Theorem algextdeglem2 34172

Description: Lemma for algextdeg 34179. 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 20941	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
3	1, 2	sylib 221	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
4	3	simp2d 1161	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
5		eqid 2765	. . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
6	5	sralmod 21358	. . . 4 ⊢ (𝐹 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
7	4, 6	syl 18	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
8		eqid 2765	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
9		eqid 2765	. . . 4 ⊢ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
10		algextdeg.f	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Field)
11	10	flddrngd 20891	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ DivRing)
12		subrgsubg 20726	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
13	8	subgss 19237	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ⊆ (Base‘𝐸))
14	4, 12, 13	3syl 19	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸))
15		algextdeglem.o	. . . . . . . . . . 11 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
16		algextdeg.k	. . . . . . . . . . 11 ⊢ 𝐾 = (𝐸 ↾s 𝐹)
17		eqid 2765	. . . . . . . . . . 11 ⊢ (0g‘𝐸) = (0g‘𝐸)
18	10	fldcrngd 20892	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ CRing)
19	15, 16, 8, 17, 18, 4	irngssv 34142	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
20		algextdeg.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
21	19, 20	sseldd 3939	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
22	21	snssd 4754	. . . . . . . 8 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸))
23	14, 22	unssd 4145	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
24	8, 11, 23	fldgensdrg 33699	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
25		issdrg 20941	. . . . . 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 221	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ DivRing))
27	26	simp2d 1161	. . . 4 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
28	8, 11, 23	fldgenssid 33698	. . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
29	28	unssad 4146	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
30	9	subsubrg 20747	. . . . . 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 483	. . . . 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 850	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘(𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
33	5, 8, 9, 27, 32	lsssra 34042	. . 3 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)))
34		algextdeglem.y	. . . . . . 7 ⊢ 𝑃 = (Poly1‘𝐾)
35	16	fveq2i 6888	. . . . . . 7 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
36	34, 35	eqtri 2788	. . . . . 6 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
37		algextdeglem.u	. . . . . 6 ⊢ 𝑈 = (Base‘𝑃)
38	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐸 ∈ Field)
39	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐹 ∈ (SubDRing‘𝐸))
40	21	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐴 ∈ (Base‘𝐸))
41		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑝 ∈ 𝑈)
42	8, 15, 36, 37, 38, 39, 40, 41	evls1fldgencl 34124	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
43	42	ralrimiva 3159	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
44		algextdeglem.g	. . . . 5 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
45	44	rnmptss 7122	. . . 4 ⊢ (∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})) → ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
46	43, 45	syl 18	. . 3 ⊢ (𝜑 → ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
47	15, 36, 8, 37, 18, 4, 21, 44, 5	evls1maplmhm 22587	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
48		eqid 2765	. . . . 5 ⊢ (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) = (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
49		eqid 2765	. . . . 5 ⊢ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)) = (LSubSp‘((subringAlg ‘𝐸)‘𝐹))
50	48, 49	reslmhm2b 21225	. . . 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 482	. . 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 1400	. 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 33700	. . . 4 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
55	8, 53, 54, 29, 10	resssra 34041	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))))
56	55	oveq2d 7435	. 2 ⊢ (𝜑 → (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)) = (𝑃 LMHom (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
57	52, 56	eleqtrrd 2868	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∪ cun 3904   ⊆ wss 3906  {csn 4591  ∪ cuni 4874   ↦ cmpt 5194  ^◡ccnv 5662  ran crn 5664   “ cima 5666  ‘cfv 6540  (class class class)co 7419  [cec 8698  Basecbs 17291   ↾_s cress 17312  0_gc0g 17514   /_s cqus 17581  SubGrpcsubg 19230   ~_QG cqg 19232  SubRingcsubrg 20718  DivRingcdr 20877  Fieldcfield 20878  SubDRingcsdrg 20939  LModclmod 21031  LSubSpclss 21102   LMHom clmhm 21190  subringAlg csra 21342  Poly₁cpl1 22387   evalSub₁ ces1 22523  deg₁cdg1 26262   fldGen cfldgen 33695   IntgRing cirng 34137   minPoly cminply 34153

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-ofr 7685  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-sup 9409  df-oi 9479  df-card 9941  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-z 12607  df-dec 12728  df-uz 12879  df-fz 13552  df-fzo 13700  df-seq 14056  df-hash 14385  df-struct 17229  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-plusg 17345  df-mulr 17346  df-sca 17348  df-vsca 17349  df-ip 17350  df-tset 17351  df-ple 17352  df-ds 17354  df-hom 17356  df-cco 17357  df-0g 17516  df-gsum 17517  df-prds 17522  df-pws 17524  df-mre 17660  df-mrc 17661  df-acs 17663  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-mhm 18878  df-submnd 18879  df-grp 19047  df-minusg 19048  df-sbg 19049  df-mulg 19178  df-subg 19233  df-ghm 19328  df-cntz 19431  df-cmn 19896  df-abl 19897  df-mgp 20261  df-rng 20275  df-ur 20308  df-srg 20313  df-ring 20361  df-cring 20362  df-oppr 20465  df-dvdsr 20485  df-unit 20486  df-invr 20516  df-dvr 20529  df-rhm 20600  df-subrng 20695  df-subrg 20719  df-drng 20879  df-field 20880  df-sdrg 20940  df-lmod 21033  df-lss 21103  df-lsp 21143  df-lmhm 21193  df-sra 21344  df-assa 22053  df-asp 22054  df-ascl 22055  df-psr 22109  df-mvr 22110  df-mpl 22111  df-opsr 22113  df-evls 22275  df-evl 22276  df-psr1 22390  df-vr1 22391  df-ply1 22392  df-coe1 22393  df-evls1 22525  df-evl1 22526  df-mon1 26339  df-fldgen 33696  df-irng 34138

This theorem is used by:  algextdeglem3  34173  algextdeglem4  34174