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Theorem rngunsnply 26790
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 2350 . 2  |-  ( ph  ->  ( V  e.  S  <->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
3 cnrng 16396 . . . . . . 7  |-fld  e.  Ring
43a1i 10 . . . . . 6  |-  ( ph  ->fld  e. 
Ring )
5 cnfldbas 16383 . . . . . . 7  |-  CC  =  ( Base ` fld )
65a1i 10 . . . . . 6  |-  ( ph  ->  CC  =  ( Base ` fld ) )
7 rngunsnply.b . . . . . . . 8  |-  ( ph  ->  B  e.  (SubRing ` fld ) )
85subrgss 15546 . . . . . . . 8  |-  ( B  e.  (SubRing ` fld )  ->  B  C_  CC )
97, 8syl 15 . . . . . . 7  |-  ( ph  ->  B  C_  CC )
10 rngunsnply.x . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
1110snssd 3760 . . . . . . 7  |-  ( ph  ->  { X }  C_  CC )
129, 11unssd 3351 . . . . . 6  |-  ( ph  ->  ( B  u.  { X } )  C_  CC )
13 eqidd 2284 . . . . . 6  |-  ( ph  ->  (RingSpan ` fld )  =  (RingSpan ` fld ) )
14 eqidd 2284 . . . . . 6  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  =  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
15 eqidd 2284 . . . . . . 7  |-  ( ph  ->  (flds  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  =  (flds  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } ) )
16 cnfld0 16398 . . . . . . . 8  |-  0  =  ( 0g ` fld )
1716a1i 10 . . . . . . 7  |-  ( ph  ->  0  =  ( 0g
` fld
) )
18 cnfldadd 16384 . . . . . . . 8  |-  +  =  ( +g  ` fld )
1918a1i 10 . . . . . . 7  |-  ( ph  ->  +  =  ( +g  ` fld ) )
20 plyf 19580 . . . . . . . . . . . 12  |-  ( p  e.  (Poly `  B
)  ->  p : CC
--> CC )
21 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( p : CC --> CC  /\  X  e.  CC )  ->  ( p `  X
)  e.  CC )
2220, 10, 21syl2anr 464 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( p `  X )  e.  CC )
23 eleq1 2343 . . . . . . . . . . 11  |-  ( a  =  ( p `  X )  ->  (
a  e.  CC  <->  ( p `  X )  e.  CC ) )
2422, 23syl5ibrcom 213 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( a  =  ( p `  X
)  ->  a  e.  CC ) )
2524rexlimdva 2667 . . . . . . . . 9  |-  ( ph  ->  ( E. p  e.  (Poly `  B )
a  =  ( p `
 X )  -> 
a  e.  CC ) )
2625ss2abdv 3246 . . . . . . . 8  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  C_  { a  |  a  e.  CC } )
27 abid2 2400 . . . . . . . . 9  |-  { a  |  a  e.  CC }  =  CC
2827, 5eqtri 2303 . . . . . . . 8  |-  { a  |  a  e.  CC }  =  ( Base ` fld )
2926, 28syl6sseq 3224 . . . . . . 7  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  C_  ( Base ` fld ) )
30 abid2 2400 . . . . . . . . 9  |-  { a  |  a  e.  B }  =  B
31 plyconst 19588 . . . . . . . . . . . . 13  |-  ( ( B  C_  CC  /\  a  e.  B )  ->  ( CC  X.  { a } )  e.  (Poly `  B ) )
329, 31sylan 457 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  ( CC  X.  { a } )  e.  (Poly `  B ) )
3310adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  B )  ->  X  e.  CC )
34 vex 2791 . . . . . . . . . . . . . . 15  |-  a  e. 
_V
3534fvconst2 5729 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
( CC  X.  {
a } ) `  X )  =  a )
3633, 35syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  B )  ->  (
( CC  X.  {
a } ) `  X )  =  a )
3736eqcomd 2288 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  a  =  ( ( CC 
X.  { a } ) `  X ) )
38 fveq1 5524 . . . . . . . . . . . . . 14  |-  ( p  =  ( CC  X.  { a } )  ->  ( p `  X )  =  ( ( CC  X.  {
a } ) `  X ) )
3938eqeq2d 2294 . . . . . . . . . . . . 13  |-  ( p  =  ( CC  X.  { a } )  ->  ( a  =  ( p `  X
)  <->  a  =  ( ( CC  X.  {
a } ) `  X ) ) )
4039rspcev 2884 . . . . . . . . . . . 12  |-  ( ( ( CC  X.  {
a } )  e.  (Poly `  B )  /\  a  =  (
( CC  X.  {
a } ) `  X ) )  ->  E. p  e.  (Poly `  B ) a  =  ( p `  X
) )
4132, 37, 40syl2anc 642 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  B )  ->  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) )
4241ex 423 . . . . . . . . . 10  |-  ( ph  ->  ( a  e.  B  ->  E. p  e.  (Poly `  B ) a  =  ( p `  X
) ) )
4342ss2abdv 3246 . . . . . . . . 9  |-  ( ph  ->  { a  |  a  e.  B }  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
4430, 43syl5eqssr 3223 . . . . . . . 8  |-  ( ph  ->  B  C_  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) } )
45 subrgsubg 15551 . . . . . . . . . 10  |-  ( B  e.  (SubRing ` fld )  ->  B  e.  (SubGrp ` fld ) )
467, 45syl 15 . . . . . . . . 9  |-  ( ph  ->  B  e.  (SubGrp ` fld )
)
4716subg0cl 14629 . . . . . . . . 9  |-  ( B  e.  (SubGrp ` fld )  ->  0  e.  B )
4846, 47syl 15 . . . . . . . 8  |-  ( ph  ->  0  e.  B )
4944, 48sseldd 3181 . . . . . . 7  |-  ( ph  ->  0  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
50 biid 227 . . . . . . . . 9  |-  ( ph  <->  ph )
51 vex 2791 . . . . . . . . . 10  |-  b  e. 
_V
52 eqeq1 2289 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
a  =  ( p `
 X )  <->  b  =  ( p `  X
) ) )
5352rexbidv 2564 . . . . . . . . . . 11  |-  ( a  =  b  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) b  =  ( p `  X
) ) )
54 fveq1 5524 . . . . . . . . . . . . 13  |-  ( p  =  e  ->  (
p `  X )  =  ( e `  X ) )
5554eqeq2d 2294 . . . . . . . . . . . 12  |-  ( p  =  e  ->  (
b  =  ( p `
 X )  <->  b  =  ( e `  X
) ) )
5655cbvrexv 2765 . . . . . . . . . . 11  |-  ( E. p  e.  (Poly `  B ) b  =  ( p `  X
)  <->  E. e  e.  (Poly `  B ) b  =  ( e `  X
) )
5753, 56syl6bb 252 . . . . . . . . . 10  |-  ( a  =  b  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. e  e.  (Poly `  B ) b  =  ( e `  X
) ) )
5851, 57elab 2914 . . . . . . . . 9  |-  ( b  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. e  e.  (Poly `  B )
b  =  ( e `
 X ) )
59 vex 2791 . . . . . . . . . 10  |-  c  e. 
_V
60 eqeq1 2289 . . . . . . . . . . . 12  |-  ( a  =  c  ->  (
a  =  ( p `
 X )  <->  c  =  ( p `  X
) ) )
6160rexbidv 2564 . . . . . . . . . . 11  |-  ( a  =  c  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) c  =  ( p `  X
) ) )
62 fveq1 5524 . . . . . . . . . . . . 13  |-  ( p  =  d  ->  (
p `  X )  =  ( d `  X ) )
6362eqeq2d 2294 . . . . . . . . . . . 12  |-  ( p  =  d  ->  (
c  =  ( p `
 X )  <->  c  =  ( d `  X
) ) )
6463cbvrexv 2765 . . . . . . . . . . 11  |-  ( E. p  e.  (Poly `  B ) c  =  ( p `  X
)  <->  E. d  e.  (Poly `  B ) c  =  ( d `  X
) )
6561, 64syl6bb 252 . . . . . . . . . 10  |-  ( a  =  c  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. d  e.  (Poly `  B ) c  =  ( d `  X
) ) )
6659, 65elab 2914 . . . . . . . . 9  |-  ( c  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. d  e.  (Poly `  B )
c  =  ( d `
 X ) )
67 simplr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  e  e.  (Poly `  B ) )
68 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  d  e.  (Poly `  B ) )
6918subrgacl 15556 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  e.  (SubRing ` fld )  /\  a  e.  B  /\  b  e.  B )  ->  (
a  +  b )  e.  B )
70693expb 1152 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  e.  (SubRing ` fld )  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  +  b )  e.  B )
717, 70sylan 457 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7271adantlr 695 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7372adantlr 695 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  e  e.  (Poly `  B
) )  /\  d  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7467, 68, 73plyadd 19599 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( e  o F  +  d )  e.  (Poly `  B
) )
75 plyf 19580 . . . . . . . . . . . . . . . . . . 19  |-  ( e  e.  (Poly `  B
)  ->  e : CC
--> CC )
76 ffn 5389 . . . . . . . . . . . . . . . . . . 19  |-  ( e : CC --> CC  ->  e  Fn  CC )
7775, 76syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( e  e.  (Poly `  B
)  ->  e  Fn  CC )
7877ad2antlr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  e  Fn  CC )
79 plyf 19580 . . . . . . . . . . . . . . . . . . 19  |-  ( d  e.  (Poly `  B
)  ->  d : CC
--> CC )
80 ffn 5389 . . . . . . . . . . . . . . . . . . 19  |-  ( d : CC --> CC  ->  d  Fn  CC )
8179, 80syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( d  e.  (Poly `  B
)  ->  d  Fn  CC )
8281adantl 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  d  Fn  CC )
83 cnex 8818 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  _V
8483a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  CC  e.  _V )
8510ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  X  e.  CC )
86 fnfvof 6090 . . . . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e  o F  +  d ) `  X )  =  ( ( e `
 X )  +  ( d `  X
) ) )
8887eqcomd 2288 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e `
 X )  +  ( d `  X
) )  =  ( ( e  o F  +  d ) `  X ) )
89 fveq1 5524 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( e  o F  +  d )  ->  ( p `  X )  =  ( ( e  o F  +  d ) `  X ) )
9089eqeq2d 2294 . . . . . . . . . . . . . . . 16  |-  ( p  =  ( e  o F  +  d )  ->  ( ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
)  <->  ( ( e `
 X )  +  ( d `  X
) )  =  ( ( e  o F  +  d ) `  X ) ) )
9190rspcev 2884 . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
) )
93 oveq2 5866 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( d `  X )  ->  (
( e `  X
)  +  c )  =  ( ( e `
 X )  +  ( d `  X
) ) )
9493eqeq1d 2291 . . . . . . . . . . . . . . 15  |-  ( c  =  ( d `  X )  ->  (
( ( e `  X )  +  c )  =  ( p `
 X )  <->  ( (
e `  X )  +  ( d `  X ) )  =  ( p `  X
) ) )
9594rexbidv 2564 . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( c  =  ( d `  X
)  ->  E. p  e.  (Poly `  B )
( ( e `  X )  +  c )  =  ( p `
 X ) ) )
9796rexlimdva 2667 . . . . . . . . . . . 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 5865 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e `  X )  ->  (
b  +  c )  =  ( ( e `
 X )  +  c ) )
9998eqeq1d 2291 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( b  +  c )  =  ( p `
 X )  <->  ( (
e `  X )  +  c )  =  ( p `  X
) ) )
10099rexbidv 2564 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  ( E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  c )  =  ( p `  X
) ) )
101100imbi2d 307 . . . . . . . . . . . 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 213 . . . . . . . . . . 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 2667 . . . . . . . . . 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 1145 . . . . . . . . 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 1225 . . . . . . . 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 5883 . . . . . . . . 9  |-  ( b  +  c )  e. 
_V
107 eqeq1 2289 . . . . . . . . . 10  |-  ( a  =  ( b  +  c )  ->  (
a  =  ( p `
 X )  <->  ( b  +  c )  =  ( p `  X
) ) )
108107rexbidv 2564 . . . . . . . . 9  |-  ( a  =  ( b  +  c )  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
) ) )
109106, 108elab 2914 . . . . . . . 8  |-  ( ( b  +  c )  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B )
( b  +  c )  =  ( p `
 X ) )
110105, 109sylibr 203 . . . . . . 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 8795 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
112 cnfldneg 16400 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  CC  ->  (
( inv g ` fld ) `  1 )  = 
-u 1 )
113111, 112mp1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( inv g ` fld ) `  1 )  =  -u 1 )
114 cnfld1 16399 . . . . . . . . . . . . . . . . . . . 20  |-  1  =  ( 1r ` fld )
115114subrg1cl 15553 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  (SubRing ` fld )  ->  1  e.  B )
1167, 115syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  1  e.  B )
117 eqid 2283 . . . . . . . . . . . . . . . . . . 19  |-  ( inv g ` fld )  =  ( inv g ` fld )
118117subginvcl 14630 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  e.  (SubGrp ` fld )  /\  1  e.  B
)  ->  ( ( inv g ` fld ) `  1 )  e.  B )
11946, 116, 118syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( inv g ` fld ) `  1 )  e.  B )
120113, 119eqeltrrd 2358 . . . . . . . . . . . . . . . 16  |-  ( ph  -> 
-u 1  e.  B
)
121 plyconst 19588 . . . . . . . . . . . . . . . 16  |-  ( ( B  C_  CC  /\  -u 1  e.  B )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B ) )
1229, 120, 121syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B )
)
123122adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B
) )
124 simpr 447 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  e  e.  (Poly `  B ) )
125 cnfldmul 16385 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
126125subrgmcl 15557 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  (SubRing ` fld )  /\  a  e.  B  /\  b  e.  B )  ->  (
a  x.  b )  e.  B )
1271263expb 1152 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  (SubRing ` fld )  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  x.  b )  e.  B
)
1287, 127sylan 457 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
129128adantlr 695 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
130123, 124, 72, 129plymul 19600 . . . . . . . . . . . . 13  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( CC 
X.  { -u 1 } )  o F  x.  e )  e.  (Poly `  B )
)
131 ffvelrn 5663 . . . . . . . . . . . . . . . 16  |-  ( ( e : CC --> CC  /\  X  e.  CC )  ->  ( e `  X
)  e.  CC )
13275, 10, 131syl2anr 464 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( e `  X )  e.  CC )
133 cnfldneg 16400 . . . . . . . . . . . . . . 15  |-  ( ( e `  X )  e.  CC  ->  (
( inv g ` fld ) `  ( e `  X
) )  =  -u ( e `  X
) )
134132, 133syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( inv g ` fld ) `  ( e `
 X ) )  =  -u ( e `  X ) )
135 negex 9050 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  _V
136 fnconstg 5429 . . . . . . . . . . . . . . . . 17  |-  ( -u
1  e.  _V  ->  ( CC  X.  { -u
1 } )  Fn  CC )
137135, 136mp1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( CC  X.  { -u 1 } )  Fn  CC )
13877adantl 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  e  Fn  CC )
13983a1i 10 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  CC  e.  _V )
14010adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  X  e.  CC )
141 fnfvof 6090 . . . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } )  o F  x.  e ) `
 X )  =  ( ( ( CC 
X.  { -u 1 } ) `  X
)  x.  ( e `
 X ) ) )
143135fvconst2 5729 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  CC  ->  (
( CC  X.  { -u 1 } ) `  X )  =  -u
1 )
144140, 143syl 15 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( CC 
X.  { -u 1 } ) `  X
)  =  -u 1
)
145144oveq1d 5873 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } ) `  X )  x.  (
e `  X )
)  =  ( -u
1  x.  ( e `
 X ) ) )
146132mulm1d 9231 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( -u 1  x.  ( e `  X
) )  =  -u ( e `  X
) )
147142, 145, 1463eqtrd 2319 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } )  o F  x.  e ) `
 X )  = 
-u ( e `  X ) )
148134, 147eqtr4d 2318 . . . . . . . . . . . . 13  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( inv g ` fld ) `  ( e `
 X ) )  =  ( ( ( CC  X.  { -u
1 } )  o F  x.  e ) `
 X ) )
149 fveq1 5524 . . . . . . . . . . . . . . 15  |-  ( p  =  ( ( CC 
X.  { -u 1 } )  o F  x.  e )  -> 
( p `  X
)  =  ( ( ( CC  X.  { -u 1 } )  o F  x.  e ) `
 X ) )
150149eqeq2d 2294 . . . . . . . . . . . . . 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 2884 . . . . . . . . . . . . 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 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( inv g ` fld ) `  ( e `
 X ) )  =  ( p `  X ) )
153 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( inv g ` fld ) `  b )  =  ( ( inv g ` fld ) `  ( e `  X
) ) )
154153eqeq1d 2291 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  (
( ( inv g ` fld ) `  b )  =  ( p `  X )  <->  ( ( inv g ` fld ) `  ( e `
 X ) )  =  ( p `  X ) ) )
155154rexbidv 2564 . . . . . . . . . . . 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 213 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( b  =  ( e `  X
)  ->  E. p  e.  (Poly `  B )
( ( inv g ` fld ) `  b )  =  ( p `  X ) ) )
157156rexlimdva 2667 . . . . . . . . . 10  |-  ( ph  ->  ( E. e  e.  (Poly `  B )
b  =  ( e `
 X )  ->  E. p  e.  (Poly `  B ) ( ( inv g ` fld ) `  b )  =  ( p `  X ) ) )
158157imp 418 . . . . . . . . 9  |-  ( (
ph  /\  E. e  e.  (Poly `  B )
b  =  ( e `
 X ) )  ->  E. p  e.  (Poly `  B ) ( ( inv g ` fld ) `  b )  =  ( p `  X ) )
15958, 158sylan2b 461 . . . . . . . 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 5539 . . . . . . . . 9  |-  ( ( inv g ` fld ) `  b )  e.  _V
161 eqeq1 2289 . . . . . . . . . 10  |-  ( a  =  ( ( inv g ` fld ) `  b )  ->  ( a  =  ( p `  X
)  <->  ( ( inv g ` fld ) `  b )  =  ( p `  X ) ) )
162161rexbidv 2564 . . . . . . . . 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 2914 . . . . . . . 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 203 . . . . . . 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 10 . . . . . . 7  |-  ( ph  ->  1  =  ( 1r
` fld
) )
166125a1i 10 . . . . . . 7  |-  ( ph  ->  x.  =  ( .r
` fld
) )
16744, 116sseldd 3181 . . . . . . 7  |-  ( ph  ->  1  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
168129adantlr 695 . . . . . . . . . . . . . . . 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 19600 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( e  o F  x.  d )  e.  (Poly `  B
) )
170 fnfvof 6090 . . . . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e  o F  x.  d
) `  X )  =  ( ( e `
 X )  x.  ( d `  X
) ) )
172171eqcomd 2288 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e `
 X )  x.  ( d `  X
) )  =  ( ( e  o F  x.  d ) `  X ) )
173 fveq1 5524 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( e  o F  x.  d )  ->  ( p `  X )  =  ( ( e  o F  x.  d ) `  X ) )
174173eqeq2d 2294 . . . . . . . . . . . . . . . 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 2884 . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  x.  ( d `  X ) )  =  ( p `  X
) )
177 oveq2 5866 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( d `  X )  ->  (
( e `  X
)  x.  c )  =  ( ( e `
 X )  x.  ( d `  X
) ) )
178177eqeq1d 2291 . . . . . . . . . . . . . . 15  |-  ( c  =  ( d `  X )  ->  (
( ( e `  X )  x.  c
)  =  ( p `
 X )  <->  ( (
e `  X )  x.  ( d `  X
) )  =  ( p `  X ) ) )
179178rexbidv 2564 . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . 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 2667 . . . . . . . . . . . 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 5865 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e `  X )  ->  (
b  x.  c )  =  ( ( e `
 X )  x.  c ) )
183182eqeq1d 2291 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( b  x.  c
)  =  ( p `
 X )  <->  ( (
e `  X )  x.  c )  =  ( p `  X ) ) )
184183rexbidv 2564 . . . . . . . . . . . . 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 307 . . . . . . . . . . . 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 213 . . . . . . . . . . 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 2667 . . . . . . . . . 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 1145 . . . . . . . . 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 1225 . . . . . . . 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 5883 . . . . . . . . 9  |-  ( b  x.  c )  e. 
_V
191 eqeq1 2289 . . . . . . . . . 10  |-  ( a  =  ( b  x.  c )  ->  (
a  =  ( p `
 X )  <->  ( b  x.  c )  =  ( p `  X ) ) )
192191rexbidv 2564 . . . . . . . . 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 2914 . . . . . . . 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 203 . . . . . . 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 15943 . . . . . 6  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  e.  (SubRing ` fld ) )
196 plyid 19591 . . . . . . . . . . 11  |-  ( ( B  C_  CC  /\  1  e.  B )  ->  X p  e.  (Poly `  B
) )
1979, 116, 196syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  X p  e.  (Poly `  B ) )
198 df-idp 19571 . . . . . . . . . . . 12  |-  X p  =  (  _I  |`  CC )
199198fveq1i 5526 . . . . . . . . . . 11  |-  ( X p `  X )  =  ( (  _I  |`  CC ) `  X
)
200 fvresi 5711 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
(  _I  |`  CC ) `
 X )  =  X )
20110, 200syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( (  _I  |`  CC ) `
 X )  =  X )
202199, 201syl5req 2328 . . . . . . . . . 10  |-  ( ph  ->  X  =  ( X p `  X ) )
203 fveq1 5524 . . . . . . . . . . . 12  |-  ( p  =  X p  -> 
( p `  X
)  =  ( X p `  X ) )
204203eqeq2d 2294 . . . . . . . . . . 11  |-  ( p  =  X p  -> 
( X  =  ( p `  X )  <-> 
X  =  ( X p `  X ) ) )
205204rspcev 2884 . . . . . . . . . 10  |-  ( ( X p  e.  (Poly `  B )  /\  X  =  ( X p `
 X ) )  ->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) )
206197, 202, 205syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) )
207 eqeq1 2289 . . . . . . . . . . . 12  |-  ( a  =  X  ->  (
a  =  ( p `
 X )  <->  X  =  ( p `  X
) ) )
208207rexbidv 2564 . . . . . . . . . . 11  |-  ( a  =  X  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) ) )
209208elabg 2915 . . . . . . . . . 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 15 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) ) )
211206, 210mpbird 223 . . . . . . . 8  |-  ( ph  ->  X  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
212211snssd 3760 . . . . . . 7  |-  ( ph  ->  { X }  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
21344, 212unssd 3351 . . . . . 6  |-  ( ph  ->  ( B  u.  { X } )  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
2144, 6, 12, 13, 14, 195, 213rgspnmin 26788 . . . . 5  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
215214sseld 3179 . . . 4  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  ->  V  e.  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } ) )
216 fvex 5539 . . . . . . 7  |-  ( p `
 X )  e. 
_V
217 eleq1 2343 . . . . . . 7  |-  ( V  =  ( p `  X )  ->  ( V  e.  _V  <->  ( p `  X )  e.  _V ) )
218216, 217mpbiri 224 . . . . . 6  |-  ( V  =  ( p `  X )  ->  V  e.  _V )
219218rexlimivw 2663 . . . . 5  |-  ( E. p  e.  (Poly `  B ) V  =  ( p `  X
)  ->  V  e.  _V )
220 eqeq1 2289 . . . . . 6  |-  ( a  =  V  ->  (
a  =  ( p `
 X )  <->  V  =  ( p `  X
) ) )
221220rexbidv 2564 . . . . 5  |-  ( a  =  V  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X
) ) )
222219, 221elab3 2921 . . . 4  |-  ( V  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) )
223215, 222syl6ib 217 . . 3  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  ->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
2244, 6, 12, 13, 14rgspncl 26786 . . . . . . 7  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  e.  (SubRing ` fld ) )
225224adantr 451 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  e.  (SubRing ` fld ) )
226 simpr 447 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  p  e.  (Poly `  B ) )
227 ssun2 3339 . . . . . . . . 9  |-  { X }  C_  ( B  u.  { X } )
2284, 6, 12, 13, 14rgspnssid 26787 . . . . . . . . 9  |-  ( ph  ->  ( B  u.  { X } )  C_  (
(RingSpan ` fld ) `  ( B  u.  { X }
) ) )
229227, 228syl5ss 3190 . . . . . . . 8  |-  ( ph  ->  { X }  C_  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
230 snidg 3665 . . . . . . . . 9  |-  ( X  e.  CC  ->  X  e.  { X } )
23110, 230syl 15 . . . . . . . 8  |-  ( ph  ->  X  e.  { X } )
232229, 231sseldd 3181 . . . . . . 7  |-  ( ph  ->  X  e.  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) )
233232adantr 451 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  X  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
234 ssun1 3338 . . . . . . . 8  |-  B  C_  ( B  u.  { X } )
235234, 228syl5ss 3190 . . . . . . 7  |-  ( ph  ->  B  C_  ( (RingSpan ` fld ) `
 ( B  u.  { X } ) ) )
236235adantr 451 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  B  C_  (
(RingSpan ` fld ) `  ( B  u.  { X }
) ) )
237225, 226, 233, 236cnsrplycl 26784 . . . . 5  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( p `  X )  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
238 eleq1 2343 . . . . 5  |-  ( V  =  ( p `  X )  ->  ( V  e.  ( (RingSpan ` fld ) `
 ( B  u.  { X } ) )  <-> 
( p `  X
)  e.  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) ) )
239237, 238syl5ibrcom 213 . . . 4  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( V  =  ( p `  X
)  ->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
240239rexlimdva 2667 . . 3  |-  ( ph  ->  ( E. p  e.  (Poly `  B ) V  =  ( p `  X )  ->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
241223, 240impbid 183 . 2  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
2422, 241bitrd 244 1  |-  ( ph  ->  ( V  e.  S  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788    u. cun 3150    C_ wss 3152   {csn 3640    _I cid 4304    X. cxp 4687    |` cres 4691    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   .rcmulr 13209   0gc0g 13400   inv gcminusg 14363  SubGrpcsubg 14615   Ringcrg 15337   1rcur 15339  SubRingcsubrg 15541  RingSpancrgspn 15542  ℂfldccnfld 16377  Polycply 19566   X pcidp 19567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-subrg 15543  df-rgspn 15544  df-cnfld 16378  df-0p 19025  df-ply 19570  df-idp 19571  df-coe 19572  df-dgr 19573
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