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Theorem evls1sca 18486
Description: Univariate polynomial evaluation maps scalars to constant functions. (Contributed by AV, 8-Sep-2019.)
Hypotheses
Ref Expression
evls1sca.q  |-  Q  =  ( S evalSub1  R )
evls1sca.w  |-  W  =  (Poly1 `  U )
evls1sca.u  |-  U  =  ( Ss  R )
evls1sca.b  |-  B  =  ( Base `  S
)
evls1sca.a  |-  A  =  (algSc `  W )
evls1sca.s  |-  ( ph  ->  S  e.  CRing )
evls1sca.r  |-  ( ph  ->  R  e.  (SubRing `  S
) )
evls1sca.x  |-  ( ph  ->  X  e.  R )
Assertion
Ref Expression
evls1sca  |-  ( ph  ->  ( Q `  ( A `  X )
)  =  ( B  X.  { X }
) )

Proof of Theorem evls1sca
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1on 7155 . . . . . . 7  |-  1o  e.  On
21a1i 11 . . . . . 6  |-  ( ph  ->  1o  e.  On )
3 evls1sca.s . . . . . 6  |-  ( ph  ->  S  e.  CRing )
4 evls1sca.r . . . . . 6  |-  ( ph  ->  R  e.  (SubRing `  S
) )
5 eqid 2457 . . . . . . 7  |-  ( ( 1o evalSub  S ) `  R
)  =  ( ( 1o evalSub  S ) `  R
)
6 eqid 2457 . . . . . . 7  |-  ( 1o mPoly  U )  =  ( 1o mPoly  U )
7 evls1sca.u . . . . . . 7  |-  U  =  ( Ss  R )
8 eqid 2457 . . . . . . 7  |-  ( S  ^s  ( B  ^m  1o ) )  =  ( S  ^s  ( B  ^m  1o ) )
9 evls1sca.b . . . . . . 7  |-  B  =  ( Base `  S
)
105, 6, 7, 8, 9evlsrhm 18316 . . . . . 6  |-  ( ( 1o  e.  On  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( ( 1o evalSub  S ) `  R
)  e.  ( ( 1o mPoly  U ) RingHom  ( S  ^s  ( B  ^m  1o ) ) ) )
112, 3, 4, 10syl3anc 1228 . . . . 5  |-  ( ph  ->  ( ( 1o evalSub  S ) `
 R )  e.  ( ( 1o mPoly  U
) RingHom  ( S  ^s  ( B  ^m  1o ) ) ) )
12 eqid 2457 . . . . . 6  |-  ( Base `  ( 1o mPoly  U )
)  =  ( Base `  ( 1o mPoly  U )
)
13 eqid 2457 . . . . . 6  |-  ( Base `  ( S  ^s  ( B  ^m  1o ) ) )  =  ( Base `  ( S  ^s  ( B  ^m  1o ) ) )
1412, 13rhmf 17501 . . . . 5  |-  ( ( ( 1o evalSub  S ) `  R )  e.  ( ( 1o mPoly  U ) RingHom  ( S  ^s  ( B  ^m  1o ) ) )  -> 
( ( 1o evalSub  S ) `
 R ) : ( Base `  ( 1o mPoly  U ) ) --> (
Base `  ( S  ^s  ( B  ^m  1o ) ) ) )
1511, 14syl 16 . . . 4  |-  ( ph  ->  ( ( 1o evalSub  S ) `
 R ) : ( Base `  ( 1o mPoly  U ) ) --> (
Base `  ( S  ^s  ( B  ^m  1o ) ) ) )
16 evls1sca.a . . . . . . 7  |-  A  =  (algSc `  W )
17 eqid 2457 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
187subrgring 17558 . . . . . . . . 9  |-  ( R  e.  (SubRing `  S
)  ->  U  e.  Ring )
194, 18syl 16 . . . . . . . 8  |-  ( ph  ->  U  e.  Ring )
20 evls1sca.w . . . . . . . . 9  |-  W  =  (Poly1 `  U )
2120ply1ring 18415 . . . . . . . 8  |-  ( U  e.  Ring  ->  W  e. 
Ring )
2219, 21syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  Ring )
2320ply1lmod 18419 . . . . . . . 8  |-  ( U  e.  Ring  ->  W  e. 
LMod )
2419, 23syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  LMod )
25 eqid 2457 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
26 eqid 2457 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
2716, 17, 22, 24, 25, 26asclf 18112 . . . . . 6  |-  ( ph  ->  A : ( Base `  (Scalar `  W )
) --> ( Base `  W
) )
289subrgss 17556 . . . . . . . . . 10  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
294, 28syl 16 . . . . . . . . 9  |-  ( ph  ->  R  C_  B )
307, 9ressbas2 14701 . . . . . . . . 9  |-  ( R 
C_  B  ->  R  =  ( Base `  U
) )
3129, 30syl 16 . . . . . . . 8  |-  ( ph  ->  R  =  ( Base `  U ) )
3220ply1sca 18420 . . . . . . . . . 10  |-  ( U  e.  Ring  ->  U  =  (Scalar `  W )
)
3319, 32syl 16 . . . . . . . . 9  |-  ( ph  ->  U  =  (Scalar `  W ) )
3433fveq2d 5876 . . . . . . . 8  |-  ( ph  ->  ( Base `  U
)  =  ( Base `  (Scalar `  W )
) )
3531, 34eqtrd 2498 . . . . . . 7  |-  ( ph  ->  R  =  ( Base `  (Scalar `  W )
) )
36 eqid 2457 . . . . . . . . . 10  |-  (PwSer1 `  U
)  =  (PwSer1 `  U
)
3720, 36, 26ply1bas 18360 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  ( 1o mPoly  U ) )
3837a1i 11 . . . . . . . 8  |-  ( ph  ->  ( Base `  W
)  =  ( Base `  ( 1o mPoly  U )
) )
3938eqcomd 2465 . . . . . . 7  |-  ( ph  ->  ( Base `  ( 1o mPoly  U ) )  =  ( Base `  W
) )
4035, 39feq23d 5732 . . . . . 6  |-  ( ph  ->  ( A : R --> ( Base `  ( 1o mPoly  U ) )  <->  A :
( Base `  (Scalar `  W
) ) --> ( Base `  W ) ) )
4127, 40mpbird 232 . . . . 5  |-  ( ph  ->  A : R --> ( Base `  ( 1o mPoly  U )
) )
42 evls1sca.x . . . . 5  |-  ( ph  ->  X  e.  R )
4341, 42ffvelrnd 6033 . . . 4  |-  ( ph  ->  ( A `  X
)  e.  ( Base `  ( 1o mPoly  U )
) )
44 fvco3 5950 . . . 4  |-  ( ( ( ( 1o evalSub  S ) `
 R ) : ( Base `  ( 1o mPoly  U ) ) --> (
Base `  ( S  ^s  ( B  ^m  1o ) ) )  /\  ( A `  X )  e.  ( Base `  ( 1o mPoly  U ) ) )  ->  ( ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  S ) `
 R ) ) `
 ( A `  X ) )  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) ) `  ( ( ( 1o evalSub  S ) `  R ) `  ( A `  X )
) ) )
4515, 43, 44syl2anc 661 . . 3  |-  ( ph  ->  ( ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )  o.  (
( 1o evalSub  S ) `  R ) ) `  ( A `  X ) )  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) ) `  ( ( ( 1o evalSub  S ) `  R
) `  ( A `  X ) ) ) )
4616a1i 11 . . . . . . . 8  |-  ( ph  ->  A  =  (algSc `  W ) )
47 eqid 2457 . . . . . . . . 9  |-  (algSc `  W )  =  (algSc `  W )
4820, 47ply1ascl 18425 . . . . . . . 8  |-  (algSc `  W )  =  (algSc `  ( 1o mPoly  U )
)
4946, 48syl6eq 2514 . . . . . . 7  |-  ( ph  ->  A  =  (algSc `  ( 1o mPoly  U ) ) )
5049fveq1d 5874 . . . . . 6  |-  ( ph  ->  ( A `  X
)  =  ( (algSc `  ( 1o mPoly  U )
) `  X )
)
5150fveq2d 5876 . . . . 5  |-  ( ph  ->  ( ( ( 1o evalSub  S ) `  R
) `  ( A `  X ) )  =  ( ( ( 1o evalSub  S ) `  R
) `  ( (algSc `  ( 1o mPoly  U )
) `  X )
) )
52 eqid 2457 . . . . . 6  |-  (algSc `  ( 1o mPoly  U ) )  =  (algSc `  ( 1o mPoly  U ) )
535, 6, 7, 9, 52, 2, 3, 4, 42evlssca 18317 . . . . 5  |-  ( ph  ->  ( ( ( 1o evalSub  S ) `  R
) `  ( (algSc `  ( 1o mPoly  U )
) `  X )
)  =  ( ( B  ^m  1o )  X.  { X }
) )
5451, 53eqtrd 2498 . . . 4  |-  ( ph  ->  ( ( ( 1o evalSub  S ) `  R
) `  ( A `  X ) )  =  ( ( B  ^m  1o )  X.  { X } ) )
5554fveq2d 5876 . . 3  |-  ( ph  ->  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) ) `  ( ( ( 1o evalSub  S ) `  R ) `  ( A `  X )
) )  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) ) `  ( ( B  ^m  1o )  X.  { X }
) ) )
56 eqidd 2458 . . . . 5  |-  ( ph  ->  ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  =  ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) ) )
57 coeq1 5170 . . . . . 6  |-  ( x  =  ( ( B  ^m  1o )  X. 
{ X } )  ->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) )  =  ( ( ( B  ^m  1o )  X.  { X }
)  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) ) )
5857adantl 466 . . . . 5  |-  ( (
ph  /\  x  =  ( ( B  ^m  1o )  X.  { X } ) )  -> 
( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) )  =  ( ( ( B  ^m  1o )  X.  { X }
)  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) ) )
5929, 42sseldd 3500 . . . . . . 7  |-  ( ph  ->  X  e.  B )
60 fconst6g 5780 . . . . . . 7  |-  ( X  e.  B  ->  (
( B  ^m  1o )  X.  { X }
) : ( B  ^m  1o ) --> B )
6159, 60syl 16 . . . . . 6  |-  ( ph  ->  ( ( B  ^m  1o )  X.  { X } ) : ( B  ^m  1o ) --> B )
62 fvex 5882 . . . . . . . . 9  |-  ( Base `  S )  e.  _V
639, 62eqeltri 2541 . . . . . . . 8  |-  B  e. 
_V
6463a1i 11 . . . . . . 7  |-  ( ph  ->  B  e.  _V )
65 ovex 6324 . . . . . . . 8  |-  ( B  ^m  1o )  e. 
_V
6665a1i 11 . . . . . . 7  |-  ( ph  ->  ( B  ^m  1o )  e.  _V )
6764, 66elmapd 7452 . . . . . 6  |-  ( ph  ->  ( ( ( B  ^m  1o )  X. 
{ X } )  e.  ( B  ^m  ( B  ^m  1o ) )  <->  ( ( B  ^m  1o )  X. 
{ X } ) : ( B  ^m  1o ) --> B ) )
6861, 67mpbird 232 . . . . 5  |-  ( ph  ->  ( ( B  ^m  1o )  X.  { X } )  e.  ( B  ^m  ( B  ^m  1o ) ) )
69 snex 4697 . . . . . . . 8  |-  { X }  e.  _V
7065, 69xpex 6603 . . . . . . 7  |-  ( ( B  ^m  1o )  X.  { X }
)  e.  _V
7170a1i 11 . . . . . 6  |-  ( ph  ->  ( ( B  ^m  1o )  X.  { X } )  e.  _V )
72 mptexg 6143 . . . . . . 7  |-  ( B  e.  _V  ->  (
y  e.  B  |->  ( 1o  X.  { y } ) )  e. 
_V )
7364, 72syl 16 . . . . . 6  |-  ( ph  ->  ( y  e.  B  |->  ( 1o  X.  {
y } ) )  e.  _V )
74 coexg 6750 . . . . . 6  |-  ( ( ( ( B  ^m  1o )  X.  { X } )  e.  _V  /\  ( y  e.  B  |->  ( 1o  X.  {
y } ) )  e.  _V )  -> 
( ( ( B  ^m  1o )  X. 
{ X } )  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) )  e.  _V )
7571, 73, 74syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( ( B  ^m  1o )  X. 
{ X } )  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) )  e.  _V )
7656, 58, 68, 75fvmptd 5961 . . . 4  |-  ( ph  ->  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) ) `  ( ( B  ^m  1o )  X.  { X }
) )  =  ( ( ( B  ^m  1o )  X.  { X } )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )
77 fconst6g 5780 . . . . . . 7  |-  ( y  e.  B  ->  ( 1o  X.  { y } ) : 1o --> B )
7877adantl 466 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  ( 1o  X.  { y } ) : 1o --> B )
7963, 1pm3.2i 455 . . . . . . . 8  |-  ( B  e.  _V  /\  1o  e.  On )
8079a1i 11 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  ( B  e.  _V  /\  1o  e.  On ) )
81 elmapg 7451 . . . . . . 7  |-  ( ( B  e.  _V  /\  1o  e.  On )  -> 
( ( 1o  X.  { y } )  e.  ( B  ^m  1o )  <->  ( 1o  X.  { y } ) : 1o --> B ) )
8280, 81syl 16 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
( 1o  X.  {
y } )  e.  ( B  ^m  1o ) 
<->  ( 1o  X.  {
y } ) : 1o --> B ) )
8378, 82mpbird 232 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  ( 1o  X.  { y } )  e.  ( B  ^m  1o ) )
84 eqidd 2458 . . . . 5  |-  ( ph  ->  ( y  e.  B  |->  ( 1o  X.  {
y } ) )  =  ( y  e.  B  |->  ( 1o  X.  { y } ) ) )
85 fconstmpt 5052 . . . . . 6  |-  ( ( B  ^m  1o )  X.  { X }
)  =  ( z  e.  ( B  ^m  1o )  |->  X )
8685a1i 11 . . . . 5  |-  ( ph  ->  ( ( B  ^m  1o )  X.  { X } )  =  ( z  e.  ( B  ^m  1o )  |->  X ) )
87 eqidd 2458 . . . . 5  |-  ( z  =  ( 1o  X.  { y } )  ->  X  =  X )
8883, 84, 86, 87fmptco 6065 . . . 4  |-  ( ph  ->  ( ( ( B  ^m  1o )  X. 
{ X } )  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) )  =  ( y  e.  B  |->  X ) )
8976, 88eqtrd 2498 . . 3  |-  ( ph  ->  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) ) `  ( ( B  ^m  1o )  X.  { X }
) )  =  ( y  e.  B  |->  X ) )
9045, 55, 893eqtrd 2502 . 2  |-  ( ph  ->  ( ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )  o.  (
( 1o evalSub  S ) `  R ) ) `  ( A `  X ) )  =  ( y  e.  B  |->  X ) )
91 elpwg 4023 . . . . . 6  |-  ( R  e.  (SubRing `  S
)  ->  ( R  e.  ~P B  <->  R  C_  B
) )
9228, 91mpbird 232 . . . . 5  |-  ( R  e.  (SubRing `  S
)  ->  R  e.  ~P B )
934, 92syl 16 . . . 4  |-  ( ph  ->  R  e.  ~P B
)
94 evls1sca.q . . . . 5  |-  Q  =  ( S evalSub1  R )
95 eqid 2457 . . . . 5  |-  ( 1o evalSub  S )  =  ( 1o evalSub  S )
9694, 95, 9evls1fval 18482 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  ~P B )  ->  Q  =  ( (
x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  S ) `
 R ) ) )
973, 93, 96syl2anc 661 . . 3  |-  ( ph  ->  Q  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  S ) `
 R ) ) )
9897fveq1d 5874 . 2  |-  ( ph  ->  ( Q `  ( A `  X )
)  =  ( ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  S ) `  R
) ) `  ( A `  X )
) )
99 fconstmpt 5052 . . 3  |-  ( B  X.  { X }
)  =  ( y  e.  B  |->  X )
10099a1i 11 . 2  |-  ( ph  ->  ( B  X.  { X } )  =  ( y  e.  B  |->  X ) )
10190, 98, 1003eqtr4d 2508 1  |-  ( ph  ->  ( Q `  ( A `  X )
)  =  ( B  X.  { X }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   {csn 4032    |-> cmpt 4515   Oncon0 4887    X. cxp 5006    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   1oc1o 7141    ^m cmap 7438   Basecbs 14643   ↾s cress 14644  Scalarcsca 14714    ^s cpws 14863   Ringcrg 17324   CRingccrg 17325   RingHom crh 17487  SubRingcsubrg 17551   LModclmod 17638  algSccascl 18086   mPoly cmpl 18128   evalSub ces 18295  PwSer1cps1 18340  Poly1cpl1 18342   evalSub1 ces1 18476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-fzo 11821  df-seq 12110  df-hash 12408  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-hom 14735  df-cco 14736  df-0g 14858  df-gsum 14859  df-prds 14864  df-pws 14866  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-mulg 16186  df-subg 16324  df-ghm 16391  df-cntz 16481  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-srg 17284  df-ring 17326  df-cring 17327  df-rnghom 17490  df-subrg 17553  df-lmod 17640  df-lss 17705  df-lsp 17744  df-assa 18087  df-asp 18088  df-ascl 18089  df-psr 18131  df-mvr 18132  df-mpl 18133  df-opsr 18135  df-evls 18297  df-psr1 18345  df-ply1 18347  df-evls1 18478
This theorem is referenced by:  evls1scasrng  18501
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