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Theorem rgspnval 27476
Description: Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
Hypotheses
Ref Expression
rgspnval.r  |-  ( ph  ->  R  e.  Ring )
rgspnval.b  |-  ( ph  ->  B  =  ( Base `  R ) )
rgspnval.ss  |-  ( ph  ->  A  C_  B )
rgspnval.n  |-  ( ph  ->  N  =  (RingSpan `  R
) )
rgspnval.sp  |-  ( ph  ->  U  =  ( N `
 A ) )
Assertion
Ref Expression
rgspnval  |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
Distinct variable groups:    ph, t    t, R    t, B    t, A
Allowed substitution hints:    U( t)    N( t)

Proof of Theorem rgspnval
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rgspnval.sp . 2  |-  ( ph  ->  U  =  ( N `
 A ) )
2 rgspnval.n . . 3  |-  ( ph  ->  N  =  (RingSpan `  R
) )
32fveq1d 5543 . 2  |-  ( ph  ->  ( N `  A
)  =  ( (RingSpan `  R ) `  A
) )
4 rgspnval.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
5 elex 2809 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
_V )
6 fveq2 5541 . . . . . . . 8  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
76pweqd 3643 . . . . . . 7  |-  ( a  =  R  ->  ~P ( Base `  a )  =  ~P ( Base `  R
) )
8 fveq2 5541 . . . . . . . . 9  |-  ( a  =  R  ->  (SubRing `  a )  =  (SubRing `  R ) )
9 rabeq 2795 . . . . . . . . 9  |-  ( (SubRing `  a )  =  (SubRing `  R )  ->  { t  e.  (SubRing `  a
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  b  C_  t } )
108, 9syl 15 . . . . . . . 8  |-  ( a  =  R  ->  { t  e.  (SubRing `  a
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  b  C_  t } )
1110inteqd 3883 . . . . . . 7  |-  ( a  =  R  ->  |^| { t  e.  (SubRing `  a
)  |  b  C_  t }  =  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } )
127, 11mpteq12dv 4114 . . . . . 6  |-  ( a  =  R  ->  (
b  e.  ~P ( Base `  a )  |->  |^|
{ t  e.  (SubRing `  a )  |  b 
C_  t } )  =  ( b  e. 
~P ( Base `  R
)  |->  |^| { t  e.  (SubRing `  R )  |  b  C_  t } ) )
13 df-rgspn 15560 . . . . . 6  |- RingSpan  =  ( a  e.  _V  |->  ( b  e.  ~P ( Base `  a )  |->  |^|
{ t  e.  (SubRing `  a )  |  b 
C_  t } ) )
14 fvex 5555 . . . . . . . 8  |-  ( Base `  R )  e.  _V
1514pwex 4209 . . . . . . 7  |-  ~P ( Base `  R )  e. 
_V
1615mptex 5762 . . . . . 6  |-  ( b  e.  ~P ( Base `  R )  |->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } )  e.  _V
1712, 13, 16fvmpt 5618 . . . . 5  |-  ( R  e.  _V  ->  (RingSpan `  R )  =  ( b  e.  ~P ( Base `  R )  |->  |^|
{ t  e.  (SubRing `  R )  |  b 
C_  t } ) )
184, 5, 173syl 18 . . . 4  |-  ( ph  ->  (RingSpan `  R )  =  ( b  e. 
~P ( Base `  R
)  |->  |^| { t  e.  (SubRing `  R )  |  b  C_  t } ) )
1918fveq1d 5543 . . 3  |-  ( ph  ->  ( (RingSpan `  R
) `  A )  =  ( ( b  e.  ~P ( Base `  R )  |->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } ) `  A
) )
20 rgspnval.ss . . . . . 6  |-  ( ph  ->  A  C_  B )
21 rgspnval.b . . . . . 6  |-  ( ph  ->  B  =  ( Base `  R ) )
2220, 21sseqtrd 3227 . . . . 5  |-  ( ph  ->  A  C_  ( Base `  R ) )
2314elpw2 4191 . . . . 5  |-  ( A  e.  ~P ( Base `  R )  <->  A  C_  ( Base `  R ) )
2422, 23sylibr 203 . . . 4  |-  ( ph  ->  A  e.  ~P ( Base `  R ) )
25 eqid 2296 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
2625subrgid 15563 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  R )  e.  (SubRing `  R ) )
274, 26syl 15 . . . . . . 7  |-  ( ph  ->  ( Base `  R
)  e.  (SubRing `  R
) )
2821, 27eqeltrd 2370 . . . . . 6  |-  ( ph  ->  B  e.  (SubRing `  R
) )
29 sseq2 3213 . . . . . . 7  |-  ( t  =  B  ->  ( A  C_  t  <->  A  C_  B
) )
3029rspcev 2897 . . . . . 6  |-  ( ( B  e.  (SubRing `  R
)  /\  A  C_  B
)  ->  E. t  e.  (SubRing `  R ) A  C_  t )
3128, 20, 30syl2anc 642 . . . . 5  |-  ( ph  ->  E. t  e.  (SubRing `  R ) A  C_  t )
32 intexrab 4186 . . . . 5  |-  ( E. t  e.  (SubRing `  R
) A  C_  t  <->  |^|
{ t  e.  (SubRing `  R )  |  A  C_  t }  e.  _V )
3331, 32sylib 188 . . . 4  |-  ( ph  ->  |^| { t  e.  (SubRing `  R )  |  A  C_  t }  e.  _V )
34 sseq1 3212 . . . . . . 7  |-  ( b  =  A  ->  (
b  C_  t  <->  A  C_  t
) )
3534rabbidv 2793 . . . . . 6  |-  ( b  =  A  ->  { t  e.  (SubRing `  R
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  A  C_  t } )
3635inteqd 3883 . . . . 5  |-  ( b  =  A  ->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t }  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
37 eqid 2296 . . . . 5  |-  ( b  e.  ~P ( Base `  R )  |->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } )  =  ( b  e.  ~P ( Base `  R )  |->  |^|
{ t  e.  (SubRing `  R )  |  b 
C_  t } )
3836, 37fvmptg 5616 . . . 4  |-  ( ( A  e.  ~P ( Base `  R )  /\  |^|
{ t  e.  (SubRing `  R )  |  A  C_  t }  e.  _V )  ->  ( ( b  e.  ~P ( Base `  R )  |->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } ) `  A
)  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
3924, 33, 38syl2anc 642 . . 3  |-  ( ph  ->  ( ( b  e. 
~P ( Base `  R
)  |->  |^| { t  e.  (SubRing `  R )  |  b  C_  t } ) `  A )  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
4019, 39eqtrd 2328 . 2  |-  ( ph  ->  ( (RingSpan `  R
) `  A )  =  |^| { t  e.  (SubRing `  R )  |  A  C_  t } )
411, 3, 403eqtrd 2332 1  |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   |^|cint 3878    e. cmpt 4093   ` cfv 5271   Basecbs 13164   Ringcrg 15353  SubRingcsubrg 15557  RingSpancrgspn 15558
This theorem is referenced by:  rgspncl  27477  rgspnssid  27478  rgspnmin  27479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559  df-rgspn 15560
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