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Theorem rngunsnply 31098
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 2513 . 2  |-  ( ph  ->  ( V  e.  S  <->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
3 cnring 18419 . . . . . . 7  |-fld  e.  Ring
43a1i 11 . . . . . 6  |-  ( ph  ->fld  e. 
Ring )
5 cnfldbas 18403 . . . . . . 7  |-  CC  =  ( Base ` fld )
65a1i 11 . . . . . 6  |-  ( ph  ->  CC  =  ( Base ` fld ) )
7 rngunsnply.b . . . . . . . 8  |-  ( ph  ->  B  e.  (SubRing ` fld ) )
85subrgss 17409 . . . . . . . 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 4160 . . . . . . 7  |-  ( ph  ->  { X }  C_  CC )
129, 11unssd 3665 . . . . . 6  |-  ( ph  ->  ( B  u.  { X } )  C_  CC )
13 eqidd 2444 . . . . . 6  |-  ( ph  ->  (RingSpan ` fld )  =  (RingSpan ` fld ) )
14 eqidd 2444 . . . . . 6  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  =  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
15 eqidd 2444 . . . . . . 7  |-  ( ph  ->  (flds  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  =  (flds  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } ) )
16 cnfld0 18421 . . . . . . . 8  |-  0  =  ( 0g ` fld )
1716a1i 11 . . . . . . 7  |-  ( ph  ->  0  =  ( 0g
` fld
) )
18 cnfldadd 18404 . . . . . . . 8  |-  +  =  ( +g  ` fld )
1918a1i 11 . . . . . . 7  |-  ( ph  ->  +  =  ( +g  ` fld ) )
20 plyf 22573 . . . . . . . . . . . 12  |-  ( p  e.  (Poly `  B
)  ->  p : CC
--> CC )
21 ffvelrn 6014 . . . . . . . . . . . 12  |-  ( ( p : CC --> CC  /\  X  e.  CC )  ->  ( p `  X
)  e.  CC )
2220, 10, 21syl2anr 478 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( p `  X )  e.  CC )
23 eleq1 2515 . . . . . . . . . . 11  |-  ( a  =  ( p `  X )  ->  (
a  e.  CC  <->  ( p `  X )  e.  CC ) )
2422, 23syl5ibrcom 222 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( a  =  ( p `  X
)  ->  a  e.  CC ) )
2524rexlimdva 2935 . . . . . . . . 9  |-  ( ph  ->  ( E. p  e.  (Poly `  B )
a  =  ( p `
 X )  -> 
a  e.  CC ) )
2625ss2abdv 3558 . . . . . . . 8  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  C_  { a  |  a  e.  CC } )
27 abid2 2583 . . . . . . . . 9  |-  { a  |  a  e.  CC }  =  CC
2827, 5eqtri 2472 . . . . . . . 8  |-  { a  |  a  e.  CC }  =  ( Base ` fld )
2926, 28syl6sseq 3535 . . . . . . 7  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  C_  ( Base ` fld ) )
30 abid2 2583 . . . . . . . . 9  |-  { a  |  a  e.  B }  =  B
31 plyconst 22581 . . . . . . . . . . . . 13  |-  ( ( B  C_  CC  /\  a  e.  B )  ->  ( CC  X.  { a } )  e.  (Poly `  B ) )
329, 31sylan 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  ( CC  X.  { a } )  e.  (Poly `  B ) )
3310adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  B )  ->  X  e.  CC )
34 vex 3098 . . . . . . . . . . . . . . 15  |-  a  e. 
_V
3534fvconst2 6111 . . . . . . . . . . . . . 14  |-  ( X  e.  CC  ->  (
( CC  X.  {
a } ) `  X )  =  a )
3633, 35syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  B )  ->  (
( CC  X.  {
a } ) `  X )  =  a )
3736eqcomd 2451 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  B )  ->  a  =  ( ( CC 
X.  { a } ) `  X ) )
38 fveq1 5855 . . . . . . . . . . . . . 14  |-  ( p  =  ( CC  X.  { a } )  ->  ( p `  X )  =  ( ( CC  X.  {
a } ) `  X ) )
3938eqeq2d 2457 . . . . . . . . . . . . 13  |-  ( p  =  ( CC  X.  { a } )  ->  ( a  =  ( p `  X
)  <->  a  =  ( ( CC  X.  {
a } ) `  X ) ) )
4039rspcev 3196 . . . . . . . . . . . 12  |-  ( ( ( CC  X.  {
a } )  e.  (Poly `  B )  /\  a  =  (
( CC  X.  {
a } ) `  X ) )  ->  E. p  e.  (Poly `  B ) a  =  ( p `  X
) )
4132, 37, 40syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  B )  ->  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) )
4241ex 434 . . . . . . . . . 10  |-  ( ph  ->  ( a  e.  B  ->  E. p  e.  (Poly `  B ) a  =  ( p `  X
) ) )
4342ss2abdv 3558 . . . . . . . . 9  |-  ( ph  ->  { a  |  a  e.  B }  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
4430, 43syl5eqssr 3534 . . . . . . . 8  |-  ( ph  ->  B  C_  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) } )
45 subrgsubg 17414 . . . . . . . . . 10  |-  ( B  e.  (SubRing ` fld )  ->  B  e.  (SubGrp ` fld ) )
467, 45syl 16 . . . . . . . . 9  |-  ( ph  ->  B  e.  (SubGrp ` fld )
)
4716subg0cl 16188 . . . . . . . . 9  |-  ( B  e.  (SubGrp ` fld )  ->  0  e.  B )
4846, 47syl 16 . . . . . . . 8  |-  ( ph  ->  0  e.  B )
4944, 48sseldd 3490 . . . . . . 7  |-  ( ph  ->  0  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
50 biid 236 . . . . . . . . 9  |-  ( ph  <->  ph )
51 vex 3098 . . . . . . . . . 10  |-  b  e. 
_V
52 eqeq1 2447 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
a  =  ( p `
 X )  <->  b  =  ( p `  X
) ) )
5352rexbidv 2954 . . . . . . . . . . 11  |-  ( a  =  b  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) b  =  ( p `  X
) ) )
54 fveq1 5855 . . . . . . . . . . . . 13  |-  ( p  =  e  ->  (
p `  X )  =  ( e `  X ) )
5554eqeq2d 2457 . . . . . . . . . . . 12  |-  ( p  =  e  ->  (
b  =  ( p `
 X )  <->  b  =  ( e `  X
) ) )
5655cbvrexv 3071 . . . . . . . . . . 11  |-  ( E. p  e.  (Poly `  B ) b  =  ( p `  X
)  <->  E. e  e.  (Poly `  B ) b  =  ( e `  X
) )
5753, 56syl6bb 261 . . . . . . . . . 10  |-  ( a  =  b  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. e  e.  (Poly `  B ) b  =  ( e `  X
) ) )
5851, 57elab 3232 . . . . . . . . 9  |-  ( b  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. e  e.  (Poly `  B )
b  =  ( e `
 X ) )
59 vex 3098 . . . . . . . . . 10  |-  c  e. 
_V
60 eqeq1 2447 . . . . . . . . . . . 12  |-  ( a  =  c  ->  (
a  =  ( p `
 X )  <->  c  =  ( p `  X
) ) )
6160rexbidv 2954 . . . . . . . . . . 11  |-  ( a  =  c  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) c  =  ( p `  X
) ) )
62 fveq1 5855 . . . . . . . . . . . . 13  |-  ( p  =  d  ->  (
p `  X )  =  ( d `  X ) )
6362eqeq2d 2457 . . . . . . . . . . . 12  |-  ( p  =  d  ->  (
c  =  ( p `
 X )  <->  c  =  ( d `  X
) ) )
6463cbvrexv 3071 . . . . . . . . . . 11  |-  ( E. p  e.  (Poly `  B ) c  =  ( p `  X
)  <->  E. d  e.  (Poly `  B ) c  =  ( d `  X
) )
6561, 64syl6bb 261 . . . . . . . . . 10  |-  ( a  =  c  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. d  e.  (Poly `  B ) c  =  ( d `  X
) ) )
6659, 65elab 3232 . . . . . . . . 9  |-  ( c  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. d  e.  (Poly `  B )
c  =  ( d `
 X ) )
67 simplr 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  e  e.  (Poly `  B ) )
68 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  d  e.  (Poly `  B ) )
6918subrgacl 17419 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( B  e.  (SubRing ` fld )  /\  a  e.  B  /\  b  e.  B )  ->  (
a  +  b )  e.  B )
70693expb 1198 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  e.  (SubRing ` fld )  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  +  b )  e.  B )
717, 70sylan 471 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7271adantlr 714 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7372adantlr 714 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  e  e.  (Poly `  B
) )  /\  d  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  +  b )  e.  B )
7467, 68, 73plyadd 22592 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( e  oF  +  d )  e.  (Poly `  B
) )
75 plyf 22573 . . . . . . . . . . . . . . . . . . 19  |-  ( e  e.  (Poly `  B
)  ->  e : CC
--> CC )
76 ffn 5721 . . . . . . . . . . . . . . . . . . 19  |-  ( e : CC --> CC  ->  e  Fn  CC )
7775, 76syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( e  e.  (Poly `  B
)  ->  e  Fn  CC )
7877ad2antlr 726 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  e  Fn  CC )
79 plyf 22573 . . . . . . . . . . . . . . . . . . 19  |-  ( d  e.  (Poly `  B
)  ->  d : CC
--> CC )
80 ffn 5721 . . . . . . . . . . . . . . . . . . 19  |-  ( d : CC --> CC  ->  d  Fn  CC )
8179, 80syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( d  e.  (Poly `  B
)  ->  d  Fn  CC )
8281adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  d  Fn  CC )
83 cnex 9576 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  _V
8483a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  CC  e.  _V )
8510ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  X  e.  CC )
86 fnfvof 6538 . . . . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e  oF  +  d ) `  X )  =  ( ( e `
 X )  +  ( d `  X
) ) )
8887eqcomd 2451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e `
 X )  +  ( d `  X
) )  =  ( ( e  oF  +  d ) `  X ) )
89 fveq1 5855 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( e  oF  +  d )  ->  ( p `  X )  =  ( ( e  oF  +  d ) `  X ) )
9089eqeq2d 2457 . . . . . . . . . . . . . . . 16  |-  ( p  =  ( e  oF  +  d )  ->  ( ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
)  <->  ( ( e `
 X )  +  ( d `  X
) )  =  ( ( e  oF  +  d ) `  X ) ) )
9190rspcev 3196 . . . . . . . . . . . . . . 15  |-  ( ( ( e  oF  +  d )  e.  (Poly `  B )  /\  ( ( e `  X )  +  ( d `  X ) )  =  ( ( e  oF  +  d ) `  X
) )  ->  E. p  e.  (Poly `  B )
( ( e `  X )  +  ( d `  X ) )  =  ( p `
 X ) )
9274, 88, 91syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
) )
93 oveq2 6289 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( d `  X )  ->  (
( e `  X
)  +  c )  =  ( ( e `
 X )  +  ( d `  X
) ) )
9493eqeq1d 2445 . . . . . . . . . . . . . . 15  |-  ( c  =  ( d `  X )  ->  (
( ( e `  X )  +  c )  =  ( p `
 X )  <->  ( (
e `  X )  +  ( d `  X ) )  =  ( p `  X
) ) )
9594rexbidv 2954 . . . . . . . . . . . . . 14  |-  ( c  =  ( d `  X )  ->  ( E. p  e.  (Poly `  B ) ( ( e `  X )  +  c )  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  ( d `  X ) )  =  ( p `  X
) ) )
9692, 95syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( c  =  ( d `  X
)  ->  E. p  e.  (Poly `  B )
( ( e `  X )  +  c )  =  ( p `
 X ) ) )
9796rexlimdva 2935 . . . . . . . . . . . 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 6288 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e `  X )  ->  (
b  +  c )  =  ( ( e `
 X )  +  c ) )
9998eqeq1d 2445 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( b  +  c )  =  ( p `
 X )  <->  ( (
e `  X )  +  c )  =  ( p `  X
) ) )
10099rexbidv 2954 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  ( E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  c )  =  ( p `  X
) ) )
101100imbi2d 316 . . . . . . . . . . . 12  |-  ( b  =  ( e `  X )  ->  (
( E. d  e.  (Poly `  B )
c  =  ( d `
 X )  ->  E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
) )  <->  ( E. d  e.  (Poly `  B
) c  =  ( d `  X )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  +  c )  =  ( p `  X
) ) ) )
10297, 101syl5ibrcom 222 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( b  =  ( e `  X
)  ->  ( E. d  e.  (Poly `  B
) c  =  ( d `  X )  ->  E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
) ) ) )
103102rexlimdva 2935 . . . . . . . . . 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 1191 . . . . . . . . 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 1272 . . . . . . . 8  |-  ( (
ph  /\  b  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  /\  c  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  ->  E. p  e.  (Poly `  B )
( b  +  c )  =  ( p `
 X ) )
106 ovex 6309 . . . . . . . . 9  |-  ( b  +  c )  e. 
_V
107 eqeq1 2447 . . . . . . . . . 10  |-  ( a  =  ( b  +  c )  ->  (
a  =  ( p `
 X )  <->  ( b  +  c )  =  ( p `  X
) ) )
108107rexbidv 2954 . . . . . . . . 9  |-  ( a  =  ( b  +  c )  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( b  +  c )  =  ( p `  X
) ) )
109106, 108elab 3232 . . . . . . . 8  |-  ( ( b  +  c )  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B )
( b  +  c )  =  ( p `
 X ) )
110105, 109sylibr 212 . . . . . . 7  |-  ( (
ph  /\  b  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  /\  c  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  ->  ( b  +  c )  e. 
{ a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } )
111 ax-1cn 9553 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
112 cnfldneg 18423 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  CC  ->  (
( invg ` fld ) `  1 )  = 
-u 1 )
113111, 112mp1i 12 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( invg ` fld ) `  1 )  =  -u 1 )
114 cnfld1 18422 . . . . . . . . . . . . . . . . . . . 20  |-  1  =  ( 1r ` fld )
115114subrg1cl 17416 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  (SubRing ` fld )  ->  1  e.  B )
1167, 115syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  1  e.  B )
117 eqid 2443 . . . . . . . . . . . . . . . . . . 19  |-  ( invg ` fld )  =  ( invg ` fld )
118117subginvcl 16189 . . . . . . . . . . . . . . . . . 18  |-  ( ( B  e.  (SubGrp ` fld )  /\  1  e.  B
)  ->  ( ( invg ` fld ) `  1 )  e.  B )
11946, 116, 118syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( invg ` fld ) `  1 )  e.  B )
120113, 119eqeltrrd 2532 . . . . . . . . . . . . . . . 16  |-  ( ph  -> 
-u 1  e.  B
)
121 plyconst 22581 . . . . . . . . . . . . . . . 16  |-  ( ( B  C_  CC  /\  -u 1  e.  B )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B ) )
1229, 120, 121syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B )
)
123122adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  B
) )
124 simpr 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  e  e.  (Poly `  B ) )
125 cnfldmul 18405 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
126125subrgmcl 17420 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  (SubRing ` fld )  /\  a  e.  B  /\  b  e.  B )  ->  (
a  x.  b )  e.  B )
1271263expb 1198 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  (SubRing ` fld )  /\  (
a  e.  B  /\  b  e.  B )
)  ->  ( a  x.  b )  e.  B
)
1287, 127sylan 471 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
129128adantlr 714 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
130123, 124, 72, 129plymul 22593 . . . . . . . . . . . . 13  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( CC 
X.  { -u 1 } )  oF  x.  e )  e.  (Poly `  B )
)
131 ffvelrn 6014 . . . . . . . . . . . . . . . 16  |-  ( ( e : CC --> CC  /\  X  e.  CC )  ->  ( e `  X
)  e.  CC )
13275, 10, 131syl2anr 478 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( e `  X )  e.  CC )
133 cnfldneg 18423 . . . . . . . . . . . . . . 15  |-  ( ( e `  X )  e.  CC  ->  (
( invg ` fld ) `  ( e `  X
) )  =  -u ( e `  X
) )
134132, 133syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( invg ` fld ) `  ( e `
 X ) )  =  -u ( e `  X ) )
135 negex 9823 . . . . . . . . . . . . . . . . 17  |-  -u 1  e.  _V
136 fnconstg 5763 . . . . . . . . . . . . . . . . 17  |-  ( -u
1  e.  _V  ->  ( CC  X.  { -u
1 } )  Fn  CC )
137135, 136mp1i 12 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( CC  X.  { -u 1 } )  Fn  CC )
13877adantl 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  e  Fn  CC )
13983a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  CC  e.  _V )
14010adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  X  e.  CC )
141 fnfvof 6538 . . . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } )  oF  x.  e ) `
 X )  =  ( ( ( CC 
X.  { -u 1 } ) `  X
)  x.  ( e `
 X ) ) )
143135fvconst2 6111 . . . . . . . . . . . . . . . . 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 6296 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } ) `  X )  x.  (
e `  X )
)  =  ( -u
1  x.  ( e `
 X ) ) )
146132mulm1d 10015 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( -u 1  x.  ( e `  X
) )  =  -u ( e `  X
) )
147142, 145, 1463eqtrd 2488 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( ( CC  X.  { -u
1 } )  oF  x.  e ) `
 X )  = 
-u ( e `  X ) )
148134, 147eqtr4d 2487 . . . . . . . . . . . . 13  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( ( invg ` fld ) `  ( e `
 X ) )  =  ( ( ( CC  X.  { -u
1 } )  oF  x.  e ) `
 X ) )
149 fveq1 5855 . . . . . . . . . . . . . . 15  |-  ( p  =  ( ( CC 
X.  { -u 1 } )  oF  x.  e )  -> 
( p `  X
)  =  ( ( ( CC  X.  { -u 1 } )  oF  x.  e ) `
 X ) )
150149eqeq2d 2457 . . . . . . . . . . . . . 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 3196 . . . . . . . . . . . . 13  |-  ( ( ( ( CC  X.  { -u 1 } )  oF  x.  e
)  e.  (Poly `  B )  /\  (
( invg ` fld ) `  ( e `  X
) )  =  ( ( ( CC  X.  { -u 1 } )  oF  x.  e
) `  X )
)  ->  E. p  e.  (Poly `  B )
( ( invg ` fld ) `  ( e `  X ) )  =  ( p `  X
) )
152130, 148, 151syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  ( e `
 X ) )  =  ( p `  X ) )
153 fveq2 5856 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( invg ` fld ) `  b )  =  ( ( invg ` fld ) `  ( e `  X
) ) )
154153eqeq1d 2445 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  (
( ( invg ` fld ) `  b )  =  ( p `  X )  <->  ( ( invg ` fld ) `  ( e `
 X ) )  =  ( p `  X ) ) )
155154rexbidv 2954 . . . . . . . . . . . 12  |-  ( b  =  ( e `  X )  ->  ( E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  b )  =  ( p `  X )  <->  E. p  e.  (Poly `  B )
( ( invg ` fld ) `  ( e `  X ) )  =  ( p `  X
) ) )
156152, 155syl5ibrcom 222 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( b  =  ( e `  X
)  ->  E. p  e.  (Poly `  B )
( ( invg ` fld ) `  b )  =  ( p `  X ) ) )
157156rexlimdva 2935 . . . . . . . . . 10  |-  ( ph  ->  ( E. e  e.  (Poly `  B )
b  =  ( e `
 X )  ->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  b )  =  ( p `  X ) ) )
158157imp 429 . . . . . . . . 9  |-  ( (
ph  /\  E. e  e.  (Poly `  B )
b  =  ( e `
 X ) )  ->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  b )  =  ( p `  X ) )
15958, 158sylan2b 475 . . . . . . . 8  |-  ( (
ph  /\  b  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  ->  E. p  e.  (Poly `  B )
( ( invg ` fld ) `  b )  =  ( p `  X ) )
160 fvex 5866 . . . . . . . . 9  |-  ( ( invg ` fld ) `  b )  e.  _V
161 eqeq1 2447 . . . . . . . . . 10  |-  ( a  =  ( ( invg ` fld ) `  b )  ->  ( a  =  ( p `  X
)  <->  ( ( invg ` fld ) `  b )  =  ( p `  X ) ) )
162161rexbidv 2954 . . . . . . . . 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 3232 . . . . . . . 8  |-  ( ( ( invg ` fld ) `  b )  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  <->  E. p  e.  (Poly `  B ) ( ( invg ` fld ) `  b )  =  ( p `  X ) )
164159, 163sylibr 212 . . . . . . 7  |-  ( (
ph  /\  b  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  ->  ( ( invg ` fld ) `  b )  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) } )
165114a1i 11 . . . . . . 7  |-  ( ph  ->  1  =  ( 1r
` fld
) )
166125a1i 11 . . . . . . 7  |-  ( ph  ->  x.  =  ( .r
` fld
) )
16744, 116sseldd 3490 . . . . . . 7  |-  ( ph  ->  1  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
168129adantlr 714 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  e  e.  (Poly `  B
) )  /\  d  e.  (Poly `  B )
)  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a  x.  b
)  e.  B )
16967, 68, 73, 168plymul 22593 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( e  oF  x.  d )  e.  (Poly `  B
) )
170 fnfvof 6538 . . . . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e  oF  x.  d
) `  X )  =  ( ( e `
 X )  x.  ( d `  X
) ) )
172171eqcomd 2451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( ( e `
 X )  x.  ( d `  X
) )  =  ( ( e  oF  x.  d ) `  X ) )
173 fveq1 5855 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( e  oF  x.  d )  ->  ( p `  X )  =  ( ( e  oF  x.  d ) `  X ) )
174173eqeq2d 2457 . . . . . . . . . . . . . . . 16  |-  ( p  =  ( e  oF  x.  d )  ->  ( ( ( e `  X )  x.  ( d `  X ) )  =  ( p `  X
)  <->  ( ( e `
 X )  x.  ( d `  X
) )  =  ( ( e  oF  x.  d ) `  X ) ) )
175174rspcev 3196 . . . . . . . . . . . . . . 15  |-  ( ( ( e  oF  x.  d )  e.  (Poly `  B )  /\  ( ( e `  X )  x.  (
d `  X )
)  =  ( ( e  oF  x.  d ) `  X
) )  ->  E. p  e.  (Poly `  B )
( ( e `  X )  x.  (
d `  X )
)  =  ( p `
 X ) )
176169, 172, 175syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  x.  ( d `  X ) )  =  ( p `  X
) )
177 oveq2 6289 . . . . . . . . . . . . . . . 16  |-  ( c  =  ( d `  X )  ->  (
( e `  X
)  x.  c )  =  ( ( e `
 X )  x.  ( d `  X
) ) )
178177eqeq1d 2445 . . . . . . . . . . . . . . 15  |-  ( c  =  ( d `  X )  ->  (
( ( e `  X )  x.  c
)  =  ( p `
 X )  <->  ( (
e `  X )  x.  ( d `  X
) )  =  ( p `  X ) ) )
179178rexbidv 2954 . . . . . . . . . . . . . 14  |-  ( c  =  ( d `  X )  ->  ( E. p  e.  (Poly `  B ) ( ( e `  X )  x.  c )  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( ( e `  X )  x.  ( d `  X ) )  =  ( p `  X
) ) )
180176, 179syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  e  e.  (Poly `  B )
)  /\  d  e.  (Poly `  B ) )  ->  ( c  =  ( d `  X
)  ->  E. p  e.  (Poly `  B )
( ( e `  X )  x.  c
)  =  ( p `
 X ) ) )
181180rexlimdva 2935 . . . . . . . . . . . 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 6288 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e `  X )  ->  (
b  x.  c )  =  ( ( e `
 X )  x.  c ) )
183182eqeq1d 2445 . . . . . . . . . . . . . 14  |-  ( b  =  ( e `  X )  ->  (
( b  x.  c
)  =  ( p `
 X )  <->  ( (
e `  X )  x.  c )  =  ( p `  X ) ) )
184183rexbidv 2954 . . . . . . . . . . . . 13  |-  ( b  =  ( e `  X )  ->  ( E. p  e.  (Poly `  B ) ( b  x.  c )  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) ( ( e `  X )  x.  c )  =  ( p `  X
) ) )
185184imbi2d 316 . . . . . . . . . . . 12  |-  ( b  =  ( e `  X )  ->  (
( E. d  e.  (Poly `  B )
c  =  ( d `
 X )  ->  E. p  e.  (Poly `  B ) ( b  x.  c )  =  ( p `  X
) )  <->  ( E. d  e.  (Poly `  B
) c  =  ( d `  X )  ->  E. p  e.  (Poly `  B ) ( ( e `  X )  x.  c )  =  ( p `  X
) ) ) )
186181, 185syl5ibrcom 222 . . . . . . . . . . 11  |-  ( (
ph  /\  e  e.  (Poly `  B ) )  ->  ( b  =  ( e `  X
)  ->  ( E. d  e.  (Poly `  B
) c  =  ( d `  X )  ->  E. p  e.  (Poly `  B ) ( b  x.  c )  =  ( p `  X
) ) ) )
187186rexlimdva 2935 . . . . . . . . . 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 1191 . . . . . . . . 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 1272 . . . . . . . 8  |-  ( (
ph  /\  b  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  /\  c  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  ->  E. p  e.  (Poly `  B )
( b  x.  c
)  =  ( p `
 X ) )
190 ovex 6309 . . . . . . . . 9  |-  ( b  x.  c )  e. 
_V
191 eqeq1 2447 . . . . . . . . . 10  |-  ( a  =  ( b  x.  c )  ->  (
a  =  ( p `
 X )  <->  ( b  x.  c )  =  ( p `  X ) ) )
192191rexbidv 2954 . . . . . . . . 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 3232 . . . . . . . 8  |-  ( ( b  x.  c )  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B )
( b  x.  c
)  =  ( p `
 X ) )
194189, 193sylibr 212 . . . . . . 7  |-  ( (
ph  /\  b  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  /\  c  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )  ->  ( b  x.  c )  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
19515, 17, 19, 29, 49, 110, 164, 165, 166, 167, 194, 4issubrngd2 17814 . . . . . 6  |-  ( ph  ->  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) }  e.  (SubRing ` fld ) )
196 plyid 22584 . . . . . . . . . . 11  |-  ( ( B  C_  CC  /\  1  e.  B )  ->  Xp  e.  (Poly `  B
) )
1979, 116, 196syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  Xp  e.  (Poly `  B ) )
198 df-idp 22564 . . . . . . . . . . . 12  |-  Xp  =  (  _I  |`  CC )
199198fveq1i 5857 . . . . . . . . . . 11  |-  ( Xp `  X )  =  ( (  _I  |`  CC ) `  X
)
200 fvresi 6082 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  (
(  _I  |`  CC ) `
 X )  =  X )
20110, 200syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( (  _I  |`  CC ) `
 X )  =  X )
202199, 201syl5req 2497 . . . . . . . . . 10  |-  ( ph  ->  X  =  ( Xp `  X ) )
203 fveq1 5855 . . . . . . . . . . . 12  |-  ( p  =  Xp  -> 
( p `  X
)  =  ( Xp `  X ) )
204203eqeq2d 2457 . . . . . . . . . . 11  |-  ( p  =  Xp  -> 
( X  =  ( p `  X )  <-> 
X  =  ( Xp `  X ) ) )
205204rspcev 3196 . . . . . . . . . 10  |-  ( ( Xp  e.  (Poly `  B )  /\  X  =  ( Xp `  X ) )  ->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) )
206197, 202, 205syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) )
207 eqeq1 2447 . . . . . . . . . . . 12  |-  ( a  =  X  ->  (
a  =  ( p `
 X )  <->  X  =  ( p `  X
) ) )
208207rexbidv 2954 . . . . . . . . . . 11  |-  ( a  =  X  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) ) )
209208elabg 3233 . . . . . . . . . 10  |-  ( X  e.  CC  ->  ( X  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X ) ) )
21010, 209syl 16 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  {
a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) }  <->  E. p  e.  (Poly `  B ) X  =  ( p `  X
) ) )
211206, 210mpbird 232 . . . . . . . 8  |-  ( ph  ->  X  e.  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
212211snssd 4160 . . . . . . 7  |-  ( ph  ->  { X }  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
21344, 212unssd 3665 . . . . . 6  |-  ( ph  ->  ( B  u.  { X } )  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
2144, 6, 12, 13, 14, 195, 213rgspnmin 31096 . . . . 5  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  C_  { a  |  E. p  e.  (Poly `  B )
a  =  ( p `
 X ) } )
215214sseld 3488 . . . 4  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  ->  V  e.  { a  |  E. p  e.  (Poly `  B
) a  =  ( p `  X ) } ) )
216 fvex 5866 . . . . . . 7  |-  ( p `
 X )  e. 
_V
217 eleq1 2515 . . . . . . 7  |-  ( V  =  ( p `  X )  ->  ( V  e.  _V  <->  ( p `  X )  e.  _V ) )
218216, 217mpbiri 233 . . . . . 6  |-  ( V  =  ( p `  X )  ->  V  e.  _V )
219218rexlimivw 2932 . . . . 5  |-  ( E. p  e.  (Poly `  B ) V  =  ( p `  X
)  ->  V  e.  _V )
220 eqeq1 2447 . . . . . 6  |-  ( a  =  V  ->  (
a  =  ( p `
 X )  <->  V  =  ( p `  X
) ) )
221220rexbidv 2954 . . . . 5  |-  ( a  =  V  ->  ( E. p  e.  (Poly `  B ) a  =  ( p `  X
)  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X
) ) )
222219, 221elab3 3239 . . . 4  |-  ( V  e.  { a  |  E. p  e.  (Poly `  B ) a  =  ( p `  X
) }  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) )
223215, 222syl6ib 226 . . 3  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  ->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
2244, 6, 12, 13, 14rgspncl 31094 . . . . . . 7  |-  ( ph  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  e.  (SubRing ` fld ) )
225224adantr 465 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  e.  (SubRing ` fld ) )
226 simpr 461 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  p  e.  (Poly `  B ) )
2274, 6, 12, 13, 14rgspnssid 31095 . . . . . . . . 9  |-  ( ph  ->  ( B  u.  { X } )  C_  (
(RingSpan ` fld ) `  ( B  u.  { X }
) ) )
228227unssbd 3667 . . . . . . . 8  |-  ( ph  ->  { X }  C_  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
229 snidg 4040 . . . . . . . . 9  |-  ( X  e.  CC  ->  X  e.  { X } )
23010, 229syl 16 . . . . . . . 8  |-  ( ph  ->  X  e.  { X } )
231228, 230sseldd 3490 . . . . . . 7  |-  ( ph  ->  X  e.  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) )
232231adantr 465 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  X  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
233227unssad 3666 . . . . . . 7  |-  ( ph  ->  B  C_  ( (RingSpan ` fld ) `
 ( B  u.  { X } ) ) )
234233adantr 465 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  B  C_  (
(RingSpan ` fld ) `  ( B  u.  { X }
) ) )
235225, 226, 232, 234cnsrplycl 31092 . . . . 5  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( p `  X )  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) )
236 eleq1 2515 . . . . 5  |-  ( V  =  ( p `  X )  ->  ( V  e.  ( (RingSpan ` fld ) `
 ( B  u.  { X } ) )  <-> 
( p `  X
)  e.  ( (RingSpan ` fld ) `  ( B  u.  { X } ) ) ) )
237235, 236syl5ibrcom 222 . . . 4  |-  ( (
ph  /\  p  e.  (Poly `  B ) )  ->  ( V  =  ( p `  X
)  ->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
238237rexlimdva 2935 . . 3  |-  ( ph  ->  ( E. p  e.  (Poly `  B ) V  =  ( p `  X )  ->  V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) ) ) )
239223, 238impbid 191 . 2  |-  ( ph  ->  ( V  e.  ( (RingSpan ` fld ) `  ( B  u.  { X }
) )  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
2402, 239bitrd 253 1  |-  ( ph  ->  ( V  e.  S  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   {cab 2428   E.wrex 2794   _Vcvv 3095    u. cun 3459    C_ wss 3461   {csn 4014    _I cid 4780    X. cxp 4987    |` cres 4991    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   -ucneg 9811   Basecbs 14614   ↾s cress 14615   +g cplusg 14679   .rcmulr 14680   0gc0g 14819   invgcminusg 16033  SubGrpcsubg 16174   1rcur 17132   Ringcrg 17177  SubRingcsubrg 17404  RingSpancrgspn 17405  ℂfldccnfld 18399  Polycply 22559   Xpcidp 22560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-rp 11232  df-fz 11684  df-fzo 11807  df-fl 11911  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-rlim 13294  df-sum 13491  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-subg 16177  df-cmn 16779  df-mgp 17121  df-ur 17133  df-ring 17179  df-cring 17180  df-subrg 17406  df-rgspn 17407  df-cnfld 18400  df-0p 22055  df-ply 22563  df-idp 22564  df-coe 22565  df-dgr 22566
This theorem is referenced by: (None)
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