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Theorem rgspnval 27044
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 5672 . 2  |-  ( ph  ->  ( N `  A
)  =  ( (RingSpan `  R ) `  A
) )
4 rgspnval.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
5 elex 2909 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
_V )
6 fveq2 5670 . . . . . . . 8  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
76pweqd 3749 . . . . . . 7  |-  ( a  =  R  ->  ~P ( Base `  a )  =  ~P ( Base `  R
) )
8 fveq2 5670 . . . . . . . . 9  |-  ( a  =  R  ->  (SubRing `  a )  =  (SubRing `  R ) )
9 rabeq 2895 . . . . . . . . 9  |-  ( (SubRing `  a )  =  (SubRing `  R )  ->  { t  e.  (SubRing `  a
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  b  C_  t } )
108, 9syl 16 . . . . . . . 8  |-  ( a  =  R  ->  { t  e.  (SubRing `  a
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  b  C_  t } )
1110inteqd 3999 . . . . . . 7  |-  ( a  =  R  ->  |^| { t  e.  (SubRing `  a
)  |  b  C_  t }  =  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } )
127, 11mpteq12dv 4230 . . . . . 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 15796 . . . . . 6  |- RingSpan  =  ( a  e.  _V  |->  ( b  e.  ~P ( Base `  a )  |->  |^|
{ t  e.  (SubRing `  a )  |  b 
C_  t } ) )
14 fvex 5684 . . . . . . . 8  |-  ( Base `  R )  e.  _V
1514pwex 4325 . . . . . . 7  |-  ~P ( Base `  R )  e. 
_V
1615mptex 5907 . . . . . 6  |-  ( b  e.  ~P ( Base `  R )  |->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t } )  e.  _V
1712, 13, 16fvmpt 5747 . . . . 5  |-  ( R  e.  _V  ->  (RingSpan `  R )  =  ( b  e.  ~P ( Base `  R )  |->  |^|
{ t  e.  (SubRing `  R )  |  b 
C_  t } ) )
184, 5, 173syl 19 . . . 4  |-  ( ph  ->  (RingSpan `  R )  =  ( b  e. 
~P ( Base `  R
)  |->  |^| { t  e.  (SubRing `  R )  |  b  C_  t } ) )
1918fveq1d 5672 . . 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 3329 . . . . 5  |-  ( ph  ->  A  C_  ( Base `  R ) )
2314elpw2 4307 . . . . 5  |-  ( A  e.  ~P ( Base `  R )  <->  A  C_  ( Base `  R ) )
2422, 23sylibr 204 . . . 4  |-  ( ph  ->  A  e.  ~P ( Base `  R ) )
25 eqid 2389 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
2625subrgid 15799 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  R )  e.  (SubRing `  R ) )
274, 26syl 16 . . . . . . 7  |-  ( ph  ->  ( Base `  R
)  e.  (SubRing `  R
) )
2821, 27eqeltrd 2463 . . . . . 6  |-  ( ph  ->  B  e.  (SubRing `  R
) )
29 sseq2 3315 . . . . . . 7  |-  ( t  =  B  ->  ( A  C_  t  <->  A  C_  B
) )
3029rspcev 2997 . . . . . 6  |-  ( ( B  e.  (SubRing `  R
)  /\  A  C_  B
)  ->  E. t  e.  (SubRing `  R ) A  C_  t )
3128, 20, 30syl2anc 643 . . . . 5  |-  ( ph  ->  E. t  e.  (SubRing `  R ) A  C_  t )
32 intexrab 4302 . . . . 5  |-  ( E. t  e.  (SubRing `  R
) A  C_  t  <->  |^|
{ t  e.  (SubRing `  R )  |  A  C_  t }  e.  _V )
3331, 32sylib 189 . . . 4  |-  ( ph  ->  |^| { t  e.  (SubRing `  R )  |  A  C_  t }  e.  _V )
34 sseq1 3314 . . . . . . 7  |-  ( b  =  A  ->  (
b  C_  t  <->  A  C_  t
) )
3534rabbidv 2893 . . . . . 6  |-  ( b  =  A  ->  { t  e.  (SubRing `  R
)  |  b  C_  t }  =  {
t  e.  (SubRing `  R
)  |  A  C_  t } )
3635inteqd 3999 . . . . 5  |-  ( b  =  A  ->  |^| { t  e.  (SubRing `  R
)  |  b  C_  t }  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
37 eqid 2389 . . . . 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 5745 . . . 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 643 . . 3  |-  ( ph  ->  ( ( b  e. 
~P ( Base `  R
)  |->  |^| { t  e.  (SubRing `  R )  |  b  C_  t } ) `  A )  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
4019, 39eqtrd 2421 . 2  |-  ( ph  ->  ( (RingSpan `  R
) `  A )  =  |^| { t  e.  (SubRing `  R )  |  A  C_  t } )
411, 3, 403eqtrd 2425 1  |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R
)  |  A  C_  t } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   E.wrex 2652   {crab 2655   _Vcvv 2901    C_ wss 3265   ~Pcpw 3744   |^|cint 3994    e. cmpt 4209   ` cfv 5396   Basecbs 13398   Ringcrg 15589  SubRingcsubrg 15793  RingSpancrgspn 15794
This theorem is referenced by:  rgspncl  27045  rgspnssid  27046  rgspnmin  27047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-mnd 14619  df-mgp 15578  df-rng 15592  df-ur 15594  df-subrg 15795  df-rgspn 15796
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