Theorem hbtlem3 37697

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.

Step	Hyp	Ref	Expression
1		hbtlem3.ij	. . . 4 |- ( ph -> I C_ J )
2		ssrexv 3667	. . . 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 ) ) ) )
3	1, 2	syl 17	. . 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 ) ) ) )
4	3	ss2abdv 3675	. 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 2622	. . . 4 |- ( deg1 ` R ) = ( deg1 ` R )
12	8, 9, 10, 11	hbtlem1 37693	. . 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 ) ) } )
13	5, 6, 7, 12	syl3anc 1326	. 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 )
15	8, 9, 10, 11	hbtlem1 37693	. . 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 ) ) } )
16	5, 14, 7, 15	syl3anc 1326	. 2 |- ( ph -> ( ( S ` J ) ` X ) = { a | E. b e. J ( ( ( deg1 ` R ) ` b ) <_ X /\ a = ( (coe1 ` b ) ` X ) ) } )
17	4, 13, 16	3sstr4d 3648	1 |- ( ph -> ( ( S ` I ) ` X ) C_ ( ( S ` J ) ` X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ` cfv 5888   <_ cle 10075   NN0cn0 11292   Ringcrg 18547  LIdealclidl 19170  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814  ldgIdlSeqcldgis 37691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-n0 11293  df-ldgis 37692

This theorem is referenced by:  hbt  37700