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

Theorem rngunsnply 36039
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 2514 . 2  |-  ( ph  ->  ( V  e.  S  <->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
3 cnring 18990 . . . . . . 7  |-fld  e.  Ring
43a1i 11 . . . . . 6  |-  ( ph  ->fld  e. 
Ring )
5 cnfldbas 18974 . . . . . . 7  |-  CC  =  ( Base ` fld )
65a1i 11 . . . . . 6  |-  ( ph  ->  CC  =  ( Base ` fld ) )
7 rngunsnply.b . . . . . . . 8  |-  ( ph  ->  B  e.  (SubRing ` fld ) )
85subrgss 18009 . . . . . . . 8  |-  ( B  e.  (SubRing ` fld )  ->  B  C_  CC )
97, 8syl 17 . . . . . . 7  |-  ( ph  ->  B  C_  CC )
10 rngunsnply.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
1110snssd 4117 . . . . . . 7  |-  ( ph  ->  { X }  C_  CC )
129, 11unssd 3610 . . . . . 6  |-  ( ph  ->  ( B  u.  { X } )  C_  CC )
13 eqidd 2452 . . . . . 6  |-  ( ph  ->  (RingSpan ` fld )  =  (RingSpan ` fld ) )
14 eqidd 2452 . . . . . 6  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  =  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
15 eqidd 2452 . . . . . . 7  |-  ( ph  ->  (flds  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  =  (flds  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } ) )
16 cnfld0 18992 . . . . . . . 8  |-  0  =  ( 0g ` fld )
1716a1i 11 . . . . . . 7  |-  ( ph  ->  0  =  ( 0g
` fld
) )
18 cnfldadd 18975 . . . . . . . 8  |-  +  =  ( +g  ` fld )
1918a1i 11 . . . . . . 7  |-  ( ph  ->  +  =  ( +g  ` fld ) )
20 plyf 23152 . . . . . . . . . . . 12  |-  ( p  e.  (Poly `  B
)  ->  p : CC
--> CC )
21 ffvelrn 6020 . . . . . . . . . . . 12  |-  ( ( p : CC --> CC  /\  X  e.  CC )  ->  ( p `  X
)  e.  CC )
2220, 10, 21syl2anr 481 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( p `  X )  e.  CC )
23 eleq1 2517 . . . . . . . . . . 11  |-  ( a  =  ( p `  X )  ->  (
a  e.  CC  <->  ( p `  X )  e.  CC ) )
2422, 23syl5ibrcom 226 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( a  =  ( p `  X
)  ->  a  e.  CC ) )
2524rexlimdva 2879 . . . . . . . . 9  |-  ( ph  ->  ( E. p  e.  (Poly `  B )
a  =  ( p `
 X )  -> 
a  e.  CC ) )
2625ss2abdv 3502 . . . . . . . 8  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  C_  { a  |  a  e.  CC } )
27 abid2 2573 . . . . . . . . 9  |-  { a  |  a  e.  CC }  =  CC
2827, 5eqtri 2473 . . . . . . . 8  |-  { a  |  a  e.  CC }  =  ( Base ` fld )
2926, 28syl6sseq 3478 . . . . . . 7  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  C_  ( Base ` fld ) )
30 abid2 2573 . . . . . . . . 9  |-  { a  |  a  e.  B }  =  B
31 plyconst 23160 . . . . . . . . . . . . 13  |-  ( ( B  C_  CC  /\  a  e.  B )  ->  ( CC  X.  { a } )  e.  (Poly `  B ) )
329, 31sylan 474 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  ( CC  X.  { a } )  e.  (Poly `  B ) )
3310adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  B )  ->  X  e.  CC )
34 vex 3048 . . . . . . . . . . . . . . 15  |-  a  e. 
_V
3534fvconst2 6120 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
( CC  X.  {
a } ) `  X )  =  a )
3633, 35syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  B )  ->  (
( CC  X.  {
a } ) `  X )  =  a )
3736eqcomd 2457 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  a  =  ( ( CC 
X.  { a } ) `  X ) )
38 fveq1 5864 . . . . . . . . . . . . . 14  |-  ( p  =  ( CC  X.  { a } )  ->  ( p `  X )  =  ( ( CC  X.  {
a } ) `  X ) )
3938eqeq2d 2461 . . . . . . . . . . . . 13  |-  ( p  =  ( CC  X.  { a } )  ->  ( a  =  ( p `  X
)  <->  a  =  ( ( CC  X.  {
a } ) `  X ) ) )
4039rspcev 3150 . . . . . . . . . . . 12  |-  ( ( ( CC  X.  {
a } )  e.  (Poly `  B )  /\  a  =  (
( CC  X.  {
a } ) `  X ) )  ->  E. p  e.  (Poly `  B ) a  =  ( p `  X
) )
4132, 37, 40syl2anc 667 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  B )  ->  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) )
4241ex 436 . . . . . . . . . 10  |-  ( ph  ->  ( a  e.  B  ->  E. p  e.  (Poly `  B ) a  =  ( p `  X
) ) )
4342ss2abdv 3502 . . . . . . . . 9  |-  ( ph  ->  { a  |  a  e.  B }  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
4430, 43syl5eqssr 3477 . . . . . . . 8  |-  ( ph  ->  B  C_  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) } )
45 subrgsubg 18014 . . . . . . . . . 10  |-  ( B  e.  (SubRing ` fld )  ->  B  e.  (SubGrp ` fld ) )
467, 45syl 17 . . . . . . . . 9  |-  ( ph  ->  B  e.  (SubGrp ` fld )
)
4716subg0cl 16825 . . . . . . . . 9  |-  ( B  e.  (SubGrp ` fld )  ->  0  e.  B )
4846, 47syl 17 . . . . . . . 8  |-  ( ph  ->  0  e.  B )
4944, 48sseldd 3433 . . . . . . 7  |-  ( ph  ->  0  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
50 biid 240 . . . . . . . . 9  |-  ( ph  <->  ph )
51 vex 3048 . . . . . . . . . 10  |-  b  e. 
_V
52 eqeq1 2455 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
a  =  ( p `
 X )  <->  b  =  ( p `  X
) ) )
5352rexbidv 2901 . . . . . . . . . . 11  |-  ( a  =  b  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) b  =  ( p `  X
) ) )
54 fveq1 5864 . . . . . . . . . . . . 13  |-  ( p  =  e  ->  (
p `  X )  =  ( e `  X ) )
5554eqeq2d 2461 . . . . . . . . . . . 12  |-  ( p  =  e  ->  (
b  =  ( p `
 X )  <->  b  =  ( e `  X
) ) )
5655cbvrexv 3020 . . . . . . . . . . 11  |-  ( E. p  e.  (Poly `  B ) b  =  ( p `  X
)  <->  E. e  e.  (Poly `  B ) b  =  ( e `  X
) )
5753, 56syl6bb 265 . . . . . . . . . 10  |-  ( a  =  b  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. e  e.  (Poly `  B ) b  =  ( e `  X
) ) )
5851, 57elab 3185 . . . . . . . . 9  |-  ( b  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. e  e.  (Poly `  B )
b  =  ( e `
 X ) )
59 vex 3048 . . . . . . . . . 10  |-  c  e. 
_V
60 eqeq1 2455 . . . . . . . . . . . 12  |-  ( a  =  c  ->  (
a  =  ( p `
 X )  <->  c  =  ( p `  X
) ) )
6160rexbidv 2901 . . . . . . . . . . 11  |-  ( a  =  c  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) c  =  ( p `  X
) ) )
62 fveq1 5864 . . . . . . . . . . . . 13  |-  ( p  =  d  ->  (
p `  X )  =  ( d `  X ) )
6362eqeq2d 2461 . . . . . . . . . . . 12  |-  ( p  =  d  ->  (
c  =  ( p `
 X )  <->  c  =  ( d `  X
) ) )
6463cbvrexv 3020 . . . . . . . . . . 11  |-  ( E. p  e.  (Poly `  B ) c  =  ( p `  X
)  <->  E. d  e.  (Poly `  B ) c  =  ( d `  X
) )
6561, 64syl6bb 265 . . . . . . . . . 10  |-  ( a  =  c  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. d  e.  (Poly `  B ) c  =  ( d `  X
) ) )
6659, 65elab 3185 . . . . . . . . 9  |-  ( c  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. d  e.  (Poly `  B )
c  =  ( d `
 X ) )
67 simplr 762 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  e  e.  (Poly `  B ) )
68 simpr 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  d  e.  (Poly `  B ) )
6918subrgacl 18019 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  e.  (SubRing ` fld )  /\  a  e.  B  /\  b  e.  B )  ->  (
a  +  b )  e.  B )
70693expb 1209 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  e.  (SubRing ` fld )  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  +  b )  e.  B )
717, 70sylan 474 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7271adantlr 721 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7372adantlr 721 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  e  e.  (Poly `  B
) )  /\  d  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7467, 68, 73plyadd 23171 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( e  oF  +  d )  e.  (Poly `  B
) )
75 plyf 23152 . . . . . . . . . . . . . . . . . . 19  |-  ( e  e.  (Poly `  B
)  ->  e : CC
--> CC )
76 ffn 5728 . . . . . . . . . . . . . . . . . . 19  |-  ( e : CC --> CC  ->  e  Fn  CC )
7775, 76syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( e  e.  (Poly `  B
)  ->  e  Fn  CC )
7877ad2antlr 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  e  Fn  CC )
79 plyf 23152 . . . . . . . . . . . . . . . . . . 19  |-  ( d  e.  (Poly `  B
)  ->  d : CC
--> CC )
80 ffn 5728 . . . . . . . . . . . . . . . . . . 19  |-  ( d : CC --> CC  ->  d  Fn  CC )
8179, 80syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( d  e.  (Poly `  B
)  ->  d  Fn  CC )
8281adantl 468 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  d  Fn  CC )
83 cnex 9620 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  _V
8483a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  CC  e.  _V )
8510ad2antrr 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  X  e.  CC )
86 fnfvof 6545 . . . . . . . . . . . . . . . . 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 1269 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e  oF  +  d ) `  X )  =  ( ( e `
 X )  +  ( d `  X
) ) )
8887eqcomd 2457 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e `
 X )  +  ( d `  X
) )  =  ( ( e  oF  +  d ) `  X ) )
89 fveq1 5864 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( e  oF  +  d )  ->  ( p `  X )  =  ( ( e  oF  +  d ) `  X ) )
9089eqeq2d 2461 . . . . . . . . . . . . . . . 16  |-  ( p  =  ( e  oF  +  d )  ->  ( ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
)  <->  ( ( e `
 X )  +  ( d `  X
) )  =  ( ( e  oF  +  d ) `  X ) ) )
9190rspcev 3150 . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
) )
93 oveq2 6298 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( d `  X )  ->  (
( e `  X
)  +  c )  =  ( ( e `
 X )  +  ( d `  X
) ) )
9493eqeq1d 2453 . . . . . . . . . . . . . . 15  |-  ( c  =  ( d `  X )  ->  (
( ( e `  X )  +  c )  =  ( p `
 X )  <->  ( (
e `  X )  +  ( d `  X ) )  =  ( p `  X
) ) )
9594rexbidv 2901 . . . . . . . . . . . . . 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 226 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( c  =  ( d `  X
)  ->  E. p  e.  (Poly `  B )
( ( e `  X )  +  c )  =  ( p `
 X ) ) )
9796rexlimdva 2879 . . . . . . . . . . . 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 6297 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e `  X )  ->  (
b  +  c )  =  ( ( e `
 X )  +  c ) )
9998eqeq1d 2453 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( b  +  c )  =  ( p `
 X )  <->  ( (
e `  X )  +  c )  =  ( p `  X
) ) )
10099rexbidv 2901 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  ( E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  c )  =  ( p `  X
) ) )
101100imbi2d 318 . . . . . . . . . . . 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 226 . . . . . . . . . . 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 2879 . . . . . . . . . 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 1202 . . . . . . . . 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 1311 . . . . . . . 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 6318 . . . . . . . . 9  |-  ( b  +  c )  e. 
_V
107 eqeq1 2455 . . . . . . . . . 10  |-  ( a  =  ( b  +  c )  ->  (
a  =  ( p `
 X )  <->  ( b  +  c )  =  ( p `  X
) ) )
108107rexbidv 2901 . . . . . . . . 9  |-  ( a  =  ( b  +  c )  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
) ) )
109106, 108elab 3185 . . . . . . . 8  |-  ( ( b  +  c )  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B )
( b  +  c )  =  ( p `
 X ) )
110105, 109sylibr 216 . . . . . . 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 9597 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
112 cnfldneg 18994 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  CC  ->  (
( invg ` fld ) `  1 )  = 
-u 1 )
113111, 112mp1i 13 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( invg ` fld ) `  1 )  =  -u 1 )
114 cnfld1 18993 . . . . . . . . . . . . . . . . . . . 20  |-  1  =  ( 1r ` fld )
115114subrg1cl 18016 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  (SubRing ` fld )  ->  1  e.  B )
1167, 115syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  1  e.  B )
117 eqid 2451 . . . . . . . . . . . . . . . . . . 19  |-  ( invg ` fld )  =  ( invg ` fld )
118117subginvcl 16826 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  e.  (SubGrp ` fld )  /\  1  e.  B
)  ->  ( ( invg ` fld ) `  1 )  e.  B )
11946, 116, 118syl2anc 667 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( invg ` fld ) `  1 )  e.  B )
120113, 119eqeltrrd 2530 . . . . . . . . . . . . . . . 16  |-  ( ph  -> 
-u 1  e.  B
)
121 plyconst 23160 . . . . . . . . . . . . . . . 16  |-  ( ( B  C_  CC  /\  -u 1  e.  B )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B ) )
1229, 120, 121syl2anc 667 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B )
)
123122adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B
) )
124 simpr 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  e  e.  (Poly `  B ) )
125 cnfldmul 18976 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
126125subrgmcl 18020 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  (SubRing ` fld )  /\  a  e.  B  /\  b  e.  B )  ->  (
a  x.  b )  e.  B )
1271263expb 1209 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  (SubRing ` fld )  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  x.  b )  e.  B
)
1287, 127sylan 474 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
129128adantlr 721 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
130123, 124, 72, 129plymul 23172 . . . . . . . . . . . . 13  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( CC 
X.  { -u 1 } )  oF  x.  e )  e.  (Poly `  B )
)
131 ffvelrn 6020 . . . . . . . . . . . . . . . 16  |-  ( ( e : CC --> CC  /\  X  e.  CC )  ->  ( e `  X
)  e.  CC )
13275, 10, 131syl2anr 481 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( e `  X )  e.  CC )
133 cnfldneg 18994 . . . . . . . . . . . . . . 15  |-  ( ( e `  X )  e.  CC  ->  (
( invg ` fld ) `  ( e `  X
) )  =  -u ( e `  X
) )
134132, 133syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( invg ` fld ) `  ( e `
 X ) )  =  -u ( e `  X ) )
135 negex 9873 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  _V
136 fnconstg 5771 . . . . . . . . . . . . . . . . 17  |-  ( -u
1  e.  _V  ->  ( CC  X.  { -u
1 } )  Fn  CC )
137135, 136mp1i 13 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( CC  X.  { -u 1 } )  Fn  CC )
13877adantl 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  e  Fn  CC )
13983a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  CC  e.  _V )
14010adantr 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  X  e.  CC )
141 fnfvof 6545 . . . . . . . . . . . . . . . 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 1269 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } )  oF  x.  e ) `
 X )  =  ( ( ( CC 
X.  { -u 1 } ) `  X
)  x.  ( e `
 X ) ) )
143135fvconst2 6120 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  CC  ->  (
( CC  X.  { -u 1 } ) `  X )  =  -u
1 )
144140, 143syl 17 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( CC 
X.  { -u 1 } ) `  X
)  =  -u 1
)
145144oveq1d 6305 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } ) `  X )  x.  (
e `  X )
)  =  ( -u
1  x.  ( e `
 X ) ) )
146132mulm1d 10070 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( -u 1  x.  ( e `  X
) )  =  -u ( e `  X
) )
147142, 145, 1463eqtrd 2489 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } )  oF  x.  e ) `
 X )  = 
-u ( e `  X ) )
148134, 147eqtr4d 2488 . . . . . . . . . . . . 13  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( invg ` fld ) `  ( e `
 X ) )  =  ( ( ( CC  X.  { -u
1 } )  oF  x.  e ) `
 X ) )
149 fveq1 5864 . . . . . . . . . . . . . . 15  |-  ( p  =  ( ( CC 
X.  { -u 1 } )  oF  x.  e )  -> 
( p `  X
)  =  ( ( ( CC  X.  { -u 1 } )  oF  x.  e ) `
 X ) )
150149eqeq2d 2461 . . . . . . . . . . . . . 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 3150 . . . . . . . . . . . . 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 667 . . . . . . . . . . . 12  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  ( e `
 X ) )  =  ( p `  X ) )
153 fveq2 5865 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( invg ` fld ) `  b )  =  ( ( invg ` fld ) `  ( e `  X
) ) )
154153eqeq1d 2453 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  (
( ( invg ` fld ) `  b )  =  ( p `  X )  <->  ( ( invg ` fld ) `  ( e `
 X ) )  =  ( p `  X ) ) )
155154rexbidv 2901 . . . . . . . . . . . 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 226 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( b  =  ( e `  X
)  ->  E. p  e.  (Poly `  B )
( ( invg ` fld ) `  b )  =  ( p `  X ) ) )
157156rexlimdva 2879 . . . . . . . . . 10  |-  ( ph  ->  ( E. e  e.  (Poly `  B )
b  =  ( e `
 X )  ->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  b )  =  ( p `  X ) ) )
158157imp 431 . . . . . . . . 9  |-  ( (
ph  /\  E. e  e.  (Poly `  B )
b  =  ( e `
 X ) )  ->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  b )  =  ( p `  X ) )
15958, 158sylan2b 478 . . . . . . . 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 5875 . . . . . . . . 9  |-  ( ( invg ` fld ) `  b )  e.  _V
161 eqeq1 2455 . . . . . . . . . 10  |-  ( a  =  ( ( invg ` fld ) `  b )  ->  ( a  =  ( p `  X
)  <->  ( ( invg ` fld ) `  b )  =  ( p `  X ) ) )
162161rexbidv 2901 . . . . . . . . 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 3185 . . . . . . . 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 216 . . . . . . 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 3433 . . . . . . 7  |-  ( ph  ->  1  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
168129adantlr 721 . . . . . . . . . . . . . . . 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 23172 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( e  oF  x.  d )  e.  (Poly `  B
) )
170 fnfvof 6545 . . . . . . . . . . . . . . . . 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 1269 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e  oF  x.  d
) `  X )  =  ( ( e `
 X )  x.  ( d `  X
) ) )
172171eqcomd 2457 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e `
 X )  x.  ( d `  X
) )  =  ( ( e  oF  x.  d ) `  X ) )
173 fveq1 5864 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( e  oF  x.  d )  ->  ( p `  X )  =  ( ( e  oF  x.  d ) `  X ) )
174173eqeq2d 2461 . . . . . . . . . . . . . . . 16  |-  ( p  =  ( e  oF  x.  d )  ->  ( ( ( e `  X )  x.  ( d `  X ) )  =  ( p `  X
)  <->  ( ( e `
 X )  x.  ( d `  X
) )  =  ( ( e  oF  x.  d ) `  X ) ) )
175174rspcev 3150 . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  x.  ( d `  X ) )  =  ( p `  X
) )
177 oveq2 6298 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( d `  X )  ->  (
( e `  X
)  x.  c )  =  ( ( e `
 X )  x.  ( d `  X
) ) )
178177eqeq1d 2453 . . . . . . . . . . . . . . 15  |-  ( c  =  ( d `  X )  ->  (
( ( e `  X )  x.  c
)  =  ( p `
 X )  <->  ( (
e `  X )  x.  ( d `  X
) )  =  ( p `  X ) ) )
179178rexbidv 2901 . . . . . . . . . . . . . 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 226 . . . . . . . . . . . . 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 2879 . . . . . . . . . . . 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 6297 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e `  X )  ->  (
b  x.  c )  =  ( ( e `
 X )  x.  c ) )
183182eqeq1d 2453 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( b  x.  c
)  =  ( p `
 X )  <->  ( (
e `  X )  x.  c )  =  ( p `  X ) ) )
184183rexbidv 2901 . . . . . . . . . . . . 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 318 . . . . . . . . . . . 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 226 . . . . . . . . . . 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 2879 . . . . . . . . . 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 1202 . . . . . . . . 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 1311 . . . . . . . 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 6318 . . . . . . . . 9  |-  ( b  x.  c )  e. 
_V
191 eqeq1 2455 . . . . . . . . . 10  |-  ( a  =  ( b  x.  c )  ->  (
a  =  ( p `
 X )  <->  ( b  x.  c )  =  ( p `  X ) ) )
192191rexbidv 2901 . . . . . . . . 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 3185 . . . . . . . 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 216 . . . . . . 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 18412 . . . . . 6  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  e.  (SubRing ` fld ) )
196 plyid 23163 . . . . . . . . . . 11  |-  ( ( B  C_  CC  /\  1  e.  B )  ->  Xp  e.  (Poly `  B
) )
1979, 116, 196syl2anc 667 . . . . . . . . . 10  |-  ( ph  ->  Xp  e.  (Poly `  B ) )
198 df-idp 23143 . . . . . . . . . . . 12  |-  Xp  =  (  _I  |`  CC )
199198fveq1i 5866 . . . . . . . . . . 11  |-  ( Xp `  X )  =  ( (  _I  |`  CC ) `  X
)
200 fvresi 6090 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
(  _I  |`  CC ) `
 X )  =  X )
20110, 200syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( (  _I  |`  CC ) `
 X )  =  X )
202199, 201syl5req 2498 . . . . . . . . . 10  |-  ( ph  ->  X  =  ( Xp `  X ) )
203 fveq1 5864 . . . . . . . . . . . 12  |-  ( p  =  Xp  -> 
( p `  X
)  =  ( Xp `  X ) )
204203eqeq2d 2461 . . . . . . . . . . 11  |-  ( p  =  Xp  -> 
( X  =  ( p `  X )  <-> 
X  =  ( Xp `  X ) ) )
205204rspcev 3150 . . . . . . . . . 10  |-  ( ( Xp  e.  (Poly `  B )  /\  X  =  ( Xp `  X ) )  ->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) )
206197, 202, 205syl2anc 667 . . . . . . . . 9  |-  ( ph  ->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) )
207 eqeq1 2455 . . . . . . . . . . . 12  |-  ( a  =  X  ->  (
a  =  ( p `
 X )  <->  X  =  ( p `  X
) ) )
208207rexbidv 2901 . . . . . . . . . . 11  |-  ( a  =  X  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) ) )
209208elabg 3186 . . . . . . . . . 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 17 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) ) )
211206, 210mpbird 236 . . . . . . . 8  |-  ( ph  ->  X  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
212211snssd 4117 . . . . . . 7  |-  ( ph  ->  { X }  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
21344, 212unssd 3610 . . . . . 6  |-  ( ph  ->  ( B  u.  { X } )  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
2144, 6, 12, 13, 14, 195, 213rgspnmin 36037 . . . . 5  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
215214sseld 3431 . . . 4  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  ->  V  e.  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } ) )
216 fvex 5875 . . . . . . 7  |-  ( p `
 X )  e. 
_V
217 eleq1 2517 . . . . . . 7  |-  ( V  =  ( p `  X )  ->  ( V  e.  _V  <->  ( p `  X )  e.  _V ) )
218216, 217mpbiri 237 . . . . . 6  |-  ( V  =  ( p `  X )  ->  V  e.  _V )
219218rexlimivw 2876 . . . . 5  |-  ( E. p  e.  (Poly `  B ) V  =  ( p `  X
)  ->  V  e.  _V )
220 eqeq1 2455 . . . . . 6  |-  ( a  =  V  ->  (
a  =  ( p `
 X )  <->  V  =  ( p `  X
) ) )
221220rexbidv 2901 . . . . 5  |-  ( a  =  V  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X
) ) )
222219, 221elab3 3192 . . . 4  |-  ( V  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) )
223215, 222syl6ib 230 . . 3  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  ->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
2244, 6, 12, 13, 14rgspncl 36035 . . . . . . 7  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  e.  (SubRing ` fld ) )
225224adantr 467 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  e.  (SubRing ` fld ) )
226 simpr 463 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  p  e.  (Poly `  B ) )
2274, 6, 12, 13, 14rgspnssid 36036 . . . . . . . . 9  |-  ( ph  ->  ( B  u.  { X } )  C_  (
(RingSpan ` fld ) `  ( B  u.  { X }
) ) )
228227unssbd 3612 . . . . . . . 8  |-  ( ph  ->  { X }  C_  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
229 snidg 3994 . . . . . . . . 9  |-  ( X  e.  CC  ->  X  e.  { X } )
23010, 229syl 17 . . . . . . . 8  |-  ( ph  ->  X  e.  { X } )
231228, 230sseldd 3433 . . . . . . 7  |-  ( ph  ->  X  e.  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) )
232231adantr 467 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  X  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
233227unssad 3611 . . . . . . 7  |-  ( ph  ->  B  C_  ( (RingSpan ` fld ) `
 ( B  u.  { X } ) ) )
234233adantr 467 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  B  C_  (
(RingSpan ` fld ) `  ( B  u.  { X }
) ) )
235225, 226, 232, 234cnsrplycl 36033 . . . . 5  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( p `  X )  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
236 eleq1 2517 . . . . 5  |-  ( V  =  ( p `  X )  ->  ( V  e.  ( (RingSpan ` fld ) `
 ( B  u.  { X } ) )  <-> 
( p `  X
)  e.  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) ) )
237235, 236syl5ibrcom 226 . . . 4  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( V  =  ( p `  X
)  ->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
238237rexlimdva 2879 . . 3  |-  ( ph  ->  ( E. p  e.  (Poly `  B ) V  =  ( p `  X )  ->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
239223, 238impbid 194 . 2  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
2402, 239bitrd 257 1  |-  ( ph  ->  ( V  e.  S  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   {cab 2437   E.wrex 2738   _Vcvv 3045    u. cun 3402    C_ wss 3404   {csn 3968    _I cid 4744    X. cxp 4832    |` cres 4836    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290    oFcof 6529   CCcc 9537   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   -ucneg 9861   Basecbs 15121   ↾s cress 15122   +g cplusg 15190   .rcmulr 15191   0gc0g 15338   invgcminusg 16670  SubGrpcsubg 16811   1rcur 17735   Ringcrg 17780  SubRingcsubrg 18004  RingSpancrgspn 18005  ℂfldccnfld 18970  Polycply 23138   Xpcidp 23139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-subg 16814  df-cmn 17432  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-subrg 18006  df-rgspn 18007  df-cnfld 18971  df-0p 22628  df-ply 23142  df-idp 23143  df-coe 23144  df-dgr 23145
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator