Theorem hbtlem4 37696

Description: The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.)

Hypotheses

Ref	Expression
hbtlem.p	|- P = (Poly1 ` R )
hbtlem.u	|- U = (LIdeal ` P )
hbtlem.s	|- S = (ldgIdlSeq ` R )
hbtlem4.r	|- ( ph -> R e. Ring )
hbtlem4.i	|- ( ph -> I e. U )
hbtlem4.x	|- ( ph -> X e. NN0 )
hbtlem4.y	|- ( ph -> Y e. NN0 )
hbtlem4.xy	|- ( ph -> X <_ Y )

Assertion

Ref	Expression
hbtlem4	|- ( ph -> ( ( S ` I ) ` X ) C_ ( ( S ` I ) ` Y ) )

Proof of Theorem hbtlem4

Dummy variables a c b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbtlem4.r	. . . . . . . . . 10 |- ( ph -> R e. Ring )
2	1	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> R e. Ring )
3		hbtlem.p	. . . . . . . . . 10 |- P = (Poly1 ` R )
4	3	ply1ring 19618	. . . . . . . . 9 |- ( R e. Ring -> P e. Ring )
5	2, 4	syl 17	. . . . . . . 8 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> P e. Ring )
6		hbtlem4.i	. . . . . . . . 9 |- ( ph -> I e. U )
7	6	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> I e. U )
8		eqid 2622	. . . . . . . . . . 11 |- (mulGrp ` P ) = (mulGrp ` P )
9	8	ringmgp 18553	. . . . . . . . . 10 |- ( P e. Ring -> (mulGrp ` P ) e. Mnd )
10	5, 9	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> (mulGrp ` P ) e. Mnd )
11		hbtlem4.x	. . . . . . . . . . 11 |- ( ph -> X e. NN0 )
12	11	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> X e. NN0 )
13		hbtlem4.y	. . . . . . . . . . 11 |- ( ph -> Y e. NN0 )
14	13	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> Y e. NN0 )
15		hbtlem4.xy	. . . . . . . . . . 11 |- ( ph -> X <_ Y )
16	15	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> X <_ Y )
17		nn0sub2 11438	. . . . . . . . . 10 |- ( ( X e. NN0 /\ Y e. NN0 /\ X <_ Y ) -> ( Y - X ) e. NN0 )
18	12, 14, 16, 17	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( Y - X ) e. NN0 )
19		eqid 2622	. . . . . . . . . . 11 |- (var1 ` R ) = (var1 ` R )
20		eqid 2622	. . . . . . . . . . 11 |- ( Base ` P ) = ( Base ` P )
21	19, 3, 20	vr1cl 19587	. . . . . . . . . 10 |- ( R e. Ring -> (var1 ` R ) e. ( Base ` P ) )
22	2, 21	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> (var1 ` R ) e. ( Base ` P ) )
23	8, 20	mgpbas 18495	. . . . . . . . . 10 |- ( Base ` P ) = ( Base ` (mulGrp ` P ) )
24		eqid 2622	. . . . . . . . . 10 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
25	23, 24	mulgnn0cl 17558	. . . . . . . . 9 |- ( ( (mulGrp ` P ) e. Mnd /\ ( Y - X ) e. NN0 /\ (var1 ` R ) e. ( Base ` P ) ) -> ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
26	10, 18, 22, 25	syl3anc 1326	. . . . . . . 8 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) )
27		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> c e. I )
28		hbtlem.u	. . . . . . . . 9 |- U = (LIdeal ` P )
29		eqid 2622	. . . . . . . . 9 |- ( .r ` P ) = ( .r ` P )
30	28, 20, 29	lidlmcl 19217	. . . . . . . 8 |- ( ( ( P e. Ring /\ I e. U ) /\ ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) e. ( Base ` P ) /\ c e. I ) ) -> ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) e. I )
31	5, 7, 26, 27, 30	syl22anc 1327	. . . . . . 7 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) e. I )
32		eqid 2622	. . . . . . . . 9 |- ( deg1 ` R ) = ( deg1 ` R )
33	20, 28	lidlss 19210	. . . . . . . . . . 11 |- ( I e. U -> I C_ ( Base ` P ) )
34	7, 33	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> I C_ ( Base ` P ) )
35	34, 27	sseldd 3604	. . . . . . . . 9 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> c e. ( Base ` P ) )
36	32, 3, 19, 8, 24	deg1pwle 23879	. . . . . . . . . 10 |- ( ( R e. Ring /\ ( Y - X ) e. NN0 ) -> ( ( deg1 ` R ) ` ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) <_ ( Y - X ) )
37	2, 18, 36	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( ( deg1 ` R ) ` ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ) <_ ( Y - X ) )
38		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( ( deg1 ` R ) ` c ) <_ X )
39	3, 32, 2, 20, 29, 26, 35, 18, 12, 37, 38	deg1mulle2 23869	. . . . . . . 8 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( ( deg1 ` R ) ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) <_ ( ( Y - X ) + X ) )
40	14	nn0cnd 11353	. . . . . . . . 9 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> Y e. CC )
41	12	nn0cnd 11353	. . . . . . . . 9 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> X e. CC )
42	40, 41	npcand 10396	. . . . . . . 8 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( ( Y - X ) + X ) = Y )
43	39, 42	breqtrd 4679	. . . . . . 7 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( ( deg1 ` R ) ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) <_ Y )
44		eqid 2622	. . . . . . . . 9 |- ( 0g ` R ) = ( 0g ` R )
45	44, 3, 19, 8, 24, 20, 29, 2, 35, 18, 12	coe1pwmulfv 19650	. . . . . . . 8 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( (coe1 ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) ` ( ( Y - X ) + X ) ) = ( (coe1 ` c ) ` X ) )
46	42	fveq2d 6195	. . . . . . . 8 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( (coe1 ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) ` ( ( Y - X ) + X ) ) = ( (coe1 ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) ` Y ) )
47	45, 46	eqtr3d 2658	. . . . . . 7 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( (coe1 ` c ) ` X ) = ( (coe1 ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) ` Y ) )
48		fveq2 6191	. . . . . . . . . 10 |- ( b = ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) -> ( ( deg1 ` R ) ` b ) = ( ( deg1 ` R ) ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) )
49	48	breq1d 4663	. . . . . . . . 9 |- ( b = ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) -> ( ( ( deg1 ` R ) ` b ) <_ Y <-> ( ( deg1 ` R ) ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) <_ Y ) )
50		fveq2 6191	. . . . . . . . . . 11 |- ( b = ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) -> (coe1 ` b ) = (coe1 ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) )
51	50	fveq1d 6193	. . . . . . . . . 10 |- ( b = ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) -> ( (coe1 ` b ) ` Y ) = ( (coe1 ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) ` Y ) )
52	51	eqeq2d 2632	. . . . . . . . 9 |- ( b = ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) -> ( ( (coe1 ` c ) ` X ) = ( (coe1 ` b ) ` Y ) <-> ( (coe1 ` c ) ` X ) = ( (coe1 ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) ` Y ) ) )
53	49, 52	anbi12d 747	. . . . . . . 8 |- ( b = ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) -> ( ( ( ( deg1 ` R ) ` b ) <_ Y /\ ( (coe1 ` c ) ` X ) = ( (coe1 ` b ) ` Y ) ) <-> ( ( ( deg1 ` R ) ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) <_ Y /\ ( (coe1 ` c ) ` X ) = ( (coe1 ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) ` Y ) ) ) )
54	53	rspcev 3309	. . . . . . 7 |- ( ( ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) e. I /\ ( ( ( deg1 ` R ) ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) <_ Y /\ ( (coe1 ` c ) ` X ) = ( (coe1 ` ( ( ( Y - X ) (.g ` (mulGrp ` P ) ) (var1 ` R ) ) ( .r ` P ) c ) ) ` Y ) ) ) -> E. b e. I ( ( ( deg1 ` R ) ` b ) <_ Y /\ ( (coe1 ` c ) ` X ) = ( (coe1 ` b ) ` Y ) ) )
55	31, 43, 47, 54	syl12anc 1324	. . . . . 6 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> E. b e. I ( ( ( deg1 ` R ) ` b ) <_ Y /\ ( (coe1 ` c ) ` X ) = ( (coe1 ` b ) ` Y ) ) )
56		eqeq1 2626	. . . . . . . 8 |- ( a = ( (coe1 ` c ) ` X ) -> ( a = ( (coe1 ` b ) ` Y ) <-> ( (coe1 ` c ) ` X ) = ( (coe1 ` b ) ` Y ) ) )
57	56	anbi2d 740	. . . . . . 7 |- ( a = ( (coe1 ` c ) ` X ) -> ( ( ( ( deg1 ` R ) ` b ) <_ Y /\ a = ( (coe1 ` b ) ` Y ) ) <-> ( ( ( deg1 ` R ) ` b ) <_ Y /\ ( (coe1 ` c ) ` X ) = ( (coe1 ` b ) ` Y ) ) ) )
58	57	rexbidv 3052	. . . . . 6 |- ( a = ( (coe1 ` c ) ` X ) -> ( E. b e. I ( ( ( deg1 ` R ) ` b ) <_ Y /\ a = ( (coe1 ` b ) ` Y ) ) <-> E. b e. I ( ( ( deg1 ` R ) ` b ) <_ Y /\ ( (coe1 ` c ) ` X ) = ( (coe1 ` b ) ` Y ) ) ) )
59	55, 58	syl5ibrcom 237	. . . . 5 |- ( ( ( ph /\ c e. I ) /\ ( ( deg1 ` R ) ` c ) <_ X ) -> ( a = ( (coe1 ` c ) ` X ) -> E. b e. I ( ( ( deg1 ` R ) ` b ) <_ Y /\ a = ( (coe1 ` b ) ` Y ) ) ) )
60	59	expimpd 629	. . . 4 |- ( ( ph /\ c e. I ) -> ( ( ( ( deg1 ` R ) ` c ) <_ X /\ a = ( (coe1 ` c ) ` X ) ) -> E. b e. I ( ( ( deg1 ` R ) ` b ) <_ Y /\ a = ( (coe1 ` b ) ` Y ) ) ) )
61	60	rexlimdva 3031	. . 3 |- ( ph -> ( E. c e. I ( ( ( deg1 ` R ) ` c ) <_ X /\ a = ( (coe1 ` c ) ` X ) ) -> E. b e. I ( ( ( deg1 ` R ) ` b ) <_ Y /\ a = ( (coe1 ` b ) ` Y ) ) ) )
62	61	ss2abdv 3675	. 2 |- ( ph -> { a | E. c e. I ( ( ( deg1 ` R ) ` c ) <_ X /\ a = ( (coe1 ` c ) ` X ) ) } C_ { a | E. b e. I ( ( ( deg1 ` R ) ` b ) <_ Y /\ a = ( (coe1 ` b ) ` Y ) ) } )
63		hbtlem.s	. . . 4 |- S = (ldgIdlSeq ` R )
64	3, 28, 63, 32	hbtlem1 37693	. . 3 |- ( ( R e. Ring /\ I e. U /\ X e. NN0 ) -> ( ( S ` I ) ` X ) = { a | E. c e. I ( ( ( deg1 ` R ) ` c ) <_ X /\ a = ( (coe1 ` c ) ` X ) ) } )
65	1, 6, 11, 64	syl3anc 1326	. 2 |- ( ph -> ( ( S ` I ) ` X ) = { a | E. c e. I ( ( ( deg1 ` R ) ` c ) <_ X /\ a = ( (coe1 ` c ) ` X ) ) } )
66	3, 28, 63, 32	hbtlem1 37693	. . 3 |- ( ( R e. Ring /\ I e. U /\ Y e. NN0 ) -> ( ( S ` I ) ` Y ) = { a | E. b e. I ( ( ( deg1 ` R ) ` b ) <_ Y /\ a = ( (coe1 ` b ) ` Y ) ) } )
67	1, 6, 13, 66	syl3anc 1326	. 2 |- ( ph -> ( ( S ` I ) ` Y ) = { a | E. b e. I ( ( ( deg1 ` R ) ` b ) <_ Y /\ a = ( (coe1 ` b ) ` Y ) ) } )
68	62, 65, 67	3sstr4d 3648	1 |- ( ph -> ( ( S ` I ) ` X ) C_ ( ( S ` I ) ` Y ) )

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  (class class class)co 6650   + caddc 9939   <_ cle 10075   - cmin 10266   NN0cn0 11292   Basecbs 15857   .rcmulr 15942   0gc0g 16100   Mndcmnd 17294  ._gcmg 17540  mulGrpcmgp 18489   Ringcrg 18547  LIdealclidl 19170  var₁cv1 19546  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-inf2 8538  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-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-int 4476  df-iun 4522  df-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-cnfld 19747  df-mdeg 23815  df-deg1 23816  df-ldgis 37692

This theorem is referenced by:  hbt  37700