Theorem algextdeglem2 33875

Description: Lemma for algextdeg 33882. 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 20721	. . . . . 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 2736	. . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
6	5	sralmod 21139	. . . 4 ⊢ (𝐹 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
7	4, 6	syl 17	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
8		eqid 2736	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
9		eqid 2736	. . . 4 ⊢ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
10		algextdeg.f	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Field)
11	10	flddrngd 20674	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ DivRing)
12		subrgsubg 20510	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
13	8	subgss 19057	. . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (0g‘𝐸) = (0g‘𝐸)
18	10	fldcrngd 20675	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ CRing)
19	15, 16, 8, 17, 18, 4	irngssv 33845	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
20		algextdeg.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
21	19, 20	sseldd 3934	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
22	21	snssd 4765	. . . . . . . 8 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸))
23	14, 22	unssd 4144	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
24	8, 11, 23	fldgensdrg 33396	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
25		issdrg 20721	. . . . . 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 33395	. . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
29	28	unssad 4145	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
30	9	subsubrg 20531	. . . . . 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 33744	. . 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 2759	. . . . . 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 33827	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
43	42	ralrimiva 3128	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
44		algextdeglem.g	. . . . 5 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
45	44	rnmptss 7068	. . . 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 22321	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
48		eqid 2736	. . . . 5 ⊢ (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) = (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
49		eqid 2736	. . . . 5 ⊢ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)) = (LSubSp‘((subringAlg ‘𝐸)‘𝐹))
50	48, 49	reslmhm2b 21006	. . . 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 33397	. . . 4 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
55	8, 53, 54, 29, 10	resssra 33743	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))))
56	55	oveq2d 7374	. 2 ⊢ (𝜑 → (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)) = (𝑃 LMHom (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
57	52, 56	eleqtrrd 2839	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∪ cun 3899   ⊆ wss 3901  {csn 4580  ∪ cuni 4863   ↦ cmpt 5179  ^◡ccnv 5623  ran crn 5625   “ cima 5627  ‘cfv 6492  (class class class)co 7358  [cec 8633  Basecbs 17136   ↾_s cress 17157  0_gc0g 17359   /_s cqus 17426  SubGrpcsubg 19050   ~_QG cqg 19052  SubRingcsubrg 20502  DivRingcdr 20662  Fieldcfield 20663  SubDRingcsdrg 20719  LModclmod 20811  LSubSpclss 20882   LMHom clmhm 20971  subringAlg csra 21123  Poly₁cpl1 22117   evalSub₁ ces1 22257  deg₁cdg1 26015   fldGen cfldgen 33392   IntgRing cirng 33840   minPoly cminply 33856

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-ghm 19142  df-cntz 19246  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-srg 20122  df-ring 20170  df-cring 20171  df-oppr 20273  df-dvdsr 20293  df-unit 20294  df-invr 20324  df-dvr 20337  df-rhm 20408  df-subrng 20479  df-subrg 20503  df-drng 20664  df-field 20665  df-sdrg 20720  df-lmod 20813  df-lss 20883  df-lsp 20923  df-lmhm 20974  df-sra 21125  df-assa 21808  df-asp 21809  df-ascl 21810  df-psr 21865  df-mvr 21866  df-mpl 21867  df-opsr 21869  df-evls 22029  df-evl 22030  df-psr1 22120  df-vr1 22121  df-ply1 22122  df-coe1 22123  df-evls1 22259  df-evl1 22260  df-mon1 26092  df-fldgen 33393  df-irng 33841

This theorem is referenced by:  algextdeglem3  33876  algextdeglem4  33877