Theorem algextdeglem2 34228

Description: Lemma for algextdeg 34235. 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 20954	. . . . . 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 2760	. . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
6	5	sralmod 21371	. . . 4 ⊢ (𝐹 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
7	4, 6	syl 18	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
8		eqid 2760	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
9		eqid 2760	. . . 4 ⊢ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
10		algextdeg.f	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Field)
11	10	flddrngd 20904	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ DivRing)
12		subrgsubg 20739	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
13	8	subgss 19250	. . . . . . . . 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 2760	. . . . . . . . . . 11 ⊢ (0g‘𝐸) = (0g‘𝐸)
18	10	fldcrngd 20905	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ CRing)
19	15, 16, 8, 17, 18, 4	irngssv 34198	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
20		algextdeg.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
21	19, 20	sseldd 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
22	21	snssd 4747	. . . . . . . 8 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸))
23	14, 22	unssd 4138	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
24	8, 11, 23	fldgensdrg 33755	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
25		issdrg 20954	. . . . . 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 33754	. . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
29	28	unssad 4139	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
30	9	subsubrg 20760	. . . . . 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 34098	. . 3 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)))
34		algextdeglem.y	. . . . . . 7 ⊢ 𝑃 = (Poly1‘𝐾)
35	16	fveq2i 6881	. . . . . . 7 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
36	34, 35	eqtri 2783	. . . . . 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 34180	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
43	42	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
44		algextdeglem.g	. . . . 5 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
45	44	rnmptss 7116	. . . 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 22602	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
48		eqid 2760	. . . . 5 ⊢ (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) = (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
49		eqid 2760	. . . . 5 ⊢ (LSubSp‘((subringAlg ‘𝐸)‘𝐹)) = (LSubSp‘((subringAlg ‘𝐸)‘𝐹))
50	48, 49	reslmhm2b 21238	. . . 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 33756	. . . 4 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
55	8, 53, 54, 29, 10	resssra 34097	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))))
56	55	oveq2d 7429	. 2 ⊢ (𝜑 → (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)) = (𝑃 LMHom (((subringAlg ‘𝐸)‘𝐹) ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))))
57	52, 56	eleqtrrd 2863	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∪ cun 3897   ⊆ wss 3899  {csn 4584  ∪ cuni 4867   ↦ cmpt 5186  ^◡ccnv 5654  ran crn 5656   “ cima 5658  ‘cfv 6533  (class class class)co 7413  [cec 8694  Basecbs 17301   ↾_s cress 17322  0_gc0g 17524   /_s cqus 17591  SubGrpcsubg 19243   ~_QG cqg 19245  SubRingcsubrg 20731  DivRingcdr 20890  Fieldcfield 20891  SubDRingcsdrg 20952  LModclmod 21044  LSubSpclss 21115   LMHom clmhm 21203  subringAlg csra 21355  Poly₁cpl1 22402   evalSub₁ ces1 22538  deg₁cdg1 26279   fldGen cfldgen 33751   IntgRing cirng 34193   minPoly cminply 34209

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-ofr 7679  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8159  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-pm 8829  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-sup 9412  df-oi 9482  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-fz 13562  df-fzo 13710  df-seq 14066  df-hash 14395  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-sca 17358  df-vsca 17359  df-ip 17360  df-tset 17361  df-ple 17362  df-ds 17364  df-hom 17366  df-cco 17367  df-0g 17526  df-gsum 17527  df-prds 17532  df-pws 17534  df-mre 17670  df-mrc 17671  df-acs 17673  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-mhm 18891  df-submnd 18892  df-grp 19060  df-minusg 19061  df-sbg 19062  df-mulg 19191  df-subg 19246  df-ghm 19341  df-cntz 19444  df-cmn 19909  df-abl 19910  df-mgp 20274  df-rng 20288  df-ur 20321  df-srg 20326  df-ring 20374  df-cring 20375  df-oppr 20478  df-dvdsr 20498  df-unit 20499  df-invr 20529  df-dvr 20542  df-rhm 20613  df-subrng 20708  df-subrg 20732  df-drng 20892  df-field 20893  df-sdrg 20953  df-lmod 21046  df-lss 21116  df-lsp 21156  df-lmhm 21206  df-sra 21357  df-assa 22068  df-asp 22069  df-ascl 22070  df-psr 22124  df-mvr 22125  df-mpl 22126  df-opsr 22128  df-evls 22290  df-evl 22291  df-psr1 22405  df-vr1 22406  df-ply1 22407  df-coe1 22408  df-evls1 22540  df-evl1 22541  df-mon1 26356  df-fldgen 33752  df-irng 34194

This theorem is used by:  algextdeglem3  34229  algextdeglem4  34230