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Theorem hbtlem3 27309
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 3409 . . . 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 16 . . 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 3417 . 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 2437 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
128, 9, 10, 11hbtlem1 27305 . . 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 1185 . 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 27305 . . 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 1185 . 2  |-  ( ph  ->  ( ( S `  J ) `  X
)  =  { a  |  E. b  e.  J  ( ( ( deg1  `  R ) `  b
)  <_  X  /\  a  =  ( (coe1 `  b ) `  X
) ) } )
174, 13, 163sstr4d 3392 1  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  J ) `  X ) )
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    <_ cle 9122   NN0cn0 10222   Ringcrg 15661  LIdealclidl 16243  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-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-i2m1 9059  ax-1ne0 9060  ax-rrecex 9063  ax-cnre 9064
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-ral 2711  df-rex 2712  df-reu 2713  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-iun 4096  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-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-ov 6085  df-recs 6634  df-rdg 6669  df-nn 10002  df-n0 10223  df-ldgis 27304
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