  X ) ) )
162161rexbidv 2973 . . . . . . . . 9  |-  ( a  =  ( ( invg ` fld ) `  b )  ->  ( E. p  e.  (Poly `  B )
a  =  ( p `
 X )  <->  E. p  e.  (Poly `  B )
( ( invg ` fld ) `  b )  =  ( p `  X ) ) )
163160, 162elab 3250 . . . . . . . 8  |-  ( ( ( invg ` fld ) `  b )  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  <->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  b )  =  ( p `  X ) )
164159, 163sylibr 212 . . . . . . 7  |-  ( (
ph  /\  b  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  ->  ( ( invg ` fld ) `  b )  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) } )
165114a1i 11 . . . . . . 7  |-  ( ph  ->  1  =  ( 1r
` fld
) )
166125a1i 11 . . . . . . 7  |-  ( ph  ->  x.  =  ( .r
` fld
) )
16744, 116sseldd 3505 . . . . . . 7  |-  ( ph  ->  1  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
168129adantlr 714 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  e  e.  (Poly `  B
) )  /\  d  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
16967, 68, 73, 168plymul 22378 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( e  oF  x.  d )  e.  (Poly `  B
) )
170 fnfvof 6537 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  Fn  CC  /\  d  Fn  CC )  /\  ( CC  e.  _V  /\  X  e.  CC ) )  ->  (
( e  oF  x.  d ) `  X )  =  ( ( e `  X
)  x.  ( d `
 X ) ) )
17178, 82, 84, 85, 170syl22anc 1229 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e  oF  x.  d
) `  X )  =  ( ( e `
 X )  x.  ( d `  X
) ) )
172171eqcomd 2475 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e `
 X )  x.  ( d `  X
) )  =  ( ( e  oF  x.  d ) `  X ) )
173 fveq1 5865 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( e  oF  x.  d )  ->  ( p `  X )  =  ( ( e  oF  x.  d ) `  X ) )
174173eqeq2d 2481 . . . . . . . . . . . . . . . 16  |-  ( p  =  ( e  oF  x.  d )  ->  ( ( ( e `  X )  x.  ( d `  X ) )  =  ( p `  X
)  <->  ( ( e `
 X )  x.  ( d `  X
) )  =  ( ( e  oF  x.  d ) `  X ) ) )
175174rspcev 3214 . . . . . . . . . . . . . . 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 ) )
176169, 172, 175syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  x.  ( d `  X ) )  =  ( p `  X
) )
177 oveq2 6292 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( d `  X )  ->  (
( e `  X
)  x.  c )  =  ( ( e `
 X )  x.  ( d `  X
) ) )
178177eqeq1d 2469 . . . . . . . . . . . . . . 15  |-  ( c  =  ( d `  X )  ->  (
( ( e `  X )  x.  c
)  =  ( p `
 X )  <->  ( (
e `  X )  x.  ( d `  X
) )  =  ( p `  X ) ) )
179178rexbidv 2973 . . . . . . . . . . . . . 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
) ) )
180176, 179syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( c  =  ( d `  X
)  ->  E. p  e.  (Poly `  B )
( ( e `  X )  x.  c
)  =  ( p `
 X ) ) )
181180rexlimdva 2955 . . . . . . . . . . . 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 6291 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e `  X )  ->  (
b  x.  c )  =  ( ( e `
 X )  x.  c ) )
183182eqeq1d 2469 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( b  x.  c
)  =  ( p `
 X )  <->  ( (
e `  X )  x.  c )  =  ( p `  X ) ) )
184183rexbidv 2973 . . . . . . . . . . . . 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
) ) )
185184imbi2d 316 . . . . . . . . . . . 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
) ) ) )
186181, 185syl5ibrcom 222 . . . . . . . . . . 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
) ) ) )
187186rexlimdva 2955 . . . . . . . . . 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
) ) ) )
1881873imp 1190 . . . . . . . . 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 ) )
18950, 58, 66, 188syl3anb 1271 . . . . . . . 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 6309 . . . . . . . . 9  |-  ( b  x.  c )  e. 
_V
191 eqeq1 2471 . . . . . . . . . 10  |-  ( a  =  ( b  x.  c )  ->  (
a  =  ( p `
 X )  <->  ( b  x.  c )  =  ( p `  X ) ) )
192191rexbidv 2973 . . . . . . . . 9  |-  ( a  =  ( b  x.  c )  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( b  x.  c )  =  ( p `  X
) ) )
193190, 192elab 3250 . . . . . . . 8  |-  ( ( b  x.  c )  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B )
( b  x.  c
)  =  ( p `
 X ) )
194189, 193sylibr 212 . . . . . . 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 ) } )
19515, 17, 19, 29, 49, 110, 164, 165, 166, 167, 194, 4issubrngd2 17635 . . . . . 6  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  e.  (SubRing ` fld ) )
196 plyid 22369 . . . . . . . . . . 11  |-  ( ( B  C_  CC  /\  1  e.  B )  ->  Xp  e.  (Poly `  B
) )
1979, 116, 196syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  Xp  e.  (Poly `  B ) )
198 df-idp 22349 . . . . . . . . . . . 12  |-  Xp  =  (  _I  |`  CC )
199198fveq1i 5867 . . . . . . . . . . 11  |-  ( Xp `  X )  =  ( (  _I  |`  CC ) `  X
)
200 fvresi 6087 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
(  _I  |`  CC ) `
 X )  =  X )
20110, 200syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( (  _I  |`  CC ) `
 X )  =  X )
202199, 201syl5req 2521 . . . . . . . . . 10  |-  ( ph  ->  X  =  ( Xp `  X ) )
203 fveq1 5865 . . . . . . . . . . . 12  |-  ( p  =  Xp  -> 
( p `  X
)  =  ( Xp `  X ) )
204203eqeq2d 2481 . . . . . . . . . . 11  |-  ( p  =  Xp  -> 
( X  =  ( p `  X )  <-> 
X  =  ( Xp `  X ) ) )
205204rspcev 3214 . . . . . . . . . 10  |-  ( ( Xp  e.  (Poly `  B )  /\  X  =  ( Xp `  X ) )  ->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) )
206197, 202, 205syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) )
207 eqeq1 2471 . . . . . . . . . . . 12  |-  ( a  =  X  ->  (
a  =  ( p `
 X )  <->  X  =  ( p `  X
) ) )
208207rexbidv 2973 . . . . . . . . . . 11  |-  ( a  =  X  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) ) )
209208elabg 3251 . . . . . . . . . 10  |-  ( X  e.  CC  ->  ( X  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X ) ) )
21010, 209syl 16 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) ) )
211206, 210mpbird 232 . . . . . . . 8  |-  ( ph  ->  X  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
212211snssd 4172 . . . . . . 7  |-  ( ph  ->  { X }  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
21344, 212unssd 3680 . . . . . 6  |-  ( ph  ->  ( B  u.  { X } )  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
2144, 6, 12, 13, 14, 195, 213rgspnmin 30753 . . . . 5  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
215214sseld 3503 . . . 4  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  ->  V  e.  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } ) )
216 fvex 5876 . . . . . . 7  |-  ( p `
 X )  e. 
_V
217 eleq1 2539 . . . . . . 7  |-  ( V  =  ( p `  X )  ->  ( V  e.  _V  <->  ( p `  X )  e.  _V ) )
218216, 217mpbiri 233 . . . . . 6  |-  ( V  =  ( p `  X )  ->  V  e.  _V )
219218rexlimivw 2952 . . . . 5  |-  ( E. p  e.  (Poly `  B ) V  =  ( p `  X
)  ->  V  e.  _V )
220 eqeq1 2471 . . . . . 6  |-  ( a  =  V  ->  (
a  =  ( p `
 X )  <->  V  =  ( p `  X
) ) )
221220rexbidv 2973 . . . . 5  |-  ( a  =  V  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X
) ) )
222219, 221elab3 3257 . . . 4  |-  ( V  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) )
223215, 222syl6ib 226 . . 3  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  ->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
2244, 6, 12, 13, 14rgspncl 30751 . . . . . . 7  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  e.  (SubRing ` fld ) )
225224adantr 465 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  e.  (SubRing ` fld ) )
226 simpr 461 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  p  e.  (Poly `  B ) )
2274, 6, 12, 13, 14rgspnssid 30752 . . . . . . . . 9  |-  ( ph  ->  ( B  u.  { X } )  C_  (
(RingSpan ` fld ) `  ( B  u.  { X }
) ) )
228227unssbd 3682 . . . . . . . 8  |-  ( ph  ->  { X }  C_  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
229 snidg 4053 . . . . . . . . 9  |-  ( X  e.  CC  ->  X  e.  { X } )
23010, 229syl 16 . . . . . . . 8  |-  ( ph  ->  X  e.  { X } )
231228, 230sseldd 3505 . . . . . . 7  |-  ( ph  ->  X  e.  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) )
232231adantr 465 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  X  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
233227unssad 3681 . . . . . . 7  |-  ( ph  ->  B  C_  ( (RingSpan ` fld ) `
 ( B  u.  { X } ) ) )
234233adantr 465 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  B  C_  (
(RingSpan ` fld ) `  ( B  u.  { X }
) ) )
235225, 226, 232, 234cnsrplycl 30749 . . . . 5  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( p `  X )  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
236 eleq1 2539 . . . . 5  |-  ( V  =  ( p `  X )  ->  ( V  e.  ( (RingSpan ` fld ) `
 ( B  u.  { X } ) )  <-> 
( p `  X
)  e.  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) ) )
237235, 236syl5ibrcom 222 . . . 4  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( V  =  ( p `  X
)  ->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
238237rexlimdva 2955 . . 3  |-  ( ph  ->  ( E. p  e.  (Poly `  B ) V  =  ( p `  X )  ->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
239223, 238impbid 191 . 2  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
2402, 239bitrd 253 1  |-  ( ph  ->  ( V  e.  S  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113    u. cun 3474    C_ wss 3476   {csn 4027    _I cid 4790    X. cxp 4997    |` cres 5001    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522   CCcc 9490   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497   -ucneg 9806   Basecbs 14490   ↾s cress 14491   +g cplusg 14555   .rcmulr 14556   0gc0g 14695   invgcminusg 15728  SubGrpcsubg 16000   1rcur 16955   Ringcrg 17000  SubRingcsubrg 17225  RingSpancrgspn 17226  ℂfldccnfld 18219  Polycply 22344   Xpcidp 22345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-subg 16003  df-cmn 16606  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-subrg 17227  df-rgspn 17228  df-cnfld 18220  df-0p 21840  df-ply 22348  df-idp 22349  df-coe 22350  df-dgr 22351
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator