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Theorem hbtlem3 27434
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 3251 . . . 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 3259 . 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 2296 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
128, 9, 10, 11hbtlem1 27430 . . 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 27430 . . 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 3234 1  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  J ) `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557    C_ wss 3165   class class class wbr 4039   ` cfv 5271    <_ cle 8884   NN0cn0 9981   Ringcrg 15353  LIdealclidl 15939  Poly1cpl1 16268  coe1cco1 16271   deg1 cdg1 19456  ldgIdlSeqcldgis 27428
This theorem is referenced by:  hbt  27437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439  df-nn 9763  df-n0 9982  df-ldgis 27429
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