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Theorem pl1cn 27915
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 2443 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
3 eqid 2443 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
4 eqid 2443 . 2  |-  ran  (eval1 `  R )  =  ran  (eval1 `  R )
5 fvex 5866 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
61, 5eqeltri 2527 . . . . . . . 8  |-  K  e. 
_V
76a1i 11 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  K  e.  _V )
8 fvex 5866 . . . . . . . 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 5866 . . . . . . . 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 997 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  ph )
13 eqid 2443 . . . . . . . . . . 11  |-  U. J  =  U. J
1413, 13cnf 19725 . . . . . . . . . 10  |-  ( f  e.  ( J  Cn  J )  ->  f : U. J --> U. J
)
15 ffn 5721 . . . . . . . . . 10  |-  ( f : U. J --> U. J  ->  f  Fn  U. J
)
1614, 15syl 16 . . . . . . . . 9  |-  ( f  e.  ( J  Cn  J )  ->  f  Fn  U. J )
17163ad2ant2 1019 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  f  Fn  U. J )
18 dffn5 5903 . . . . . . . . . 10  |-  ( f  Fn  K  <->  f  =  ( x  e.  K  |->  ( f `  x
) ) )
19 pl1cn.2 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  TopRing )
20 trgtgp 20648 . . . . . . . . . . . . 13  |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
21 pl1cn.j . . . . . . . . . . . . . 14  |-  J  =  ( TopOpen `  R )
2221, 1tgptopon 20559 . . . . . . . . . . . . 13  |-  ( R  e.  TopGrp  ->  J  e.  (TopOn `  K ) )
2319, 20, 223syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  (TopOn `  K ) )
24 toponuni 19406 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  K
)  ->  K  =  U. J )
2523, 24syl 16 . . . . . . . . . . 11  |-  ( ph  ->  K  =  U. J
)
2625fneq2d 5662 . . . . . . . . . 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 19725 . . . . . . . . . 10  |-  ( g  e.  ( J  Cn  J )  ->  g : U. J --> U. J
)
31 ffn 5721 . . . . . . . . . 10  |-  ( g : U. J --> U. J  ->  g  Fn  U. J
)
3230, 31syl 16 . . . . . . . . 9  |-  ( g  e.  ( J  Cn  J )  ->  g  Fn  U. J )
33323ad2ant3 1020 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  g  Fn  U. J )
34 dffn5 5903 . . . . . . . . . 10  |-  ( g  Fn  K  <->  g  =  ( x  e.  K  |->  ( g `  x
) ) )
3525fneq2d 5662 . . . . . . . . . 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 6541 . . . . . 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 1018 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  J  e.  (TopOn `  K )
)
41 simp2 998 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  f  e.  ( J  Cn  J
) )
4229, 41eqeltrrd 2532 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
x  e.  K  |->  ( f `  x ) )  e.  ( J  Cn  J ) )
43 simp3 999 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  g  e.  ( J  Cn  J
) )
4438, 43eqeltrrd 2532 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
x  e.  K  |->  ( g `  x ) )  e.  ( J  Cn  J ) )
45 eqid 2443 . . . . . . . . . 10  |-  ( +f `  R )  =  ( +f `  R )
461, 2, 45plusffval 15856 . . . . . . . . 9  |-  ( +f `  R )  =  ( y  e.  K ,  z  e.  K  |->  ( y ( +g  `  R ) z ) )
4721, 45tgpcn 20561 . . . . . . . . . 10  |-  ( R  e.  TopGrp  ->  ( +f `  R )  e.  ( ( J  tX  J
)  Cn  J ) )
4819, 20, 473syl 20 . . . . . . . . 9  |-  ( ph  ->  ( +f `  R )  e.  ( ( J  tX  J
)  Cn  J ) )
4946, 48syl5eqelr 2536 . . . . . . . 8  |-  ( ph  ->  ( y  e.  K ,  z  e.  K  |->  ( y ( +g  `  R ) z ) )  e.  ( ( J  tX  J )  Cn  J ) )
50493ad2ant1 1018 . . . . . . 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 ) )
51 oveq12 6290 . . . . . . 7  |-  ( ( y  =  ( f `
 x )  /\  z  =  ( g `  x ) )  -> 
( y ( +g  `  R ) z )  =  ( ( f `
 x ) ( +g  `  R ) ( g `  x
) ) )
5240, 42, 44, 40, 40, 50, 51cnmpt12 20146 . . . . . 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 ) )
5339, 52eqeltrd 2531 . . . . 5  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
f  oF ( +g  `  R ) g )  e.  ( J  Cn  J ) )
54533adant2l 1223 . . . 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
) )
55543adant3l 1225 . . 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
) )
56553expb 1198 . 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
) )
577, 9, 11, 29, 38offval2 6541 . . . . . 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 ) ) ) )
58 eqid 2443 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
5958, 1mgpbas 17126 . . . . . . . . . 10  |-  K  =  ( Base `  (mulGrp `  R ) )
6058, 3mgpplusg 17124 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
61 eqid 2443 . . . . . . . . . 10  |-  ( +f `  (mulGrp `  R ) )  =  ( +f `  (mulGrp `  R ) )
6259, 60, 61plusffval 15856 . . . . . . . . 9  |-  ( +f `  (mulGrp `  R ) )  =  ( y  e.  K ,  z  e.  K  |->  ( y ( .r
`  R ) z ) )
6321, 61mulrcn 20659 . . . . . . . . . 10  |-  ( R  e.  TopRing  ->  ( +f `  (mulGrp `  R )
)  e.  ( ( J  tX  J )  Cn  J ) )
6419, 63syl 16 . . . . . . . . 9  |-  ( ph  ->  ( +f `  (mulGrp `  R ) )  e.  ( ( J 
tX  J )  Cn  J ) )
6562, 64syl5eqelr 2536 . . . . . . . 8  |-  ( ph  ->  ( y  e.  K ,  z  e.  K  |->  ( y ( .r
`  R ) z ) )  e.  ( ( J  tX  J
)  Cn  J ) )
66653ad2ant1 1018 . . . . . . 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 ) )
67 oveq12 6290 . . . . . . 7  |-  ( ( y  =  ( f `
 x )  /\  z  =  ( g `  x ) )  -> 
( y ( .r
`  R ) z )  =  ( ( f `  x ) ( .r `  R
) ( g `  x ) ) )
6840, 42, 44, 40, 40, 66, 67cnmpt12 20146 . . . . . 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 ) )
6957, 68eqeltrd 2531 . . . . 5  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
f  oF ( .r `  R ) g )  e.  ( J  Cn  J ) )
70693adant2l 1223 . . . 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
) )
71703adant3l 1225 . . 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
) )
72713expb 1198 . 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 ) )
73 eleq1 2515 . 2  |-  ( h  =  ( K  X.  { f } )  ->  ( h  e.  ( J  Cn  J
)  <->  ( K  X.  { f } )  e.  ( J  Cn  J ) ) )
74 eleq1 2515 . 2  |-  ( h  =  (  _I  |`  K )  ->  ( h  e.  ( J  Cn  J
)  <->  (  _I  |`  K )  e.  ( J  Cn  J ) ) )
75 eleq1 2515 . 2  |-  ( h  =  f  ->  (
h  e.  ( J  Cn  J )  <->  f  e.  ( J  Cn  J
) ) )
76 eleq1 2515 . 2  |-  ( h  =  g  ->  (
h  e.  ( J  Cn  J )  <->  g  e.  ( J  Cn  J
) ) )
77 eleq1 2515 . 2  |-  ( h  =  ( f  oF ( +g  `  R
) g )  -> 
( h  e.  ( J  Cn  J )  <-> 
( f  oF ( +g  `  R
) g )  e.  ( J  Cn  J
) ) )
78 eleq1 2515 . 2  |-  ( h  =  ( f  oF ( .r `  R ) g )  ->  ( h  e.  ( J  Cn  J
)  <->  ( f  oF ( .r `  R ) g )  e.  ( J  Cn  J ) ) )
79 eleq1 2515 . 2  |-  ( h  =  ( E `  F )  ->  (
h  e.  ( J  Cn  J )  <->  ( E `  F )  e.  ( J  Cn  J ) ) )
8023adantr 465 . . 3  |-  ( (
ph  /\  f  e.  K )  ->  J  e.  (TopOn `  K )
)
81 simpr 461 . . 3  |-  ( (
ph  /\  f  e.  K )  ->  f  e.  K )
82 cnconst2 19762 . . 3  |-  ( ( J  e.  (TopOn `  K )  /\  J  e.  (TopOn `  K )  /\  f  e.  K
)  ->  ( K  X.  { f } )  e.  ( J  Cn  J ) )
8380, 80, 81, 82syl3anc 1229 . 2  |-  ( (
ph  /\  f  e.  K )  ->  ( K  X.  { f } )  e.  ( J  Cn  J ) )
84 idcn 19736 . . 3  |-  ( J  e.  (TopOn `  K
)  ->  (  _I  |`  K )  e.  ( J  Cn  J ) )
8523, 84syl 16 . 2  |-  ( ph  ->  (  _I  |`  K )  e.  ( J  Cn  J ) )
86 pl1cn.1 . . . . 5  |-  ( ph  ->  R  e.  CRing )
87 pl1cn.e . . . . . . 7  |-  E  =  (eval1 `  R )
88 pl1cn.p . . . . . . 7  |-  P  =  (Poly1 `  R )
89 eqid 2443 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
9087, 88, 89, 1evl1rhm 18347 . . . . . 6  |-  ( R  e.  CRing  ->  E  e.  ( P RingHom  ( R  ^s  K
) ) )
91 pl1cn.b . . . . . . 7  |-  B  =  ( Base `  P
)
92 eqid 2443 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
9391, 92rhmf 17354 . . . . . 6  |-  ( E  e.  ( P RingHom  ( R  ^s  K ) )  ->  E : B --> ( Base `  ( R  ^s  K ) ) )
94 ffn 5721 . . . . . 6  |-  ( E : B --> ( Base `  ( R  ^s  K ) )  ->  E  Fn  B )
95 dffn3 5728 . . . . . . 7  |-  ( E  Fn  B  <->  E : B
--> ran  E )
9695biimpi 194 . . . . . 6  |-  ( E  Fn  B  ->  E : B --> ran  E )
9790, 93, 94, 964syl 21 . . . . 5  |-  ( R  e.  CRing  ->  E : B
--> ran  E )
9886, 97syl 16 . . . 4  |-  ( ph  ->  E : B --> ran  E
)
99 pl1cn.3 . . . 4  |-  ( ph  ->  F  e.  B )
10098, 99ffvelrnd 6017 . . 3  |-  ( ph  ->  ( E `  F
)  e.  ran  E
)
10187rneqi 5219 . . 3  |-  ran  E  =  ran  (eval1 `  R )
102100, 101syl6eleq 2541 . 2  |-  ( ph  ->  ( E `  F
)  e.  ran  (eval1 `  R ) )
1031, 2, 3, 4, 56, 72, 73, 74, 75, 76, 77, 78, 79, 83, 85, 102pf1ind 18370 1  |-  ( ph  ->  ( E `  F
)  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   _Vcvv 3095   {csn 4014   U.cuni 4234    |-> cmpt 4495    _I cid 4780    X. cxp 4987   ran crn 4990    |` cres 4991    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283    oFcof 6523   Basecbs 14614   +g cplusg 14679   .rcmulr 14680   TopOpenctopn 14801    ^s cpws 14826   +fcplusf 15848  mulGrpcmgp 17120   CRingccrg 17178   RingHom crh 17340  Poly1cpl1 18195  eval1ce1 18330  TopOnctopon 19373    Cn ccn 19703    tX ctx 20039   TopGrpctgp 20548   TopRingctrg 20636
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  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-iin 4318  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-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  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-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-fz 11684  df-fzo 11807  df-seq 12090  df-hash 12388  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-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-prds 14827  df-pws 14829  df-mre 14965  df-mrc 14966  df-acs 14968  df-plusf 15850  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-mhm 15945  df-submnd 15946  df-grp 16036  df-minusg 16037  df-sbg 16038  df-mulg 16039  df-subg 16177  df-ghm 16244  df-cntz 16334  df-cmn 16779  df-abl 16780  df-mgp 17121  df-ur 17133  df-srg 17137  df-ring 17179  df-cring 17180  df-rnghom 17343  df-subrg 17406  df-lmod 17493  df-lss 17558  df-lsp 17597  df-assa 17940  df-asp 17941  df-ascl 17942  df-psr 17984  df-mvr 17985  df-mpl 17986  df-opsr 17988  df-evls 18150  df-evl 18151  df-psr1 18198  df-ply1 18200  df-evl1 18332  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cn 19706  df-cnp 19707  df-tx 20041  df-tmd 20549  df-tgp 20550  df-trg 20640
This theorem is referenced by: (None)
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