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Theorem hbtlem4 27330
Description: The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Hypotheses
Ref Expression
hbtlem.p  |-  P  =  (Poly1 `  R )
hbtlem.u  |-  U  =  (LIdeal `  P )
hbtlem.s  |-  S  =  (ldgIdlSeq `  R )
hbtlem4.r  |-  ( ph  ->  R  e.  Ring )
hbtlem4.i  |-  ( ph  ->  I  e.  U )
hbtlem4.x  |-  ( ph  ->  X  e.  NN0 )
hbtlem4.y  |-  ( ph  ->  Y  e.  NN0 )
hbtlem4.xy  |-  ( ph  ->  X  <_  Y )
Assertion
Ref Expression
hbtlem4  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  I ) `  Y ) )

Proof of Theorem hbtlem4
Dummy variables  a 
c  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbtlem4.r . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
21ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  R  e.  Ring )
3 hbtlem.p . . . . . . . . . 10  |-  P  =  (Poly1 `  R )
43ply1rng 16326 . . . . . . . . 9  |-  ( R  e.  Ring  ->  P  e. 
Ring )
52, 4syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  P  e.  Ring )
6 hbtlem4.i . . . . . . . . 9  |-  ( ph  ->  I  e.  U )
76ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  I  e.  U )
8 eqid 2283 . . . . . . . . . . 11  |-  (mulGrp `  P )  =  (mulGrp `  P )
98rngmgp 15347 . . . . . . . . . 10  |-  ( P  e.  Ring  ->  (mulGrp `  P )  e.  Mnd )
105, 9syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  (mulGrp `  P
)  e.  Mnd )
11 hbtlem4.x . . . . . . . . . . 11  |-  ( ph  ->  X  e.  NN0 )
1211ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  e.  NN0 )
13 hbtlem4.y . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  NN0 )
1413ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  Y  e.  NN0 )
15 hbtlem4.xy . . . . . . . . . . 11  |-  ( ph  ->  X  <_  Y )
1615ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  <_  Y )
17 nn0sub2 10077 . . . . . . . . . 10  |-  ( ( X  e.  NN0  /\  Y  e.  NN0  /\  X  <_  Y )  ->  ( Y  -  X )  e.  NN0 )
1812, 14, 16, 17syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( Y  -  X )  e.  NN0 )
19 eqid 2283 . . . . . . . . . . 11  |-  (var1 `  R
)  =  (var1 `  R
)
20 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  P )
2119, 3, 20vr1cl 16294 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (var1 `  R
)  e.  ( Base `  P ) )
222, 21syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  (var1 `  R
)  e.  ( Base `  P ) )
238, 20mgpbas 15331 . . . . . . . . . 10  |-  ( Base `  P )  =  (
Base `  (mulGrp `  P
) )
24 eqid 2283 . . . . . . . . . 10  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
2523, 24mulgnn0cl 14583 . . . . . . . . 9  |-  ( ( (mulGrp `  P )  e.  Mnd  /\  ( Y  -  X )  e. 
NN0  /\  (var1 `  R
)  e.  ( Base `  P ) )  -> 
( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) )  e.  ( Base `  P
) )
2610, 18, 22, 25syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( Y  -  X )
(.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
27 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  c  e.  I )
28 hbtlem.u . . . . . . . . 9  |-  U  =  (LIdeal `  P )
29 eqid 2283 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
3028, 20, 29lidlmcl 15969 . . . . . . . 8  |-  ( ( ( P  e.  Ring  /\  I  e.  U )  /\  ( ( ( Y  -  X ) (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )  /\  c  e.  I
) )  ->  (
( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) ( .r `  P ) c )  e.  I
)
315, 7, 26, 27, 30syl22anc 1183 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( (
( Y  -  X
) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c )  e.  I )
32 eqid 2283 . . . . . . . . 9  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
3320, 28lidlss 15961 . . . . . . . . . . 11  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
347, 33syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  I  C_  ( Base `  P ) )
3534, 27sseldd 3181 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  c  e.  ( Base `  P )
)
3632, 3, 19, 8, 24deg1pwle 19505 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  ( Y  -  X )  e.  NN0 )  ->  (
( deg1  `
 R ) `  ( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )  <_  ( Y  -  X ) )
372, 18, 36syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  (
( Y  -  X
) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  <_ 
( Y  -  X
) )
38 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  c
)  <_  X )
393, 32, 2, 20, 29, 26, 35, 18, 12, 37, 38deg1mulle2 19495 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  (
( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) ( .r `  P ) c ) )  <_ 
( ( Y  -  X )  +  X
) )
4014nn0cnd 10020 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  Y  e.  CC )
4112nn0cnd 10020 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  e.  CC )
4240, 41npcand 9161 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( Y  -  X )  +  X )  =  Y )
4339, 42breqtrd 4047 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  (
( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) ( .r `  P ) c ) )  <_  Y )
44 eqid 2283 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
4544, 3, 19, 8, 24, 20, 29, 2, 35, 18, 12coe1pwmulfv 16356 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( (coe1 `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P )
) (var1 `  R ) ) ( .r `  P
) c ) ) `
 ( ( Y  -  X )  +  X ) )  =  ( (coe1 `  c ) `  X ) )
4642fveq2d 5529 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( (coe1 `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P )
) (var1 `  R ) ) ( .r `  P
) c ) ) `
 ( ( Y  -  X )  +  X ) )  =  ( (coe1 `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c ) ) `  Y
) )
4745, 46eqtr3d 2317 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( (coe1 `  c ) `  X
)  =  ( (coe1 `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P )
) (var1 `  R ) ) ( .r `  P
) c ) ) `
 Y ) )
48 fveq2 5525 . . . . . . . . . 10  |-  ( b  =  ( ( ( Y  -  X ) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c )  ->  ( ( deg1  `  R ) `  b
)  =  ( ( deg1  `  R ) `  (
( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) ( .r `  P ) c ) ) )
4948breq1d 4033 . . . . . . . . 9  |-  ( b  =  ( ( ( Y  -  X ) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c )  ->  ( (
( deg1  `
 R ) `  b )  <_  Y  <->  ( ( deg1  `  R ) `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P )
) (var1 `  R ) ) ( .r `  P
) c ) )  <_  Y ) )
50 fveq2 5525 . . . . . . . . . . 11  |-  ( b  =  ( ( ( Y  -  X ) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c )  ->  (coe1 `  b
)  =  (coe1 `  (
( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) ( .r `  P ) c ) ) )
5150fveq1d 5527 . . . . . . . . . 10  |-  ( b  =  ( ( ( Y  -  X ) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c )  ->  ( (coe1 `  b ) `  Y
)  =  ( (coe1 `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P )
) (var1 `  R ) ) ( .r `  P
) c ) ) `
 Y ) )
5251eqeq2d 2294 . . . . . . . . 9  |-  ( b  =  ( ( ( Y  -  X ) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c )  ->  ( (
(coe1 `  c ) `  X )  =  ( (coe1 `  b ) `  Y )  <->  ( (coe1 `  c ) `  X
)  =  ( (coe1 `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P )
) (var1 `  R ) ) ( .r `  P
) c ) ) `
 Y ) ) )
5349, 52anbi12d 691 . . . . . . . 8  |-  ( b  =  ( ( ( Y  -  X ) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c )  ->  ( (
( ( deg1  `  R ) `  b )  <_  Y  /\  ( (coe1 `  c ) `  X )  =  ( (coe1 `  b ) `  Y ) )  <->  ( (
( deg1  `
 R ) `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P )
) (var1 `  R ) ) ( .r `  P
) c ) )  <_  Y  /\  (
(coe1 `  c ) `  X )  =  ( (coe1 `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c ) ) `  Y
) ) ) )
5453rspcev 2884 . . . . . . 7  |-  ( ( ( ( ( Y  -  X ) (.g `  (mulGrp `  P )
) (var1 `  R ) ) ( .r `  P
) c )  e.  I  /\  ( ( ( deg1  `  R ) `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P )
) (var1 `  R ) ) ( .r `  P
) c ) )  <_  Y  /\  (
(coe1 `  c ) `  X )  =  ( (coe1 `  ( ( ( Y  -  X ) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c ) ) `  Y
) ) )  ->  E. b  e.  I 
( ( ( deg1  `  R
) `  b )  <_  Y  /\  ( (coe1 `  c ) `  X
)  =  ( (coe1 `  b ) `  Y
) ) )
5531, 43, 47, 54syl12anc 1180 . . . . . 6  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  E. b  e.  I  ( (
( deg1  `
 R ) `  b )  <_  Y  /\  ( (coe1 `  c ) `  X )  =  ( (coe1 `  b ) `  Y ) ) )
56 eqeq1 2289 . . . . . . . 8  |-  ( a  =  ( (coe1 `  c
) `  X )  ->  ( a  =  ( (coe1 `  b ) `  Y )  <->  ( (coe1 `  c ) `  X
)  =  ( (coe1 `  b ) `  Y
) ) )
5756anbi2d 684 . . . . . . 7  |-  ( a  =  ( (coe1 `  c
) `  X )  ->  ( ( ( ( deg1  `  R ) `  b
)  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y
) )  <->  ( (
( deg1  `
 R ) `  b )  <_  Y  /\  ( (coe1 `  c ) `  X )  =  ( (coe1 `  b ) `  Y ) ) ) )
5857rexbidv 2564 . . . . . 6  |-  ( a  =  ( (coe1 `  c
) `  X )  ->  ( E. b  e.  I  ( ( ( deg1  `  R ) `  b
)  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y
) )  <->  E. b  e.  I  ( (
( deg1  `
 R ) `  b )  <_  Y  /\  ( (coe1 `  c ) `  X )  =  ( (coe1 `  b ) `  Y ) ) ) )
5955, 58syl5ibrcom 213 . . . . 5  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( a  =  ( (coe1 `  c
) `  X )  ->  E. b  e.  I 
( ( ( deg1  `  R
) `  b )  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y ) ) ) )
6059expimpd 586 . . . 4  |-  ( (
ph  /\  c  e.  I )  ->  (
( ( ( deg1  `  R
) `  c )  <_  X  /\  a  =  ( (coe1 `  c ) `  X ) )  ->  E. b  e.  I 
( ( ( deg1  `  R
) `  b )  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y ) ) ) )
6160rexlimdva 2667 . . 3  |-  ( ph  ->  ( E. c  e.  I  ( ( ( deg1  `  R ) `  c
)  <_  X  /\  a  =  ( (coe1 `  c ) `  X
) )  ->  E. b  e.  I  ( (
( deg1  `
 R ) `  b )  <_  Y  /\  a  =  (
(coe1 `  b ) `  Y ) ) ) )
6261ss2abdv 3246 . 2  |-  ( ph  ->  { a  |  E. c  e.  I  (
( ( deg1  `  R ) `  c )  <_  X  /\  a  =  (
(coe1 `  c ) `  X ) ) } 
C_  { a  |  E. b  e.  I 
( ( ( deg1  `  R
) `  b )  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y ) ) } )
63 hbtlem.s . . . 4  |-  S  =  (ldgIdlSeq `  R )
643, 28, 63, 32hbtlem1 27327 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  U  /\  X  e. 
NN0 )  ->  (
( S `  I
) `  X )  =  { a  |  E. c  e.  I  (
( ( deg1  `  R ) `  c )  <_  X  /\  a  =  (
(coe1 `  c ) `  X ) ) } )
651, 6, 11, 64syl3anc 1182 . 2  |-  ( ph  ->  ( ( S `  I ) `  X
)  =  { a  |  E. c  e.  I  ( ( ( deg1  `  R ) `  c
)  <_  X  /\  a  =  ( (coe1 `  c ) `  X
) ) } )
663, 28, 63, 32hbtlem1 27327 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  U  /\  Y  e. 
NN0 )  ->  (
( S `  I
) `  Y )  =  { a  |  E. b  e.  I  (
( ( deg1  `  R ) `  b )  <_  Y  /\  a  =  (
(coe1 `  b ) `  Y ) ) } )
671, 6, 13, 66syl3anc 1182 . 2  |-  ( ph  ->  ( ( S `  I ) `  Y
)  =  { a  |  E. b  e.  I  ( ( ( deg1  `  R ) `  b
)  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y
) ) } )
6862, 65, 673sstr4d 3221 1  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  I ) `  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    + caddc 8740    <_ cle 8868    - cmin 9037   NN0cn0 9965   Basecbs 13148   .rcmulr 13209   0gc0g 13400   Mndcmnd 14361  .gcmg 14366  mulGrpcmgp 15325   Ringcrg 15337  LIdealclidl 15923  var1cv1 16251  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440  ldgIdlSeqcldgis 27325
This theorem is referenced by:  hbt  27334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-psr 16098  df-mvr 16099  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442  df-ldgis 27326
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