Theorem algextdeglem2 33878

Description: Lemma for algextdeg 33885. 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 20756	. . . . . 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 1144	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
5		eqid 2737	. . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
6	5	sralmod 21174	. . . 4 ⊢ (𝐹 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
7	4, 6	syl 17	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
8		eqid 2737	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
9		eqid 2737	. . . 4 ⊢ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
10		algextdeg.f	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Field)
11	10	flddrngd 20709	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ DivRing)
12		subrgsubg 20545	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
13	8	subgss 19094	. . . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (0g‘𝐸) = (0g‘𝐸)
18	10	fldcrngd 20710	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ CRing)
19	15, 16, 8, 17, 18, 4	irngssv 33848	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
20		algextdeg.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
21	19, 20	sseldd 3923	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
22	21	snssd 4753	. . . . . . . 8 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸))
23	14, 22	unssd 4133	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
24	8, 11, 23	fldgensdrg 33390	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
25		issdrg 20756	. . . . . 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 1144	. . . 4 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
28	8, 11, 23	fldgenssid 33389	. . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
29	28	unssad 4134	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
30	9	subsubrg 20566	. . . . . 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 837	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘(𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
33	5, 8, 9, 27, 32	lsssra 33747	. . 3 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)))
34		algextdeglem.y	. . . . . . 7 ⊢ 𝑃 = (Poly1‘𝐾)
35	16	fveq2i 6837	. . . . . . 7 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
36	34, 35	eqtri 2760	. . . . . 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 33830	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
43	42	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
44		algextdeglem.g	. . . . 5 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
45	44	rnmptss 7069	. . . 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 22352	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
48		eqid 2737	. . . . 5 ⊢ (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) = (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
49		eqid 2737	. . . . 5 ⊢ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)) = (LSubSp‘((subringAlg ‘𝐸)‘𝐹))
50	48, 49	reslmhm2b 21041	. . . 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 1376	. 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 33391	. . . 4 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
55	8, 53, 54, 29, 10	resssra 33746	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))))
56	55	oveq2d 7376	. 2 ⊢ (𝜑 → (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)) = (𝑃 LMHom (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
57	52, 56	eleqtrrd 2840	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∪ cun 3888   ⊆ wss 3890  {csn 4568  ∪ cuni 4851   ↦ cmpt 5167  ^◡ccnv 5623  ran crn 5625   “ cima 5627  ‘cfv 6492  (class class class)co 7360  [cec 8634  Basecbs 17170   ↾_s cress 17191  0_gc0g 17393   /_s cqus 17460  SubGrpcsubg 19087   ~_QG cqg 19089  SubRingcsubrg 20537  DivRingcdr 20697  Fieldcfield 20698  SubDRingcsdrg 20754  LModclmod 20846  LSubSpclss 20917   LMHom clmhm 21006  subringAlg csra 21158  Poly₁cpl1 22150   evalSub₁ ces1 22288  deg₁cdg1 26029   fldGen cfldgen 33386   IntgRing cirng 33843   minPoly cminply 33859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-ofr 7625  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-tpos 8169  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-sup 9348  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-hom 17235  df-cco 17236  df-0g 17395  df-gsum 17396  df-prds 17401  df-pws 17403  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-mulg 19035  df-subg 19090  df-ghm 19179  df-cntz 19283  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-srg 20159  df-ring 20207  df-cring 20208  df-oppr 20308  df-dvdsr 20328  df-unit 20329  df-invr 20359  df-dvr 20372  df-rhm 20443  df-subrng 20514  df-subrg 20538  df-drng 20699  df-field 20700  df-sdrg 20755  df-lmod 20848  df-lss 20918  df-lsp 20958  df-lmhm 21009  df-sra 21160  df-assa 21843  df-asp 21844  df-ascl 21845  df-psr 21899  df-mvr 21900  df-mpl 21901  df-opsr 21903  df-evls 22062  df-evl 22063  df-psr1 22153  df-vr1 22154  df-ply1 22155  df-coe1 22156  df-evls1 22290  df-evl1 22291  df-mon1 26106  df-fldgen 33387  df-irng 33844

This theorem is referenced by:  algextdeglem3  33879  algextdeglem4  33880