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Theorem hbtlem3 31244
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 3479 . . . 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 3487 . 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 2382 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
128, 9, 10, 11hbtlem1 31240 . . 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 1226 . 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 31240 . . 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 1226 . 2  |-  ( ph  ->  ( ( S `  J ) `  X
)  =  { a  |  E. b  e.  J  ( ( ( deg1  `  R ) `  b
)  <_  X  /\  a  =  ( (coe1 `  b ) `  X
) ) } )
174, 13, 163sstr4d 3460 1  |-  ( ph  ->  ( ( S `  I ) `  X
)  C_  ( ( S `  J ) `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   {cab 2367   E.wrex 2733    C_ wss 3389   class class class wbr 4367   ` cfv 5496    <_ cle 9540   NN0cn0 10712   Ringcrg 17311  LIdealclidl 17929  Poly1cpl1 18329  coe1cco1 18330   deg1 cdg1 22537  ldgIdlSeqcldgis 31238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-i2m1 9471  ax-1ne0 9472  ax-rrecex 9475  ax-cnre 9476
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-om 6600  df-recs 6960  df-rdg 6994  df-nn 10453  df-n0 10713  df-ldgis 31239
This theorem is referenced by:  hbt  31247
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