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Theorem pl1cn 27559
Description: A univariate polynomial is continuous. (Contributed by Thierry Arnoux, 17-Sep-2018.)
Hypotheses
Ref Expression
pl1cn.p  |-  P  =  (Poly1 `  R )
pl1cn.e  |-  E  =  (eval1 `  R )
pl1cn.b  |-  B  =  ( Base `  P
)
pl1cn.k  |-  K  =  ( Base `  R
)
pl1cn.j  |-  J  =  ( TopOpen `  R )
pl1cn.1  |-  ( ph  ->  R  e.  CRing )
pl1cn.2  |-  ( ph  ->  R  e.  TopRing )
pl1cn.3  |-  ( ph  ->  F  e.  B )
Assertion
Ref Expression
pl1cn  |-  ( ph  ->  ( E `  F
)  e.  ( J  Cn  J ) )

Proof of Theorem pl1cn
Dummy variables  h  f  g  x  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pl1cn.k . 2  |-  K  =  ( Base `  R
)
2 eqid 2460 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
3 eqid 2460 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
4 eqid 2460 . 2  |-  ran  (eval1 `  R )  =  ran  (eval1 `  R )
5 fvex 5867 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
61, 5eqeltri 2544 . . . . . . . 8  |-  K  e. 
_V
76a1i 11 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  K  e.  _V )
8 fvex 5867 . . . . . . . 8  |-  ( f `
 x )  e. 
_V
98a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  /\  x  e.  K )  ->  (
f `  x )  e.  _V )
10 fvex 5867 . . . . . . . 8  |-  ( g `
 x )  e. 
_V
1110a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  /\  x  e.  K )  ->  (
g `  x )  e.  _V )
12 simp1 991 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  ph )
13 eqid 2460 . . . . . . . . . . 11  |-  U. J  =  U. J
1413, 13cnf 19506 . . . . . . . . . 10  |-  ( f  e.  ( J  Cn  J )  ->  f : U. J --> U. J
)
15 ffn 5722 . . . . . . . . . 10  |-  ( f : U. J --> U. J  ->  f  Fn  U. J
)
1614, 15syl 16 . . . . . . . . 9  |-  ( f  e.  ( J  Cn  J )  ->  f  Fn  U. J )
17163ad2ant2 1013 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  f  Fn  U. J )
18 dffn5 5904 . . . . . . . . . 10  |-  ( f  Fn  K  <->  f  =  ( x  e.  K  |->  ( f `  x
) ) )
19 pl1cn.2 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  TopRing )
20 trgtgp 20398 . . . . . . . . . . . . 13  |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
21 pl1cn.j . . . . . . . . . . . . . 14  |-  J  =  ( TopOpen `  R )
2221, 1tgptopon 20309 . . . . . . . . . . . . 13  |-  ( R  e.  TopGrp  ->  J  e.  (TopOn `  K ) )
2319, 20, 223syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  (TopOn `  K ) )
24 toponuni 19188 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  K
)  ->  K  =  U. J )
2523, 24syl 16 . . . . . . . . . . 11  |-  ( ph  ->  K  =  U. J
)
2625fneq2d 5663 . . . . . . . . . 10  |-  ( ph  ->  ( f  Fn  K  <->  f  Fn  U. J ) )
2718, 26syl5rbbr 260 . . . . . . . . 9  |-  ( ph  ->  ( f  Fn  U. J 
<->  f  =  ( x  e.  K  |->  ( f `
 x ) ) ) )
2827biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  f  Fn  U. J )  ->  f  =  ( x  e.  K  |->  ( f `  x ) ) )
2912, 17, 28syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  f  =  ( x  e.  K  |->  ( f `  x ) ) )
3013, 13cnf 19506 . . . . . . . . . 10  |-  ( g  e.  ( J  Cn  J )  ->  g : U. J --> U. J
)
31 ffn 5722 . . . . . . . . . 10  |-  ( g : U. J --> U. J  ->  g  Fn  U. J
)
3230, 31syl 16 . . . . . . . . 9  |-  ( g  e.  ( J  Cn  J )  ->  g  Fn  U. J )
33323ad2ant3 1014 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  g  Fn  U. J )
34 dffn5 5904 . . . . . . . . . 10  |-  ( g  Fn  K  <->  g  =  ( x  e.  K  |->  ( g `  x
) ) )
3525fneq2d 5663 . . . . . . . . . 10  |-  ( ph  ->  ( g  Fn  K  <->  g  Fn  U. J ) )
3634, 35syl5rbbr 260 . . . . . . . . 9  |-  ( ph  ->  ( g  Fn  U. J 
<->  g  =  ( x  e.  K  |->  ( g `
 x ) ) ) )
3736biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  g  Fn  U. J )  ->  g  =  ( x  e.  K  |->  ( g `  x ) ) )
3812, 33, 37syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  g  =  ( x  e.  K  |->  ( g `  x ) ) )
397, 9, 11, 29, 38offval2 6531 . . . . . 6  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
f  oF ( +g  `  R ) g )  =  ( x  e.  K  |->  ( ( f `  x
) ( +g  `  R
) ( g `  x ) ) ) )
40233ad2ant1 1012 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  J  e.  (TopOn `  K )
)
41 simp2 992 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  f  e.  ( J  Cn  J
) )
42 eleq1 2532 . . . . . . . . 9  |-  ( f  =  ( x  e.  K  |->  ( f `  x ) )  -> 
( f  e.  ( J  Cn  J )  <-> 
( x  e.  K  |->  ( f `  x
) )  e.  ( J  Cn  J ) ) )
4342biimpa 484 . . . . . . . 8  |-  ( ( f  =  ( x  e.  K  |->  ( f `
 x ) )  /\  f  e.  ( J  Cn  J ) )  ->  ( x  e.  K  |->  ( f `
 x ) )  e.  ( J  Cn  J ) )
4429, 41, 43syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
x  e.  K  |->  ( f `  x ) )  e.  ( J  Cn  J ) )
45 simp3 993 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  g  e.  ( J  Cn  J
) )
46 eleq1 2532 . . . . . . . . 9  |-  ( g  =  ( x  e.  K  |->  ( g `  x ) )  -> 
( g  e.  ( J  Cn  J )  <-> 
( x  e.  K  |->  ( g `  x
) )  e.  ( J  Cn  J ) ) )
4746biimpa 484 . . . . . . . 8  |-  ( ( g  =  ( x  e.  K  |->  ( g `
 x ) )  /\  g  e.  ( J  Cn  J ) )  ->  ( x  e.  K  |->  ( g `
 x ) )  e.  ( J  Cn  J ) )
4838, 45, 47syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
x  e.  K  |->  ( g `  x ) )  e.  ( J  Cn  J ) )
49 eqid 2460 . . . . . . . . . 10  |-  ( +f `  R )  =  ( +f `  R )
501, 2, 49plusffval 15733 . . . . . . . . 9  |-  ( +f `  R )  =  ( y  e.  K ,  z  e.  K  |->  ( y ( +g  `  R ) z ) )
5121, 49tgpcn 20311 . . . . . . . . . 10  |-  ( R  e.  TopGrp  ->  ( +f `  R )  e.  ( ( J  tX  J
)  Cn  J ) )
5219, 20, 513syl 20 . . . . . . . . 9  |-  ( ph  ->  ( +f `  R )  e.  ( ( J  tX  J
)  Cn  J ) )
5350, 52syl5eqelr 2553 . . . . . . . 8  |-  ( ph  ->  ( y  e.  K ,  z  e.  K  |->  ( y ( +g  `  R ) z ) )  e.  ( ( J  tX  J )  Cn  J ) )
54533ad2ant1 1012 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
y  e.  K , 
z  e.  K  |->  ( y ( +g  `  R
) z ) )  e.  ( ( J 
tX  J )  Cn  J ) )
55 oveq12 6284 . . . . . . 7  |-  ( ( y  =  ( f `
 x )  /\  z  =  ( g `  x ) )  -> 
( y ( +g  `  R ) z )  =  ( ( f `
 x ) ( +g  `  R ) ( g `  x
) ) )
5640, 44, 48, 40, 40, 54, 55cnmpt12 19896 . . . . . 6  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
x  e.  K  |->  ( ( f `  x
) ( +g  `  R
) ( g `  x ) ) )  e.  ( J  Cn  J ) )
5739, 56eqeltrd 2548 . . . . 5  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
f  oF ( +g  `  R ) g )  e.  ( J  Cn  J ) )
58573adant2l 1217 . . . 4  |-  ( (
ph  /\  ( f  e.  ran  (eval1 `  R )  /\  f  e.  ( J  Cn  J ) )  /\  g  e.  ( J  Cn  J ) )  -> 
( f  oF ( +g  `  R
) g )  e.  ( J  Cn  J
) )
59583adant3l 1219 . . 3  |-  ( (
ph  /\  ( f  e.  ran  (eval1 `  R )  /\  f  e.  ( J  Cn  J ) )  /\  ( g  e.  ran  (eval1 `  R )  /\  g  e.  ( J  Cn  J
) ) )  -> 
( f  oF ( +g  `  R
) g )  e.  ( J  Cn  J
) )
60593expb 1192 . 2  |-  ( (
ph  /\  ( (
f  e.  ran  (eval1 `  R )  /\  f  e.  ( J  Cn  J
) )  /\  (
g  e.  ran  (eval1 `  R )  /\  g  e.  ( J  Cn  J
) ) ) )  ->  ( f  oF ( +g  `  R
) g )  e.  ( J  Cn  J
) )
617, 9, 11, 29, 38offval2 6531 . . . . . 6  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
f  oF ( .r `  R ) g )  =  ( x  e.  K  |->  ( ( f `  x
) ( .r `  R ) ( g `
 x ) ) ) )
62 eqid 2460 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
6362, 1mgpbas 16930 . . . . . . . . . 10  |-  K  =  ( Base `  (mulGrp `  R ) )
6462, 3mgpplusg 16928 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
65 eqid 2460 . . . . . . . . . 10  |-  ( +f `  (mulGrp `  R ) )  =  ( +f `  (mulGrp `  R ) )
6663, 64, 65plusffval 15733 . . . . . . . . 9  |-  ( +f `  (mulGrp `  R ) )  =  ( y  e.  K ,  z  e.  K  |->  ( y ( .r
`  R ) z ) )
6721, 65mulrcn 20409 . . . . . . . . . 10  |-  ( R  e.  TopRing  ->  ( +f `  (mulGrp `  R )
)  e.  ( ( J  tX  J )  Cn  J ) )
6819, 67syl 16 . . . . . . . . 9  |-  ( ph  ->  ( +f `  (mulGrp `  R ) )  e.  ( ( J 
tX  J )  Cn  J ) )
6966, 68syl5eqelr 2553 . . . . . . . 8  |-  ( ph  ->  ( y  e.  K ,  z  e.  K  |->  ( y ( .r
`  R ) z ) )  e.  ( ( J  tX  J
)  Cn  J ) )
70693ad2ant1 1012 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
y  e.  K , 
z  e.  K  |->  ( y ( .r `  R ) z ) )  e.  ( ( J  tX  J )  Cn  J ) )
71 oveq12 6284 . . . . . . 7  |-  ( ( y  =  ( f `
 x )  /\  z  =  ( g `  x ) )  -> 
( y ( .r
`  R ) z )  =  ( ( f `  x ) ( .r `  R
) ( g `  x ) ) )
7240, 44, 48, 40, 40, 70, 71cnmpt12 19896 . . . . . 6  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
x  e.  K  |->  ( ( f `  x
) ( .r `  R ) ( g `
 x ) ) )  e.  ( J  Cn  J ) )
7361, 72eqeltrd 2548 . . . . 5  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
f  oF ( .r `  R ) g )  e.  ( J  Cn  J ) )
74733adant2l 1217 . . . 4  |-  ( (
ph  /\  ( f  e.  ran  (eval1 `  R )  /\  f  e.  ( J  Cn  J ) )  /\  g  e.  ( J  Cn  J ) )  -> 
( f  oF ( .r `  R
) g )  e.  ( J  Cn  J
) )
75743adant3l 1219 . . 3  |-  ( (
ph  /\  ( f  e.  ran  (eval1 `  R )  /\  f  e.  ( J  Cn  J ) )  /\  ( g  e.  ran  (eval1 `  R )  /\  g  e.  ( J  Cn  J
) ) )  -> 
( f  oF ( .r `  R
) g )  e.  ( J  Cn  J
) )
76753expb 1192 . 2  |-  ( (
ph  /\  ( (
f  e.  ran  (eval1 `  R )  /\  f  e.  ( J  Cn  J
) )  /\  (
g  e.  ran  (eval1 `  R )  /\  g  e.  ( J  Cn  J
) ) ) )  ->  ( f  oF ( .r `  R ) g )  e.  ( J  Cn  J ) )
77 eleq1 2532 . 2  |-  ( h  =  ( K  X.  { f } )  ->  ( h  e.  ( J  Cn  J
)  <->  ( K  X.  { f } )  e.  ( J  Cn  J ) ) )
78 eleq1 2532 . 2  |-  ( h  =  (  _I  |`  K )  ->  ( h  e.  ( J  Cn  J
)  <->  (  _I  |`  K )  e.  ( J  Cn  J ) ) )
79 eleq1 2532 . 2  |-  ( h  =  f  ->  (
h  e.  ( J  Cn  J )  <->  f  e.  ( J  Cn  J
) ) )
80 eleq1 2532 . 2  |-  ( h  =  g  ->  (
h  e.  ( J  Cn  J )  <->  g  e.  ( J  Cn  J
) ) )
81 eleq1 2532 . 2  |-  ( h  =  ( f  oF ( +g  `  R
) g )  -> 
( h  e.  ( J  Cn  J )  <-> 
( f  oF ( +g  `  R
) g )  e.  ( J  Cn  J
) ) )
82 eleq1 2532 . 2  |-  ( h  =  ( f  oF ( .r `  R ) g )  ->  ( h  e.  ( J  Cn  J
)  <->  ( f  oF ( .r `  R ) g )  e.  ( J  Cn  J ) ) )
83 eleq1 2532 . 2  |-  ( h  =  ( E `  F )  ->  (
h  e.  ( J  Cn  J )  <->  ( E `  F )  e.  ( J  Cn  J ) ) )
8423adantr 465 . . 3  |-  ( (
ph  /\  f  e.  K )  ->  J  e.  (TopOn `  K )
)
85 simpr 461 . . 3  |-  ( (
ph  /\  f  e.  K )  ->  f  e.  K )
86 cnconst2 19543 . . 3  |-  ( ( J  e.  (TopOn `  K )  /\  J  e.  (TopOn `  K )  /\  f  e.  K
)  ->  ( K  X.  { f } )  e.  ( J  Cn  J ) )
8784, 84, 85, 86syl3anc 1223 . 2  |-  ( (
ph  /\  f  e.  K )  ->  ( K  X.  { f } )  e.  ( J  Cn  J ) )
88 idcn 19517 . . 3  |-  ( J  e.  (TopOn `  K
)  ->  (  _I  |`  K )  e.  ( J  Cn  J ) )
8923, 88syl 16 . 2  |-  ( ph  ->  (  _I  |`  K )  e.  ( J  Cn  J ) )
90 pl1cn.1 . . . . 5  |-  ( ph  ->  R  e.  CRing )
91 pl1cn.e . . . . . . 7  |-  E  =  (eval1 `  R )
92 pl1cn.p . . . . . . 7  |-  P  =  (Poly1 `  R )
93 eqid 2460 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
9491, 92, 93, 1evl1rhm 18132 . . . . . 6  |-  ( R  e.  CRing  ->  E  e.  ( P RingHom  ( R  ^s  K
) ) )
95 pl1cn.b . . . . . . 7  |-  B  =  ( Base `  P
)
96 eqid 2460 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
9795, 96rhmf 17152 . . . . . 6  |-  ( E  e.  ( P RingHom  ( R  ^s  K ) )  ->  E : B --> ( Base `  ( R  ^s  K ) ) )
98 ffn 5722 . . . . . 6  |-  ( E : B --> ( Base `  ( R  ^s  K ) )  ->  E  Fn  B )
99 dffn3 5729 . . . . . . 7  |-  ( E  Fn  B  <->  E : B
--> ran  E )
10099biimpi 194 . . . . . 6  |-  ( E  Fn  B  ->  E : B --> ran  E )
10194, 97, 98, 1004syl 21 . . . . 5  |-  ( R  e.  CRing  ->  E : B
--> ran  E )
10290, 101syl 16 . . . 4  |-  ( ph  ->  E : B --> ran  E
)
103 pl1cn.3 . . . 4  |-  ( ph  ->  F  e.  B )
104102, 103ffvelrnd 6013 . . 3  |-  ( ph  ->  ( E `  F
)  e.  ran  E
)
10591rneqi 5220 . . 3  |-  ran  E  =  ran  (eval1 `  R )
106104, 105syl6eleq 2558 . 2  |-  ( ph  ->  ( E `  F
)  e.  ran  (eval1 `  R ) )
1071, 2, 3, 4, 60, 76, 77, 78, 79, 80, 81, 82, 83, 87, 89, 106pf1ind 18155 1  |-  ( ph  ->  ( E `  F
)  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3106   {csn 4020   U.cuni 4238    |-> cmpt 4498    _I cid 4783    X. cxp 4990   ran crn 4993    |` cres 4994    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277    oFcof 6513   Basecbs 14479   +g cplusg 14544   .rcmulr 14545   TopOpenctopn 14666    ^s cpws 14691   +fcplusf 15718  mulGrpcmgp 16924   CRingccrg 16980   RingHom crh 17138  Poly1cpl1 17980  eval1ce1 18115  TopOnctopon 19155    Cn ccn 19484    tX ctx 19789   TopGrpctgp 20298   TopRingctrg 20386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-ofr 6516  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-prds 14692  df-pws 14694  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-plusf 15722  df-mhm 15770  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mulg 15854  df-subg 15986  df-ghm 16053  df-cntz 16143  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-srg 16941  df-rng 16981  df-cring 16982  df-rnghom 17141  df-subrg 17203  df-lmod 17290  df-lss 17355  df-lsp 17394  df-assa 17725  df-asp 17726  df-ascl 17727  df-psr 17769  df-mvr 17770  df-mpl 17771  df-opsr 17773  df-evls 17935  df-evl 17936  df-psr1 17983  df-ply1 17985  df-evl1 18117  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cn 19487  df-cnp 19488  df-tx 19791  df-tmd 20299  df-tgp 20300  df-trg 20390
This theorem is referenced by: (None)
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