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Theorem pl1cn 26407
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 5722 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
61, 5eqeltri 2513 . . . . . . . 8  |-  K  e. 
_V
76a1i 11 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  K  e.  _V )
8 fvex 5722 . . . . . . . 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 5722 . . . . . . . 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 988 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  ph )
13 eqid 2443 . . . . . . . . . . 11  |-  U. J  =  U. J
1413, 13cnf 18872 . . . . . . . . . 10  |-  ( f  e.  ( J  Cn  J )  ->  f : U. J --> U. J
)
15 ffn 5580 . . . . . . . . . 10  |-  ( f : U. J --> U. J  ->  f  Fn  U. J
)
1614, 15syl 16 . . . . . . . . 9  |-  ( f  e.  ( J  Cn  J )  ->  f  Fn  U. J )
17163ad2ant2 1010 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  f  Fn  U. J )
18 dffn5 5758 . . . . . . . . . 10  |-  ( f  Fn  K  <->  f  =  ( x  e.  K  |->  ( f `  x
) ) )
19 pl1cn.2 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  TopRing )
20 trgtgp 19764 . . . . . . . . . . . . 13  |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
21 pl1cn.j . . . . . . . . . . . . . 14  |-  J  =  ( TopOpen `  R )
2221, 1tgptopon 19675 . . . . . . . . . . . . 13  |-  ( R  e.  TopGrp  ->  J  e.  (TopOn `  K ) )
2319, 20, 223syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  (TopOn `  K ) )
24 toponuni 18554 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  K
)  ->  K  =  U. J )
2523, 24syl 16 . . . . . . . . . . 11  |-  ( ph  ->  K  =  U. J
)
2625fneq2d 5523 . . . . . . . . . 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 18872 . . . . . . . . . 10  |-  ( g  e.  ( J  Cn  J )  ->  g : U. J --> U. J
)
31 ffn 5580 . . . . . . . . . 10  |-  ( g : U. J --> U. J  ->  g  Fn  U. J
)
3230, 31syl 16 . . . . . . . . 9  |-  ( g  e.  ( J  Cn  J )  ->  g  Fn  U. J )
33323ad2ant3 1011 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  g  Fn  U. J )
34 dffn5 5758 . . . . . . . . . 10  |-  ( g  Fn  K  <->  g  =  ( x  e.  K  |->  ( g `  x
) ) )
3525fneq2d 5523 . . . . . . . . . 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 6357 . . . . . 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 1009 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  J  e.  (TopOn `  K )
)
41 simp2 989 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  f  e.  ( J  Cn  J
) )
42 eleq1 2503 . . . . . . . . 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 990 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  g  e.  ( J  Cn  J
) )
46 eleq1 2503 . . . . . . . . 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 2443 . . . . . . . . . 10  |-  ( +f `  R )  =  ( +f `  R )
501, 2, 49plusffval 15448 . . . . . . . . 9  |-  ( +f `  R )  =  ( y  e.  K ,  z  e.  K  |->  ( y ( +g  `  R ) z ) )
5121, 49tgpcn 19677 . . . . . . . . . 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 2528 . . . . . . . 8  |-  ( ph  ->  ( y  e.  K ,  z  e.  K  |->  ( y ( +g  `  R ) z ) )  e.  ( ( J  tX  J )  Cn  J ) )
54533ad2ant1 1009 . . . . . . 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 6121 . . . . . . 7  |-  ( ( y  =  ( f `
 x )  /\  z  =  ( g `  x ) )  -> 
( y ( +g  `  R ) z )  =  ( ( f `
 x ) ( +g  `  R ) ( g `  x
) ) )
5640, 44, 48, 40, 40, 54, 55cnmpt12 19262 . . . . . 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 2517 . . . . 5  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
f  oF ( +g  `  R ) g )  e.  ( J  Cn  J ) )
58573adant2l 1212 . . . 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 1214 . . 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 1188 . 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 6357 . . . . . 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 2443 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
6362, 1mgpbas 16619 . . . . . . . . . 10  |-  K  =  ( Base `  (mulGrp `  R ) )
6462, 3mgpplusg 16617 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
65 eqid 2443 . . . . . . . . . 10  |-  ( +f `  (mulGrp `  R ) )  =  ( +f `  (mulGrp `  R ) )
6663, 64, 65plusffval 15448 . . . . . . . . 9  |-  ( +f `  (mulGrp `  R ) )  =  ( y  e.  K ,  z  e.  K  |->  ( y ( .r
`  R ) z ) )
6721, 65mulrcn 19775 . . . . . . . . . 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 2528 . . . . . . . 8  |-  ( ph  ->  ( y  e.  K ,  z  e.  K  |->  ( y ( .r
`  R ) z ) )  e.  ( ( J  tX  J
)  Cn  J ) )
70693ad2ant1 1009 . . . . . . 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 6121 . . . . . . 7  |-  ( ( y  =  ( f `
 x )  /\  z  =  ( g `  x ) )  -> 
( y ( .r
`  R ) z )  =  ( ( f `  x ) ( .r `  R
) ( g `  x ) ) )
7240, 44, 48, 40, 40, 70, 71cnmpt12 19262 . . . . . 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 2517 . . . . 5  |-  ( (
ph  /\  f  e.  ( J  Cn  J
)  /\  g  e.  ( J  Cn  J
) )  ->  (
f  oF ( .r `  R ) g )  e.  ( J  Cn  J ) )
74733adant2l 1212 . . . 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 1214 . . 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 1188 . 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 2503 . 2  |-  ( h  =  ( K  X.  { f } )  ->  ( h  e.  ( J  Cn  J
)  <->  ( K  X.  { f } )  e.  ( J  Cn  J ) ) )
78 eleq1 2503 . 2  |-  ( h  =  (  _I  |`  K )  ->  ( h  e.  ( J  Cn  J
)  <->  (  _I  |`  K )  e.  ( J  Cn  J ) ) )
79 eleq1 2503 . 2  |-  ( h  =  f  ->  (
h  e.  ( J  Cn  J )  <->  f  e.  ( J  Cn  J
) ) )
80 eleq1 2503 . 2  |-  ( h  =  g  ->  (
h  e.  ( J  Cn  J )  <->  g  e.  ( J  Cn  J
) ) )
81 eleq1 2503 . 2  |-  ( h  =  ( f  oF ( +g  `  R
) g )  -> 
( h  e.  ( J  Cn  J )  <-> 
( f  oF ( +g  `  R
) g )  e.  ( J  Cn  J
) ) )
82 eleq1 2503 . 2  |-  ( h  =  ( f  oF ( .r `  R ) g )  ->  ( h  e.  ( J  Cn  J
)  <->  ( f  oF ( .r `  R ) g )  e.  ( J  Cn  J ) ) )
83 eleq1 2503 . 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 18909 . . 3  |-  ( ( J  e.  (TopOn `  K )  /\  J  e.  (TopOn `  K )  /\  f  e.  K
)  ->  ( K  X.  { f } )  e.  ( J  Cn  J ) )
8784, 84, 85, 86syl3anc 1218 . 2  |-  ( (
ph  /\  f  e.  K )  ->  ( K  X.  { f } )  e.  ( J  Cn  J ) )
88 idcn 18883 . . 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 2443 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
9491, 92, 93, 1evl1rhm 17788 . . . . . 6  |-  ( R  e.  CRing  ->  E  e.  ( P RingHom  ( R  ^s  K
) ) )
95 pl1cn.b . . . . . . 7  |-  B  =  ( Base `  P
)
96 eqid 2443 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
9795, 96rhmf 16838 . . . . . 6  |-  ( E  e.  ( P RingHom  ( R  ^s  K ) )  ->  E : B --> ( Base `  ( R  ^s  K ) ) )
98 ffn 5580 . . . . . 6  |-  ( E : B --> ( Base `  ( R  ^s  K ) )  ->  E  Fn  B )
99 dffn3 5587 . . . . . . 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 5865 . . 3  |-  ( ph  ->  ( E `  F
)  e.  ran  E
)
10591rneqi 5087 . . 3  |-  ran  E  =  ran  (eval1 `  R )
106104, 105syl6eleq 2533 . 2  |-  ( ph  ->  ( E `  F
)  e.  ran  (eval1 `  R ) )
1071, 2, 3, 4, 60, 76, 77, 78, 79, 80, 81, 82, 83, 87, 89, 106pf1ind 17811 1  |-  ( ph  ->  ( E `  F
)  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2993   {csn 3898   U.cuni 4112    e. cmpt 4371    _I cid 4652    X. cxp 4859   ran crn 4862    |` cres 4863    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114    oFcof 6339   Basecbs 14195   +g cplusg 14259   .rcmulr 14260   TopOpenctopn 14381    ^s cpws 14406   +fcplusf 15433  mulGrpcmgp 16613   CRingccrg 16668   RingHom crh 16826  Poly1cpl1 17655  eval1ce1 17771  TopOnctopon 18521    Cn ccn 18850    tX ctx 19155   TopGrpctgp 19664   TopRingctrg 19752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-ofr 6342  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-fz 11459  df-fzo 11570  df-seq 11828  df-hash 12125  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-hom 14283  df-cco 14284  df-rest 14382  df-topn 14383  df-0g 14401  df-gsum 14402  df-topgen 14403  df-prds 14407  df-pws 14409  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-plusf 15437  df-mhm 15485  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-mulg 15569  df-subg 15699  df-ghm 15766  df-cntz 15856  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-srg 16630  df-rng 16669  df-cring 16670  df-rnghom 16828  df-subrg 16885  df-lmod 16972  df-lss 17036  df-lsp 17075  df-assa 17406  df-asp 17407  df-ascl 17408  df-psr 17445  df-mvr 17446  df-mpl 17447  df-opsr 17449  df-evls 17610  df-evl 17611  df-psr1 17658  df-ply1 17660  df-evl1 17773  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cn 18853  df-cnp 18854  df-tx 19157  df-tmd 19665  df-tgp 19666  df-trg 19756
This theorem is referenced by: (None)
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