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 27346
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 2502 . 2  |-  ( ph  ->  ( V  e.  S  <->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
3 cnrng 16715 . . . . . . 7  |-fld  e.  Ring
43a1i 11 . . . . . 6  |-  ( ph  ->fld  e. 
Ring )
5 cnfldbas 16699 . . . . . . 7  |-  CC  =  ( Base ` fld )
65a1i 11 . . . . . 6  |-  ( ph  ->  CC  =  ( Base ` fld ) )
7 rngunsnply.b . . . . . . . 8  |-  ( ph  ->  B  e.  (SubRing ` fld ) )
85subrgss 15861 . . . . . . . 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 3935 . . . . . . 7  |-  ( ph  ->  { X }  C_  CC )
129, 11unssd 3515 . . . . . 6  |-  ( ph  ->  ( B  u.  { X } )  C_  CC )
13 eqidd 2436 . . . . . 6  |-  ( ph  ->  (RingSpan ` fld )  =  (RingSpan ` fld ) )
14 eqidd 2436 . . . . . 6  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  =  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
15 eqidd 2436 . . . . . . 7  |-  ( ph  ->  (flds  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  =  (flds  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } ) )
16 cnfld0 16717 . . . . . . . 8  |-  0  =  ( 0g ` fld )
1716a1i 11 . . . . . . 7  |-  ( ph  ->  0  =  ( 0g
` fld
) )
18 cnfldadd 16700 . . . . . . . 8  |-  +  =  ( +g  ` fld )
1918a1i 11 . . . . . . 7  |-  ( ph  ->  +  =  ( +g  ` fld ) )
20 plyf 20109 . . . . . . . . . . . 12  |-  ( p  e.  (Poly `  B
)  ->  p : CC
--> CC )
21 ffvelrn 5860 . . . . . . . . . . . 12  |-  ( ( p : CC --> CC  /\  X  e.  CC )  ->  ( p `  X
)  e.  CC )
2220, 10, 21syl2anr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( p `  X )  e.  CC )
23 eleq1 2495 . . . . . . . . . . 11  |-  ( a  =  ( p `  X )  ->  (
a  e.  CC  <->  ( p `  X )  e.  CC ) )
2422, 23syl5ibrcom 214 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( a  =  ( p `  X
)  ->  a  e.  CC ) )
2524rexlimdva 2822 . . . . . . . . 9  |-  ( ph  ->  ( E. p  e.  (Poly `  B )
a  =  ( p `
 X )  -> 
a  e.  CC ) )
2625ss2abdv 3408 . . . . . . . 8  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  C_  { a  |  a  e.  CC } )
27 abid2 2552 . . . . . . . . 9  |-  { a  |  a  e.  CC }  =  CC
2827, 5eqtri 2455 . . . . . . . 8  |-  { a  |  a  e.  CC }  =  ( Base ` fld )
2926, 28syl6sseq 3386 . . . . . . 7  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  C_  ( Base ` fld ) )
30 abid2 2552 . . . . . . . . 9  |-  { a  |  a  e.  B }  =  B
31 plyconst 20117 . . . . . . . . . . . . 13  |-  ( ( B  C_  CC  /\  a  e.  B )  ->  ( CC  X.  { a } )  e.  (Poly `  B ) )
329, 31sylan 458 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  ( CC  X.  { a } )  e.  (Poly `  B ) )
3310adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  B )  ->  X  e.  CC )
34 vex 2951 . . . . . . . . . . . . . . 15  |-  a  e. 
_V
3534fvconst2 5939 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
( CC  X.  {
a } ) `  X )  =  a )
3633, 35syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  B )  ->  (
( CC  X.  {
a } ) `  X )  =  a )
3736eqcomd 2440 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  a  =  ( ( CC 
X.  { a } ) `  X ) )
38 fveq1 5719 . . . . . . . . . . . . . 14  |-  ( p  =  ( CC  X.  { a } )  ->  ( p `  X )  =  ( ( CC  X.  {
a } ) `  X ) )
3938eqeq2d 2446 . . . . . . . . . . . . 13  |-  ( p  =  ( CC  X.  { a } )  ->  ( a  =  ( p `  X
)  <->  a  =  ( ( CC  X.  {
a } ) `  X ) ) )
4039rspcev 3044 . . . . . . . . . . . 12  |-  ( ( ( CC  X.  {
a } )  e.  (Poly `  B )  /\  a  =  (
( CC  X.  {
a } ) `  X ) )  ->  E. p  e.  (Poly `  B ) a  =  ( p `  X
) )
4132, 37, 40syl2anc 643 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  B )  ->  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) )
4241ex 424 . . . . . . . . . 10  |-  ( ph  ->  ( a  e.  B  ->  E. p  e.  (Poly `  B ) a  =  ( p `  X
) ) )
4342ss2abdv 3408 . . . . . . . . 9  |-  ( ph  ->  { a  |  a  e.  B }  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
4430, 43syl5eqssr 3385 . . . . . . . 8  |-  ( ph  ->  B  C_  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) } )
45 subrgsubg 15866 . . . . . . . . . 10  |-  ( B  e.  (SubRing ` fld )  ->  B  e.  (SubGrp ` fld ) )
467, 45syl 16 . . . . . . . . 9  |-  ( ph  ->  B  e.  (SubGrp ` fld )
)
4716subg0cl 14944 . . . . . . . . 9  |-  ( B  e.  (SubGrp ` fld )  ->  0  e.  B )
4846, 47syl 16 . . . . . . . 8  |-  ( ph  ->  0  e.  B )
4944, 48sseldd 3341 . . . . . . 7  |-  ( ph  ->  0  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
50 biid 228 . . . . . . . . 9  |-  ( ph  <->  ph )
51 vex 2951 . . . . . . . . . 10  |-  b  e. 
_V
52 eqeq1 2441 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
a  =  ( p `
 X )  <->  b  =  ( p `  X
) ) )
5352rexbidv 2718 . . . . . . . . . . 11  |-  ( a  =  b  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) b  =  ( p `  X
) ) )
54 fveq1 5719 . . . . . . . . . . . . 13  |-  ( p  =  e  ->  (
p `  X )  =  ( e `  X ) )
5554eqeq2d 2446 . . . . . . . . . . . 12  |-  ( p  =  e  ->  (
b  =  ( p `
 X )  <->  b  =  ( e `  X
) ) )
5655cbvrexv 2925 . . . . . . . . . . 11  |-  ( E. p  e.  (Poly `  B ) b  =  ( p `  X
)  <->  E. e  e.  (Poly `  B ) b  =  ( e `  X
) )
5753, 56syl6bb 253 . . . . . . . . . 10  |-  ( a  =  b  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. e  e.  (Poly `  B ) b  =  ( e `  X
) ) )
5851, 57elab 3074 . . . . . . . . 9  |-  ( b  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. e  e.  (Poly `  B )
b  =  ( e `
 X ) )
59 vex 2951 . . . . . . . . . 10  |-  c  e. 
_V
60 eqeq1 2441 . . . . . . . . . . . 12  |-  ( a  =  c  ->  (
a  =  ( p `
 X )  <->  c  =  ( p `  X
) ) )
6160rexbidv 2718 . . . . . . . . . . 11  |-  ( a  =  c  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) c  =  ( p `  X
) ) )
62 fveq1 5719 . . . . . . . . . . . . 13  |-  ( p  =  d  ->  (
p `  X )  =  ( d `  X ) )
6362eqeq2d 2446 . . . . . . . . . . . 12  |-  ( p  =  d  ->  (
c  =  ( p `
 X )  <->  c  =  ( d `  X
) ) )
6463cbvrexv 2925 . . . . . . . . . . 11  |-  ( E. p  e.  (Poly `  B ) c  =  ( p `  X
)  <->  E. d  e.  (Poly `  B ) c  =  ( d `  X
) )
6561, 64syl6bb 253 . . . . . . . . . 10  |-  ( a  =  c  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. d  e.  (Poly `  B ) c  =  ( d `  X
) ) )
6659, 65elab 3074 . . . . . . . . 9  |-  ( c  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. d  e.  (Poly `  B )
c  =  ( d `
 X ) )
67 simplr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  e  e.  (Poly `  B ) )
68 simpr 448 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  d  e.  (Poly `  B ) )
6918subrgacl 15871 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  e.  (SubRing ` fld )  /\  a  e.  B  /\  b  e.  B )  ->  (
a  +  b )  e.  B )
70693expb 1154 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  e.  (SubRing ` fld )  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  +  b )  e.  B )
717, 70sylan 458 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7271adantlr 696 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7372adantlr 696 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  e  e.  (Poly `  B
) )  /\  d  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7467, 68, 73plyadd 20128 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( e  o F  +  d )  e.  (Poly `  B
) )
75 plyf 20109 . . . . . . . . . . . . . . . . . . 19  |-  ( e  e.  (Poly `  B
)  ->  e : CC
--> CC )
76 ffn 5583 . . . . . . . . . . . . . . . . . . 19  |-  ( e : CC --> CC  ->  e  Fn  CC )
7775, 76syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( e  e.  (Poly `  B
)  ->  e  Fn  CC )
7877ad2antlr 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  e  Fn  CC )
79 plyf 20109 . . . . . . . . . . . . . . . . . . 19  |-  ( d  e.  (Poly `  B
)  ->  d : CC
--> CC )
80 ffn 5583 . . . . . . . . . . . . . . . . . . 19  |-  ( d : CC --> CC  ->  d  Fn  CC )
8179, 80syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( d  e.  (Poly `  B
)  ->  d  Fn  CC )
8281adantl 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  d  Fn  CC )
83 cnex 9063 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  _V
8483a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  CC  e.  _V )
8510ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  X  e.  CC )
86 fnfvof 6309 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  Fn  CC  /\  d  Fn  CC )  /\  ( CC  e.  _V  /\  X  e.  CC ) )  ->  (
( e  o F  +  d ) `  X )  =  ( ( e `  X
)  +  ( d `
 X ) ) )
8778, 82, 84, 85, 86syl22anc 1185 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e  o F  +  d ) `  X )  =  ( ( e `
 X )  +  ( d `  X
) ) )
8887eqcomd 2440 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e `
 X )  +  ( d `  X
) )  =  ( ( e  o F  +  d ) `  X ) )
89 fveq1 5719 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( e  o F  +  d )  ->  ( p `  X )  =  ( ( e  o F  +  d ) `  X ) )
9089eqeq2d 2446 . . . . . . . . . . . . . . . 16  |-  ( p  =  ( e  o F  +  d )  ->  ( ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
)  <->  ( ( e `
 X )  +  ( d `  X
) )  =  ( ( e  o F  +  d ) `  X ) ) )
9190rspcev 3044 . . . . . . . . . . . . . . 15  |-  ( ( ( e  o F  +  d )  e.  (Poly `  B )  /\  ( ( e `  X )  +  ( d `  X ) )  =  ( ( e  o F  +  d ) `  X
) )  ->  E. p  e.  (Poly `  B )
( ( e `  X )  +  ( d `  X ) )  =  ( p `
 X ) )
9274, 88, 91syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
) )
93 oveq2 6081 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( d `  X )  ->  (
( e `  X
)  +  c )  =  ( ( e `
 X )  +  ( d `  X
) ) )
9493eqeq1d 2443 . . . . . . . . . . . . . . 15  |-  ( c  =  ( d `  X )  ->  (
( ( e `  X )  +  c )  =  ( p `
 X )  <->  ( (
e `  X )  +  ( d `  X ) )  =  ( p `  X
) ) )
9594rexbidv 2718 . . . . . . . . . . . . . 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 214 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( c  =  ( d `  X
)  ->  E. p  e.  (Poly `  B )
( ( e `  X )  +  c )  =  ( p `
 X ) ) )
9796rexlimdva 2822 . . . . . . . . . . . 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 6080 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e `  X )  ->  (
b  +  c )  =  ( ( e `
 X )  +  c ) )
9998eqeq1d 2443 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( b  +  c )  =  ( p `
 X )  <->  ( (
e `  X )  +  c )  =  ( p `  X
) ) )
10099rexbidv 2718 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  ( E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  c )  =  ( p `  X
) ) )
101100imbi2d 308 . . . . . . . . . . . 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 214 . . . . . . . . . . 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 2822 . . . . . . . . . 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 1147 . . . . . . . . 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 1227 . . . . . . . 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 6098 . . . . . . . . 9  |-  ( b  +  c )  e. 
_V
107 eqeq1 2441 . . . . . . . . . 10  |-  ( a  =  ( b  +  c )  ->  (
a  =  ( p `
 X )  <->  ( b  +  c )  =  ( p `  X
) ) )
108107rexbidv 2718 . . . . . . . . 9  |-  ( a  =  ( b  +  c )  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
) ) )
109106, 108elab 3074 . . . . . . . 8  |-  ( ( b  +  c )  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B )
( b  +  c )  =  ( p `
 X ) )
110105, 109sylibr 204 . . . . . . 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 9040 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
112 cnfldneg 16719 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  CC  ->  (
( inv g ` fld ) `  1 )  = 
-u 1 )
113111, 112mp1i 12 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( inv g ` fld ) `  1 )  =  -u 1 )
114 cnfld1 16718 . . . . . . . . . . . . . . . . . . . 20  |-  1  =  ( 1r ` fld )
115114subrg1cl 15868 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  (SubRing ` fld )  ->  1  e.  B )
1167, 115syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  1  e.  B )
117 eqid 2435 . . . . . . . . . . . . . . . . . . 19  |-  ( inv g ` fld )  =  ( inv g ` fld )
118117subginvcl 14945 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  e.  (SubGrp ` fld )  /\  1  e.  B
)  ->  ( ( inv g ` fld ) `  1 )  e.  B )
11946, 116, 118syl2anc 643 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( inv g ` fld ) `  1 )  e.  B )
120113, 119eqeltrrd 2510 . . . . . . . . . . . . . . . 16  |-  ( ph  -> 
-u 1  e.  B
)
121 plyconst 20117 . . . . . . . . . . . . . . . 16  |-  ( ( B  C_  CC  /\  -u 1  e.  B )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B ) )
1229, 120, 121syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B )
)
123122adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B
) )
124 simpr 448 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  e  e.  (Poly `  B ) )
125 cnfldmul 16701 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
126125subrgmcl 15872 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  (SubRing ` fld )  /\  a  e.  B  /\  b  e.  B )  ->  (
a  x.  b )  e.  B )
1271263expb 1154 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  (SubRing ` fld )  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  x.  b )  e.  B
)
1287, 127sylan 458 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
129128adantlr 696 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
130123, 124, 72, 129plymul 20129 . . . . . . . . . . . . 13  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( CC 
X.  { -u 1 } )  o F  x.  e )  e.  (Poly `  B )
)
131 ffvelrn 5860 . . . . . . . . . . . . . . . 16  |-  ( ( e : CC --> CC  /\  X  e.  CC )  ->  ( e `  X
)  e.  CC )
13275, 10, 131syl2anr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( e `  X )  e.  CC )
133 cnfldneg 16719 . . . . . . . . . . . . . . 15  |-  ( ( e `  X )  e.  CC  ->  (
( inv g ` fld ) `  ( e `  X
) )  =  -u ( e `  X
) )
134132, 133syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( inv g ` fld ) `  ( e `
 X ) )  =  -u ( e `  X ) )
135 negex 9296 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  _V
136 fnconstg 5623 . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  e  Fn  CC )
13983a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  CC  e.  _V )
14010adantr 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  X  e.  CC )
141 fnfvof 6309 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( CC  X.  { -u 1 } )  Fn  CC  /\  e  Fn  CC )  /\  ( CC  e.  _V  /\  X  e.  CC ) )  -> 
( ( ( CC 
X.  { -u 1 } )  o F  x.  e ) `  X )  =  ( ( ( CC  X.  { -u 1 } ) `
 X )  x.  ( e `  X
) ) )
142137, 138, 139, 140, 141syl22anc 1185 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } )  o F  x.  e ) `
 X )  =  ( ( ( CC 
X.  { -u 1 } ) `  X
)  x.  ( e `
 X ) ) )
143135fvconst2 5939 . . . . . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } ) `  X )  x.  (
e `  X )
)  =  ( -u
1  x.  ( e `
 X ) ) )
146132mulm1d 9477 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( -u 1  x.  ( e `  X
) )  =  -u ( e `  X
) )
147142, 145, 1463eqtrd 2471 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } )  o F  x.  e ) `
 X )  = 
-u ( e `  X ) )
148134, 147eqtr4d 2470 . . . . . . . . . . . . 13  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( inv g ` fld ) `  ( e `
 X ) )  =  ( ( ( CC  X.  { -u
1 } )  o F  x.  e ) `
 X ) )
149 fveq1 5719 . . . . . . . . . . . . . . 15  |-  ( p  =  ( ( CC 
X.  { -u 1 } )  o F  x.  e )  -> 
( p `  X
)  =  ( ( ( CC  X.  { -u 1 } )  o F  x.  e ) `
 X ) )
150149eqeq2d 2446 . . . . . . . . . . . . . 14  |-  ( p  =  ( ( CC 
X.  { -u 1 } )  o F  x.  e )  -> 
( ( ( inv g ` fld ) `  ( e `
 X ) )  =  ( p `  X )  <->  ( ( inv g ` fld ) `  ( e `
 X ) )  =  ( ( ( CC  X.  { -u
1 } )  o F  x.  e ) `
 X ) ) )
151150rspcev 3044 . . . . . . . . . . . . 13  |-  ( ( ( ( CC  X.  { -u 1 } )  o F  x.  e
)  e.  (Poly `  B )  /\  (
( inv g ` fld ) `  ( e `  X
) )  =  ( ( ( CC  X.  { -u 1 } )  o F  x.  e
) `  X )
)  ->  E. p  e.  (Poly `  B )
( ( inv g ` fld ) `  ( e `  X ) )  =  ( p `  X
) )
152130, 148, 151syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( inv g ` fld ) `  ( e `
 X ) )  =  ( p `  X ) )
153 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( inv g ` fld ) `  b )  =  ( ( inv g ` fld ) `  ( e `  X
) ) )
154153eqeq1d 2443 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  (
( ( inv g ` fld ) `  b )  =  ( p `  X )  <->  ( ( inv g ` fld ) `  ( e `
 X ) )  =  ( p `  X ) ) )
155154rexbidv 2718 . . . . . . . . . . . 12  |-  ( b  =  ( e `  X )  ->  ( E. p  e.  (Poly `  B ) ( ( inv g ` fld ) `  b )  =  ( p `  X )  <->  E. p  e.  (Poly `  B )
( ( inv g ` fld ) `  ( e `  X ) )  =  ( p `  X
) ) )
156152, 155syl5ibrcom 214 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( b  =  ( e `  X
)  ->  E. p  e.  (Poly `  B )
( ( inv g ` fld ) `  b )  =  ( p `  X ) ) )
157156rexlimdva 2822 . . . . . . . . . 10  |-  ( ph  ->  ( E. e  e.  (Poly `  B )
b  =  ( e `
 X )  ->  E. p  e.  (Poly `  B ) ( ( inv g ` fld ) `  b )  =  ( p `  X ) ) )
158157imp 419 . . . . . . . . 9  |-  ( (
ph  /\  E. e  e.  (Poly `  B )
b  =  ( e `
 X ) )  ->  E. p  e.  (Poly `  B ) ( ( inv g ` fld ) `  b )  =  ( p `  X ) )
15958, 158sylan2b 462 . . . . . . . 8  |-  ( (
ph  /\  b  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  ->  E. p  e.  (Poly `  B )
( ( inv g ` fld ) `  b )  =  ( p `  X ) )
160 fvex 5734 . . . . . . . . 9  |-  ( ( inv g ` fld ) `  b )  e.  _V
161 eqeq1 2441 . . . . . . . . . 10  |-  ( a  =  ( ( inv g ` fld ) `  b )  ->  ( a  =  ( p `  X
)  <->  ( ( inv g ` fld ) `  b )  =  ( p `  X ) ) )
162161rexbidv 2718 . . . . . . . . 9  |-  ( a  =  ( ( inv g ` fld ) `  b )  ->  ( E. p  e.  (Poly `  B )
a  =  ( p `
 X )  <->  E. p  e.  (Poly `  B )
( ( inv g ` fld ) `  b )  =  ( p `  X ) ) )
163160, 162elab 3074 . . . . . . . 8  |-  ( ( ( inv g ` fld ) `  b )  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  <->  E. p  e.  (Poly `  B ) ( ( inv g ` fld ) `  b )  =  ( p `  X ) )
164159, 163sylibr 204 . . . . . . 7  |-  ( (
ph  /\  b  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  ->  ( ( inv g ` 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 3341 . . . . . . 7  |-  ( ph  ->  1  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
168129adantlr 696 . . . . . . . . . . . . . . . 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 20129 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( e  o F  x.  d )  e.  (Poly `  B
) )
170 fnfvof 6309 . . . . . . . . . . . . . . . . 17  |-  ( ( ( e  Fn  CC  /\  d  Fn  CC )  /\  ( CC  e.  _V  /\  X  e.  CC ) )  ->  (
( e  o F  x.  d ) `  X )  =  ( ( e `  X
)  x.  ( d `
 X ) ) )
17178, 82, 84, 85, 170syl22anc 1185 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e  o F  x.  d
) `  X )  =  ( ( e `
 X )  x.  ( d `  X
) ) )
172171eqcomd 2440 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e `
 X )  x.  ( d `  X
) )  =  ( ( e  o F  x.  d ) `  X ) )
173 fveq1 5719 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( e  o F  x.  d )  ->  ( p `  X )  =  ( ( e  o F  x.  d ) `  X ) )
174173eqeq2d 2446 . . . . . . . . . . . . . . . 16  |-  ( p  =  ( e  o F  x.  d )  ->  ( ( ( e `  X )  x.  ( d `  X ) )  =  ( p `  X
)  <->  ( ( e `
 X )  x.  ( d `  X
) )  =  ( ( e  o F  x.  d ) `  X ) ) )
175174rspcev 3044 . . . . . . . . . . . . . . 15  |-  ( ( ( e  o F  x.  d )  e.  (Poly `  B )  /\  ( ( e `  X )  x.  (
d `  X )
)  =  ( ( e  o F  x.  d ) `  X
) )  ->  E. p  e.  (Poly `  B )
( ( e `  X )  x.  (
d `  X )
)  =  ( p `
 X ) )
176169, 172, 175syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  x.  ( d `  X ) )  =  ( p `  X
) )
177 oveq2 6081 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( d `  X )  ->  (
( e `  X
)  x.  c )  =  ( ( e `
 X )  x.  ( d `  X
) ) )
178177eqeq1d 2443 . . . . . . . . . . . . . . 15  |-  ( c  =  ( d `  X )  ->  (
( ( e `  X )  x.  c
)  =  ( p `
 X )  <->  ( (
e `  X )  x.  ( d `  X
) )  =  ( p `  X ) ) )
179178rexbidv 2718 . . . . . . . . . . . . . 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 214 . . . . . . . . . . . . 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 2822 . . . . . . . . . . . 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 6080 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e `  X )  ->  (
b  x.  c )  =  ( ( e `
 X )  x.  c ) )
183182eqeq1d 2443 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( b  x.  c
)  =  ( p `
 X )  <->  ( (
e `  X )  x.  c )  =  ( p `  X ) ) )
184183rexbidv 2718 . . . . . . . . . . . . 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 308 . . . . . . . . . . . 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 214 . . . . . . . . . . 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 2822 . . . . . . . . . 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 1147 . . . . . . . . 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 1227 . . . . . . . 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 6098 . . . . . . . . 9  |-  ( b  x.  c )  e. 
_V
191 eqeq1 2441 . . . . . . . . . 10  |-  ( a  =  ( b  x.  c )  ->  (
a  =  ( p `
 X )  <->  ( b  x.  c )  =  ( p `  X ) ) )
192191rexbidv 2718 . . . . . . . . 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 3074 . . . . . . . 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 204 . . . . . . 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 16254 . . . . . 6  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  e.  (SubRing ` fld ) )
196 plyid 20120 . . . . . . . . . . 11  |-  ( ( B  C_  CC  /\  1  e.  B )  ->  X p  e.  (Poly `  B
) )
1979, 116, 196syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  X p  e.  (Poly `  B ) )
198 df-idp 20100 . . . . . . . . . . . 12  |-  X p  =  (  _I  |`  CC )
199198fveq1i 5721 . . . . . . . . . . 11  |-  ( X p `  X )  =  ( (  _I  |`  CC ) `  X
)
200 fvresi 5916 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
(  _I  |`  CC ) `
 X )  =  X )
20110, 200syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( (  _I  |`  CC ) `
 X )  =  X )
202199, 201syl5req 2480 . . . . . . . . . 10  |-  ( ph  ->  X  =  ( X p `  X ) )
203 fveq1 5719 . . . . . . . . . . . 12  |-  ( p  =  X p  -> 
( p `  X
)  =  ( X p `  X ) )
204203eqeq2d 2446 . . . . . . . . . . 11  |-  ( p  =  X p  -> 
( X  =  ( p `  X )  <-> 
X  =  ( X p `  X ) ) )
205204rspcev 3044 . . . . . . . . . 10  |-  ( ( X p  e.  (Poly `  B )  /\  X  =  ( X p `
 X ) )  ->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) )
206197, 202, 205syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) )
207 eqeq1 2441 . . . . . . . . . . . 12  |-  ( a  =  X  ->  (
a  =  ( p `
 X )  <->  X  =  ( p `  X
) ) )
208207rexbidv 2718 . . . . . . . . . . 11  |-  ( a  =  X  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) ) )
209208elabg 3075 . . . . . . . . . 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 224 . . . . . . . 8  |-  ( ph  ->  X  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
212211snssd 3935 . . . . . . 7  |-  ( ph  ->  { X }  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
21344, 212unssd 3515 . . . . . 6  |-  ( ph  ->  ( B  u.  { X } )  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
2144, 6, 12, 13, 14, 195, 213rgspnmin 27344 . . . . 5  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
215214sseld 3339 . . . 4  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  ->  V  e.  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } ) )
216 fvex 5734 . . . . . . 7  |-  ( p `
 X )  e. 
_V
217 eleq1 2495 . . . . . . 7  |-  ( V  =  ( p `  X )  ->  ( V  e.  _V  <->  ( p `  X )  e.  _V ) )
218216, 217mpbiri 225 . . . . . 6  |-  ( V  =  ( p `  X )  ->  V  e.  _V )
219218rexlimivw 2818 . . . . 5  |-  ( E. p  e.  (Poly `  B ) V  =  ( p `  X
)  ->  V  e.  _V )
220 eqeq1 2441 . . . . . 6  |-  ( a  =  V  ->  (
a  =  ( p `
 X )  <->  V  =  ( p `  X
) ) )
221220rexbidv 2718 . . . . 5  |-  ( a  =  V  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X
) ) )
222219, 221elab3 3081 . . . 4  |-  ( V  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) )
223215, 222syl6ib 218 . . 3  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  ->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
2244, 6, 12, 13, 14rgspncl 27342 . . . . . . 7  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  e.  (SubRing ` fld ) )
225224adantr 452 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  e.  (SubRing ` fld ) )
226 simpr 448 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  p  e.  (Poly `  B ) )
2274, 6, 12, 13, 14rgspnssid 27343 . . . . . . . . 9  |-  ( ph  ->  ( B  u.  { X } )  C_  (
(RingSpan ` fld ) `  ( B  u.  { X }
) ) )
228227unssbd 3517 . . . . . . . 8  |-  ( ph  ->  { X }  C_  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
229 snidg 3831 . . . . . . . . 9  |-  ( X  e.  CC  ->  X  e.  { X } )
23010, 229syl 16 . . . . . . . 8  |-  ( ph  ->  X  e.  { X } )
231228, 230sseldd 3341 . . . . . . 7  |-  ( ph  ->  X  e.  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) )
232231adantr 452 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  X  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
233227unssad 3516 . . . . . . 7  |-  ( ph  ->  B  C_  ( (RingSpan ` fld ) `
 ( B  u.  { X } ) ) )
234233adantr 452 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  B  C_  (
(RingSpan ` fld ) `  ( B  u.  { X }
) ) )
235225, 226, 232, 234cnsrplycl 27340 . . . . 5  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( p `  X )  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
236 eleq1 2495 . . . . 5  |-  ( V  =  ( p `  X )  ->  ( V  e.  ( (RingSpan ` fld ) `
 ( B  u.  { X } ) )  <-> 
( p `  X
)  e.  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) ) )
237235, 236syl5ibrcom 214 . . . 4  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( V  =  ( p `  X
)  ->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
238237rexlimdva 2822 . . 3  |-  ( ph  ->  ( E. p  e.  (Poly `  B ) V  =  ( p `  X )  ->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
239223, 238impbid 184 . 2  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
2402, 239bitrd 245 1  |-  ( ph  ->  ( V  e.  S  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948    u. cun 3310    C_ wss 3312   {csn 3806    _I cid 4485    X. cxp 4868    |` cres 4872    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987   -ucneg 9284   Basecbs 13461   ↾s cress 13462   +g cplusg 13521   .rcmulr 13522   0gc0g 13715   inv gcminusg 14678  SubGrpcsubg 14930   Ringcrg 15652   1rcur 15654  SubRingcsubrg 15856  RingSpancrgspn 15857  ℂfldccnfld 16695  Polycply 20095   X pcidp 20096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-subg 14933  df-cmn 15406  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-subrg 15858  df-rgspn 15859  df-cnfld 16696  df-0p 19554  df-ply 20099  df-idp 20100  df-coe 20101  df-dgr 20102
  Copyright terms: Public domain W3C validator