Theorem algextdeglem2 33723

Description: Lemma for algextdeg 33730. Both the ring of polynomials 𝑃 and the field 𝐿 generated by 𝐾 and the algebraic element 𝐴 can be considered as modules over the elements of 𝐹. Then, the evaluation map 𝐺, mapping polynomials to their evaluation at 𝐴, is a module homomorphism between those modules. (Contributed by Thierry Arnoux, 2-Apr-2025.)

Hypotheses

Ref	Expression
algextdeg.k	⊢ 𝐾 = (𝐸 ↾s 𝐹)
algextdeg.l	⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
algextdeg.d	⊢ 𝐷 = (deg1‘𝐸)
algextdeg.m	⊢ 𝑀 = (𝐸 minPoly 𝐹)
algextdeg.f	⊢ (𝜑 → 𝐸 ∈ Field)
algextdeg.e	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
algextdeg.a	⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
algextdeglem.o	⊢ 𝑂 = (𝐸 evalSub1 𝐹)
algextdeglem.y	⊢ 𝑃 = (Poly1‘𝐾)
algextdeglem.u	⊢ 𝑈 = (Base‘𝑃)
algextdeglem.g	⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
algextdeglem.n	⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))
algextdeglem.z	⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})
algextdeglem.q	⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))
algextdeglem.j	⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))

Assertion

Ref	Expression
algextdeglem2	⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))

Distinct variable groups:   𝐴,𝑝   𝐸,𝑝   𝐹,𝑝,𝑥   𝐺,𝑝,𝑥   𝐽,𝑝,𝑥   𝐾,𝑝   𝐿,𝑝,𝑥   𝑥,𝑁   𝑂,𝑝   𝑃,𝑝,𝑥   𝑄,𝑝,𝑥   𝑈,𝑝,𝑥   𝑍,𝑝,𝑥   𝜑,𝑝,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐷(𝑥,𝑝)   𝐸(𝑥)   𝐾(𝑥)   𝑀(𝑥,𝑝)   𝑁(𝑝)   𝑂(𝑥)

Proof of Theorem algextdeglem2

Step	Hyp	Ref	Expression
1		algextdeg.e	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
2		issdrg 20698	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
3	1, 2	sylib 218	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
4	3	simp2d 1143	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
5		eqid 2731	. . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
6	5	sralmod 21116	. . . 4 ⊢ (𝐹 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
7	4, 6	syl 17	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
8		eqid 2731	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
9		eqid 2731	. . . 4 ⊢ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
10		algextdeg.f	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Field)
11	10	flddrngd 20651	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ DivRing)
12		subrgsubg 20487	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
13	8	subgss 19035	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ⊆ (Base‘𝐸))
14	4, 12, 13	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸))
15		algextdeglem.o	. . . . . . . . . . 11 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
16		algextdeg.k	. . . . . . . . . . 11 ⊢ 𝐾 = (𝐸 ↾s 𝐹)
17		eqid 2731	. . . . . . . . . . 11 ⊢ (0g‘𝐸) = (0g‘𝐸)
18	10	fldcrngd 20652	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ CRing)
19	15, 16, 8, 17, 18, 4	irngssv 33693	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
20		algextdeg.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
21	19, 20	sseldd 3930	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
22	21	snssd 4756	. . . . . . . 8 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸))
23	14, 22	unssd 4137	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
24	8, 11, 23	fldgensdrg 33272	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
25		issdrg 20698	. . . . . 6 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ DivRing))
26	24, 25	sylib 218	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ DivRing))
27	26	simp2d 1143	. . . 4 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
28	8, 11, 23	fldgenssid 33271	. . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
29	28	unssad 4138	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
30	9	subsubrg 20508	. . . . . 6 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) → (𝐹 ∈ (SubRing‘(𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))) ↔ (𝐹 ∈ (SubRing‘𝐸) ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))))
31	30	biimpar 477	. . . . 5 ⊢ (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐹 ∈ (SubRing‘𝐸) ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))) → 𝐹 ∈ (SubRing‘(𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
32	27, 4, 29, 31	syl12anc 836	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘(𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
33	5, 8, 9, 27, 32	lsssra 33592	. . 3 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)))
34		algextdeglem.y	. . . . . . 7 ⊢ 𝑃 = (Poly1‘𝐾)
35	16	fveq2i 6820	. . . . . . 7 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
36	34, 35	eqtri 2754	. . . . . 6 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
37		algextdeglem.u	. . . . . 6 ⊢ 𝑈 = (Base‘𝑃)
38	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐸 ∈ Field)
39	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐹 ∈ (SubDRing‘𝐸))
40	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐴 ∈ (Base‘𝐸))
41		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑝 ∈ 𝑈)
42	8, 15, 36, 37, 38, 39, 40, 41	evls1fldgencl 33675	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
43	42	ralrimiva 3124	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
44		algextdeglem.g	. . . . 5 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
45	44	rnmptss 7051	. . . 4 ⊢ (∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})) → ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
46	43, 45	syl 17	. . 3 ⊢ (𝜑 → ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
47	15, 36, 8, 37, 18, 4, 21, 44, 5	evls1maplmhm 22287	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
48		eqid 2731	. . . . 5 ⊢ (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) = (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
49		eqid 2731	. . . . 5 ⊢ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)) = (LSubSp‘((subringAlg ‘𝐸)‘𝐹))
50	48, 49	reslmhm2b 20983	. . . 4 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LMod ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)) ∧ ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) → (𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)) ↔ 𝐺 ∈ (𝑃 LMHom (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))))))
51	50	biimpa 476	. . 3 ⊢ (((((subringAlg ‘𝐸)‘𝐹) ∈ LMod ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)) ∧ ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∧ 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹))) → 𝐺 ∈ (𝑃 LMHom (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
52	7, 33, 46, 47, 51	syl31anc 1375	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
53		algextdeg.l	. . . 4 ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
54	8, 11, 23	fldgenssv 33273	. . . 4 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
55	8, 53, 54, 29, 10	resssra 33591	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))))
56	55	oveq2d 7357	. 2 ⊢ (𝜑 → (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)) = (𝑃 LMHom (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
57	52, 56	eleqtrrd 2834	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∪ cun 3895   ⊆ wss 3897  {csn 4571  ∪ cuni 4854   ↦ cmpt 5167  ^◡ccnv 5610  ran crn 5612   “ cima 5614  ‘cfv 6476  (class class class)co 7341  [cec 8615  Basecbs 17115   ↾_s cress 17136  0_gc0g 17338   /_s cqus 17404  SubGrpcsubg 19028   ~_QG cqg 19030  SubRingcsubrg 20479  DivRingcdr 20639  Fieldcfield 20640  SubDRingcsdrg 20696  LModclmod 20788  LSubSpclss 20859   LMHom clmhm 20948  subringAlg csra 21100  Poly₁cpl1 22084   evalSub₁ ces1 22223  deg₁cdg1 25981   fldGen cfldgen 33268   IntgRing cirng 33688   minPoly cminply 33704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-ofr 7606  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-pm 8748  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-sup 9321  df-oi 9391  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-z 12464  df-dec 12584  df-uz 12728  df-fz 13403  df-fzo 13550  df-seq 13904  df-hash 14233  df-struct 17053  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-mulr 17170  df-sca 17172  df-vsca 17173  df-ip 17174  df-tset 17175  df-ple 17176  df-ds 17178  df-hom 17180  df-cco 17181  df-0g 17340  df-gsum 17341  df-prds 17346  df-pws 17348  df-mre 17483  df-mrc 17484  df-acs 17486  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-submnd 18687  df-grp 18844  df-minusg 18845  df-sbg 18846  df-mulg 18976  df-subg 19031  df-ghm 19120  df-cntz 19224  df-cmn 19689  df-abl 19690  df-mgp 20054  df-rng 20066  df-ur 20095  df-srg 20100  df-ring 20148  df-cring 20149  df-oppr 20250  df-dvdsr 20270  df-unit 20271  df-invr 20301  df-dvr 20314  df-rhm 20385  df-subrng 20456  df-subrg 20480  df-drng 20641  df-field 20642  df-sdrg 20697  df-lmod 20790  df-lss 20860  df-lsp 20900  df-lmhm 20951  df-sra 21102  df-assa 21785  df-asp 21786  df-ascl 21787  df-psr 21841  df-mvr 21842  df-mpl 21843  df-opsr 21845  df-evls 22004  df-evl 22005  df-psr1 22087  df-vr1 22088  df-ply1 22089  df-coe1 22090  df-evls1 22225  df-evl1 22226  df-mon1 26058  df-fldgen 33269  df-irng 33689

This theorem is referenced by:  algextdeglem3  33724  algextdeglem4  33725