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Theorem hbtlem4 29324
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 718 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  R  e.  Ring )
3 hbtlem.p . . . . . . . . . 10  |-  P  =  (Poly1 `  R )
43ply1rng 17600 . . . . . . . . 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 718 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  I  e.  U )
8 eqid 2433 . . . . . . . . . . 11  |-  (mulGrp `  P )  =  (mulGrp `  P )
98rngmgp 16586 . . . . . . . . . 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 718 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  e.  NN0 )
13 hbtlem4.y . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  NN0 )
1413ad2antrr 718 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  Y  e.  NN0 )
15 hbtlem4.xy . . . . . . . . . . 11  |-  ( ph  ->  X  <_  Y )
1615ad2antrr 718 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  <_  Y )
17 nn0sub2 10692 . . . . . . . . . 10  |-  ( ( X  e.  NN0  /\  Y  e.  NN0  /\  X  <_  Y )  ->  ( Y  -  X )  e.  NN0 )
1812, 14, 16, 17syl3anc 1211 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( Y  -  X )  e.  NN0 )
19 eqid 2433 . . . . . . . . . . 11  |-  (var1 `  R
)  =  (var1 `  R
)
20 eqid 2433 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  P )
2119, 3, 20vr1cl 17569 . . . . . . . . . 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 16570 . . . . . . . . . 10  |-  ( Base `  P )  =  (
Base `  (mulGrp `  P
) )
24 eqid 2433 . . . . . . . . . 10  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
2523, 24mulgnn0cl 15622 . . . . . . . . 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 1211 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( Y  -  X )
(.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
27 simplr 747 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  c  e.  I )
28 hbtlem.u . . . . . . . . 9  |-  U  =  (LIdeal `  P )
29 eqid 2433 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
3028, 20, 29lidlmcl 17220 . . . . . . . 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 1212 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( (
( Y  -  X
) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ( .r
`  P ) c )  e.  I )
32 eqid 2433 . . . . . . . . 9  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
3320, 28lidlss 17212 . . . . . . . . . . 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 3345 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  c  e.  ( Base `  P )
)
3632, 3, 19, 8, 24deg1pwle 21475 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  ( Y  -  X )  e.  NN0 )  ->  (
( deg1  `
 R ) `  ( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )  <_  ( Y  -  X ) )
372, 18, 36syl2anc 654 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  (
( Y  -  X
) (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  <_ 
( Y  -  X
) )
38 simpr 458 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  c
)  <_  X )
393, 32, 2, 20, 29, 26, 35, 18, 12, 37, 38deg1mulle2 21465 . . . . . . . 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 10625 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  Y  e.  CC )
4112nn0cnd 10625 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  X  e.  CC )
4240, 41npcand 9710 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( Y  -  X )  +  X )  =  Y )
4339, 42breqtrd 4304 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  (
( deg1  `
 R ) `  c )  <_  X
)  ->  ( ( deg1  `  R ) `  (
( ( Y  -  X ) (.g `  (mulGrp `  P ) ) (var1 `  R ) ) ( .r `  P ) c ) )  <_  Y )
44 eqid 2433 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
4544, 3, 19, 8, 24, 20, 29, 2, 35, 18, 12coe1pwmulfv 17630 . . . . . . . 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 5683 . . . . . . . 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 2467 . . . . . . 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 5679 . . . . . . . . . 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 4290 . . . . . . . . 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 5679 . . . . . . . . . . 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 5681 . . . . . . . . . 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 2444 . . . . . . . . 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 703 . . . . . . . 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 3062 . . . . . . 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 1209 . . . . . 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 2439 . . . . . . . 8  |-  ( a  =  ( (coe1 `  c
) `  X )  ->  ( a  =  ( (coe1 `  b ) `  Y )  <->  ( (coe1 `  c ) `  X
)  =  ( (coe1 `  b ) `  Y
) ) )
5756anbi2d 696 . . . . . . 7  |-  ( a  =  ( (coe1 `  c
) `  X )  ->  ( ( ( ( deg1  `  R ) `  b
)  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y
) )  <->  ( (
( deg1  `
 R ) `  b )  <_  Y  /\  ( (coe1 `  c ) `  X )  =  ( (coe1 `  b ) `  Y ) ) ) )
5857rexbidv 2726 . . . . . 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 598 . . . 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 3413 . 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 29321 . . 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 1211 . 2  |-  ( ph  ->  ( ( S `  I ) `  X
)  =  { a  |  E. c  e.  I  ( ( ( deg1  `  R ) `  c
)  <_  X  /\  a  =  ( (coe1 `  c ) `  X
) ) } )
663, 28, 63, 32hbtlem1 29321 . . 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 1211 . 2  |-  ( ph  ->  ( ( S `  I ) `  Y
)  =  { a  |  E. b  e.  I  ( ( ( deg1  `  R ) `  b
)  <_  Y  /\  a  =  ( (coe1 `  b ) `  Y
) ) } )
6862, 65, 673sstr4d 3387 1  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  I ) `  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   {cab 2419   E.wrex 2706    C_ wss 3316   class class class wbr 4280   ` cfv 5406  (class class class)co 6080    + caddc 9272    <_ cle 9406    - cmin 9582   NN0cn0 10566   Basecbs 14156   .rcmulr 14221   0gc0g 14360   Mndcmnd 15391  .gcmg 15396  mulGrpcmgp 16564   Ringcrg 16576  LIdealclidl 17172  var1cv1 17526  Poly1cpl1 17527  coe1cco1 17530   deg1 cdg1 21407  ldgIdlSeqcldgis 29319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-ofr 6310  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-fz 11424  df-fzo 11532  df-seq 11790  df-hash 12087  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-starv 14235  df-sca 14236  df-vsca 14237  df-ip 14238  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-0g 14362  df-gsum 14363  df-mre 14506  df-mrc 14507  df-acs 14509  df-mnd 15397  df-mhm 15446  df-submnd 15447  df-grp 15524  df-minusg 15525  df-sbg 15526  df-mulg 15527  df-subg 15657  df-ghm 15724  df-cntz 15814  df-cmn 16258  df-abl 16259  df-mgp 16565  df-rng 16579  df-cring 16580  df-ur 16581  df-subrg 16786  df-lmod 16873  df-lss 16935  df-sra 17174  df-rgmod 17175  df-lidl 17176  df-psr 17350  df-mvr 17351  df-mpl 17352  df-opsr 17358  df-psr1 17532  df-vr1 17533  df-ply1 17534  df-coe1 17537  df-cnfld 17662  df-mdeg 21408  df-deg1 21409  df-ldgis 29320
This theorem is referenced by:  hbt  29328
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