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Theorem hbtlem3 29506
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 3436 . . . 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 3444 . 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 2443 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
128, 9, 10, 11hbtlem1 29502 . . 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 1218 . 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 29502 . . 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 1218 . 2  |-  ( ph  ->  ( ( S `  J ) `  X
)  =  { a  |  E. b  e.  J  ( ( ( deg1  `  R ) `  b
)  <_  X  /\  a  =  ( (coe1 `  b ) `  X
) ) } )
174, 13, 163sstr4d 3418 1  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  J ) `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2735    C_ wss 3347   class class class wbr 4311   ` cfv 5437    <_ cle 9438   NN0cn0 10598   Ringcrg 16664  LIdealclidl 17270  Poly1cpl1 17652  coe1cco1 17653   deg1 cdg1 21542  ldgIdlSeqcldgis 29500
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 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-i2m1 9369  ax-1ne0 9370  ax-rrecex 9373  ax-cnre 9374
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-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-om 6496  df-recs 6851  df-rdg 6885  df-nn 10342  df-n0 10599  df-ldgis 29501
This theorem is referenced by:  hbt  29509
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