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Theorem hbtlem3 27331
Description: The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Hypotheses
Ref Expression
hbtlem.p  |-  P  =  (Poly1 `  R )
hbtlem.u  |-  U  =  (LIdeal `  P )
hbtlem.s  |-  S  =  (ldgIdlSeq `  R )
hbtlem3.r  |-  ( ph  ->  R  e.  Ring )
hbtlem3.i  |-  ( ph  ->  I  e.  U )
hbtlem3.j  |-  ( ph  ->  J  e.  U )
hbtlem3.ij  |-  ( ph  ->  I  C_  J )
hbtlem3.x  |-  ( ph  ->  X  e.  NN0 )
Assertion
Ref Expression
hbtlem3  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  J ) `  X ) )

Proof of Theorem hbtlem3
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbtlem3.ij . . . 4  |-  ( ph  ->  I  C_  J )
2 ssrexv 3238 . . . 4  |-  ( I 
C_  J  ->  ( E. b  e.  I 
( ( ( deg1  `  R
) `  b )  <_  X  /\  a  =  ( (coe1 `  b ) `  X ) )  ->  E. b  e.  J  ( ( ( deg1  `  R
) `  b )  <_  X  /\  a  =  ( (coe1 `  b ) `  X ) ) ) )
31, 2syl 15 . . 3  |-  ( ph  ->  ( E. b  e.  I  ( ( ( deg1  `  R ) `  b
)  <_  X  /\  a  =  ( (coe1 `  b ) `  X
) )  ->  E. b  e.  J  ( (
( deg1  `
 R ) `  b )  <_  X  /\  a  =  (
(coe1 `  b ) `  X ) ) ) )
43ss2abdv 3246 . 2  |-  ( ph  ->  { a  |  E. b  e.  I  (
( ( deg1  `  R ) `  b )  <_  X  /\  a  =  (
(coe1 `  b ) `  X ) ) } 
C_  { a  |  E. b  e.  J  ( ( ( deg1  `  R
) `  b )  <_  X  /\  a  =  ( (coe1 `  b ) `  X ) ) } )
5 hbtlem3.r . . 3  |-  ( ph  ->  R  e.  Ring )
6 hbtlem3.i . . 3  |-  ( ph  ->  I  e.  U )
7 hbtlem3.x . . 3  |-  ( ph  ->  X  e.  NN0 )
8 hbtlem.p . . . 4  |-  P  =  (Poly1 `  R )
9 hbtlem.u . . . 4  |-  U  =  (LIdeal `  P )
10 hbtlem.s . . . 4  |-  S  =  (ldgIdlSeq `  R )
11 eqid 2283 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
128, 9, 10, 11hbtlem1 27327 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  U  /\  X  e. 
NN0 )  ->  (
( S `  I
) `  X )  =  { a  |  E. b  e.  I  (
( ( deg1  `  R ) `  b )  <_  X  /\  a  =  (
(coe1 `  b ) `  X ) ) } )
135, 6, 7, 12syl3anc 1182 . 2  |-  ( ph  ->  ( ( S `  I ) `  X
)  =  { a  |  E. b  e.  I  ( ( ( deg1  `  R ) `  b
)  <_  X  /\  a  =  ( (coe1 `  b ) `  X
) ) } )
14 hbtlem3.j . . 3  |-  ( ph  ->  J  e.  U )
158, 9, 10, 11hbtlem1 27327 . . 3  |-  ( ( R  e.  Ring  /\  J  e.  U  /\  X  e. 
NN0 )  ->  (
( S `  J
) `  X )  =  { a  |  E. b  e.  J  (
( ( deg1  `  R ) `  b )  <_  X  /\  a  =  (
(coe1 `  b ) `  X ) ) } )
165, 14, 7, 15syl3anc 1182 . 2  |-  ( ph  ->  ( ( S `  J ) `  X
)  =  { a  |  E. b  e.  J  ( ( ( deg1  `  R ) `  b
)  <_  X  /\  a  =  ( (coe1 `  b ) `  X
) ) } )
174, 13, 163sstr4d 3221 1  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  J ) `  X ) )
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    <_ cle 8868   NN0cn0 9965   Ringcrg 15337  LIdealclidl 15923  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-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
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-ral 2548  df-rex 2549  df-reu 2550  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-iun 3907  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-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-ov 5861  df-recs 6388  df-rdg 6423  df-nn 9747  df-n0 9966  df-ldgis 27326
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