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Theorem hbtlem4 27308
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 708 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  R  e.  Ring )
3 hbtlem.p . . . . . . . . . 10  |-  P  =  (Poly1 `  R )
43ply1rng 16643 . . . . . . . . 9  |-  ( R  e.  Ring  ->  P  e. 
Ring )
52, 4syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  P  e.  Ring )
6 hbtlem4.i . . . . . . . . 9  |-  ( ph  ->  I  e.  U )
76ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  I  e.  U )
8 eqid 2437 . . . . . . . . . . 11  |-  (mulGrp `  P )  =  (mulGrp `  P )
98rngmgp 15671 . . . . . . . . . 10  |-  ( P  e.  Ring  ->  (mulGrp `  P )  e.  Mnd )
105, 9syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  (mulGrp `  P
)  e.  Mnd )
11 hbtlem4.x . . . . . . . . . . 11  |-  ( ph  ->  X  e.  NN0 )
1211ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  e.  NN0 )
13 hbtlem4.y . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  NN0 )
1413ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  Y  e.  NN0 )
15 hbtlem4.xy . . . . . . . . . . 11  |-  ( ph  ->  X  <_  Y )
1615ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  <_  Y )
17 nn0sub2 10336 . . . . . . . . . 10  |-  ( ( X  e.  NN0  /\  Y  e.  NN0  /\  X  <_  Y )  ->  ( Y  -  X )  e.  NN0 )
1812, 14, 16, 17syl3anc 1185 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( Y  -  X )  e.  NN0 )
19 eqid 2437 . . . . . . . . . . 11  |-  (var1 `  R
)  =  (var1 `  R
)
20 eqid 2437 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  P )
2119, 3, 20vr1cl 16612 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (var1 `  R
)  e.  ( Base `  P ) )
222, 21syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  (var1 `  R
)  e.  ( Base `  P ) )
238, 20mgpbas 15655 . . . . . . . . . 10  |-  ( Base `  P )  =  (
Base `  (mulGrp `  P
) )
24 eqid 2437 . . . . . . . . . 10  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
2523, 24mulgnn0cl 14907 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( Y  -  X )
(.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
27 simplr 733 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  c  e.  I )
28 hbtlem.u . . . . . . . . 9  |-  U  =  (LIdeal `  P )
29 eqid 2437 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
3028, 20, 29lidlmcl 16289 . . . . . . . 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 1186 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( (
( Y  -  X
) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c )  e.  I )
32 eqid 2437 . . . . . . . . 9  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
3320, 28lidlss 16281 . . . . . . . . . . 11  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
347, 33syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  I  C_  ( Base `  P ) )
3534, 27sseldd 3350 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  c  e.  ( Base `  P )
)
3632, 3, 19, 8, 24deg1pwle 20043 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  ( Y  -  X )  e.  NN0 )  ->  (
( deg1  `
 R ) `  ( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )  <_  ( Y  -  X ) )
372, 18, 36syl2anc 644 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  (
( Y  -  X
) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  <_ 
( Y  -  X
) )
38 simpr 449 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  c
)  <_  X )
393, 32, 2, 20, 29, 26, 35, 18, 12, 37, 38deg1mulle2 20033 . . . . . . . 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 10277 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  Y  e.  CC )
4112nn0cnd 10277 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  e.  CC )
4240, 41npcand 9416 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( Y  -  X )  +  X )  =  Y )
4339, 42breqtrd 4237 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  (
( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) ( .r `  P ) c ) )  <_  Y )
44 eqid 2437 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
4544, 3, 19, 8, 24, 20, 29, 2, 35, 18, 12coe1pwmulfv 16673 . . . . . . . 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 5733 . . . . . . . 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 2471 . . . . . . 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 5729 . . . . . . . . . 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 4223 . . . . . . . . 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 5729 . . . . . . . . . . 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 5731 . . . . . . . . . 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 2448 . . . . . . . . 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 693 . . . . . . . 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 3053 . . . . . . 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 1183 . . . . . 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 2443 . . . . . . . 8  |-  ( a  =  ( (coe1 `  c
) `  X )  ->  ( a  =  ( (coe1 `  b ) `  Y )  <->  ( (coe1 `  c ) `  X
)  =  ( (coe1 `  b ) `  Y
) ) )
5756anbi2d 686 . . . . . . 7  |-  ( a  =  ( (coe1 `  c
) `  X )  ->  ( ( ( ( deg1  `  R ) `  b
)  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y
) )  <->  ( (
( deg1  `
 R ) `  b )  <_  Y  /\  ( (coe1 `  c ) `  X )  =  ( (coe1 `  b ) `  Y ) ) ) )
5857rexbidv 2727 . . . . . 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 215 . . . . 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 588 . . . 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 2831 . . 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 3417 . 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 27305 . . 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 1185 . 2  |-  ( ph  ->  ( ( S `  I ) `  X
)  =  { a  |  E. c  e.  I  ( ( ( deg1  `  R ) `  c
)  <_  X  /\  a  =  ( (coe1 `  c ) `  X
) ) } )
663, 28, 63, 32hbtlem1 27305 . . 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 1185 . 2  |-  ( ph  ->  ( ( S `  I ) `  Y
)  =  { a  |  E. b  e.  I  ( ( ( deg1  `  R ) `  b
)  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y
) ) } )
6862, 65, 673sstr4d 3392 1  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  I ) `  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2423   E.wrex 2707    C_ wss 3321   class class class wbr 4213   ` cfv 5455  (class class class)co 6082    + caddc 8994    <_ cle 9122    - cmin 9292   NN0cn0 10222   Basecbs 13470   .rcmulr 13531   0gc0g 13724   Mndcmnd 14685  .gcmg 14690  mulGrpcmgp 15649   Ringcrg 15661  LIdealclidl 16243  var1cv1 16571  Poly1cpl1 16572  coe1cco1 16575   deg1 cdg1 19978  ldgIdlSeqcldgis 27303
This theorem is referenced by:  hbt  27312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-ofr 6307  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-oi 7480  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-fz 11045  df-fzo 11137  df-seq 11325  df-hash 11620  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-starv 13545  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-0g 13728  df-gsum 13729  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-mhm 14739  df-submnd 14740  df-grp 14813  df-minusg 14814  df-sbg 14815  df-mulg 14816  df-subg 14942  df-ghm 15005  df-cntz 15117  df-cmn 15415  df-abl 15416  df-mgp 15650  df-rng 15664  df-cring 15665  df-ur 15666  df-subrg 15867  df-lmod 15953  df-lss 16010  df-sra 16245  df-rgmod 16246  df-lidl 16247  df-psr 16418  df-mvr 16419  df-mpl 16420  df-opsr 16426  df-psr1 16577  df-vr1 16578  df-ply1 16579  df-coe1 16582  df-cnfld 16705  df-mdeg 19979  df-deg1 19980  df-ldgis 27304
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