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Theorem hbtlem4 29508
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 725 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  R  e.  Ring )
3 hbtlem.p . . . . . . . . . 10  |-  P  =  (Poly1 `  R )
43ply1rng 17725 . . . . . . . . 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 725 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  I  e.  U )
8 eqid 2443 . . . . . . . . . . 11  |-  (mulGrp `  P )  =  (mulGrp `  P )
98rngmgp 16673 . . . . . . . . . 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 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  e.  NN0 )
13 hbtlem4.y . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  NN0 )
1413ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  Y  e.  NN0 )
15 hbtlem4.xy . . . . . . . . . . 11  |-  ( ph  ->  X  <_  Y )
1615ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  <_  Y )
17 nn0sub2 10726 . . . . . . . . . 10  |-  ( ( X  e.  NN0  /\  Y  e.  NN0  /\  X  <_  Y )  ->  ( Y  -  X )  e.  NN0 )
1812, 14, 16, 17syl3anc 1218 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( Y  -  X )  e.  NN0 )
19 eqid 2443 . . . . . . . . . . 11  |-  (var1 `  R
)  =  (var1 `  R
)
20 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  P )
2119, 3, 20vr1cl 17693 . . . . . . . . . 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 16619 . . . . . . . . . 10  |-  ( Base `  P )  =  (
Base `  (mulGrp `  P
) )
24 eqid 2443 . . . . . . . . . 10  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
2523, 24mulgnn0cl 15664 . . . . . . . . 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 1218 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( Y  -  X )
(.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
27 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  c  e.  I )
28 hbtlem.u . . . . . . . . 9  |-  U  =  (LIdeal `  P )
29 eqid 2443 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
3028, 20, 29lidlmcl 17321 . . . . . . . 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 1219 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( (
( Y  -  X
) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c )  e.  I )
32 eqid 2443 . . . . . . . . 9  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
3320, 28lidlss 17313 . . . . . . . . . . 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 3378 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  c  e.  ( Base `  P )
)
3632, 3, 19, 8, 24deg1pwle 21613 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  ( Y  -  X )  e.  NN0 )  ->  (
( deg1  `
 R ) `  ( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )  <_  ( Y  -  X ) )
372, 18, 36syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  (
( Y  -  X
) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  <_ 
( Y  -  X
) )
38 simpr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  c
)  <_  X )
393, 32, 2, 20, 29, 26, 35, 18, 12, 37, 38deg1mulle2 21603 . . . . . . . 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 10659 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  Y  e.  CC )
4112nn0cnd 10659 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  e.  CC )
4240, 41npcand 9744 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( Y  -  X )  +  X )  =  Y )
4339, 42breqtrd 4337 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  (
( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) ( .r `  P ) c ) )  <_  Y )
44 eqid 2443 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
4544, 3, 19, 8, 24, 20, 29, 2, 35, 18, 12coe1pwmulfv 17755 . . . . . . . 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 5716 . . . . . . . 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 2477 . . . . . . 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 5712 . . . . . . . . . 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 4323 . . . . . . . . 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 5712 . . . . . . . . . . 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 5714 . . . . . . . . . 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 2454 . . . . . . . . 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 710 . . . . . . . 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 3094 . . . . . . 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 1216 . . . . . 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 2449 . . . . . . . 8  |-  ( a  =  ( (coe1 `  c
) `  X )  ->  ( a  =  ( (coe1 `  b ) `  Y )  <->  ( (coe1 `  c ) `  X
)  =  ( (coe1 `  b ) `  Y
) ) )
5756anbi2d 703 . . . . . . 7  |-  ( a  =  ( (coe1 `  c
) `  X )  ->  ( ( ( ( deg1  `  R ) `  b
)  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y
) )  <->  ( (
( deg1  `
 R ) `  b )  <_  Y  /\  ( (coe1 `  c ) `  X )  =  ( (coe1 `  b ) `  Y ) ) ) )
5857rexbidv 2757 . . . . . 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 222 . . . . 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 603 . . . 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 2862 . . 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 3446 . 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 29505 . . 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 1218 . 2  |-  ( ph  ->  ( ( S `  I ) `  X
)  =  { a  |  E. c  e.  I  ( ( ( deg1  `  R ) `  c
)  <_  X  /\  a  =  ( (coe1 `  c ) `  X
) ) } )
663, 28, 63, 32hbtlem1 29505 . . 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 1218 . 2  |-  ( ph  ->  ( ( S `  I ) `  Y
)  =  { a  |  E. b  e.  I  ( ( ( deg1  `  R ) `  b
)  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y
) ) } )
6862, 65, 673sstr4d 3420 1  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  I ) `  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2737    C_ wss 3349   class class class wbr 4313   ` cfv 5439  (class class class)co 6112    + caddc 9306    <_ cle 9440    - cmin 9616   NN0cn0 10600   Basecbs 14195   .rcmulr 14260   0gc0g 14399   Mndcmnd 15430  .gcmg 15435  mulGrpcmgp 16613   Ringcrg 16667  LIdealclidl 17273  var1cv1 17654  Poly1cpl1 17655  coe1cco1 17656   deg1 cdg1 21545  ldgIdlSeqcldgis 29503
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  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
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-starv 14274  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-0g 14401  df-gsum 14402  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  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-rng 16669  df-cring 16670  df-subrg 16885  df-lmod 16972  df-lss 17036  df-sra 17275  df-rgmod 17276  df-lidl 17277  df-psr 17445  df-mvr 17446  df-mpl 17447  df-opsr 17449  df-psr1 17658  df-vr1 17659  df-ply1 17660  df-coe1 17661  df-cnfld 17841  df-mdeg 21546  df-deg1 21547  df-ldgis 29504
This theorem is referenced by:  hbt  29512
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