Theorem selvval2lem4 40223

Description: The fourth argument passed to evalSub is in the domain (a polynomial in (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))). (Contributed by SN, 5-Nov-2023.)

Hypotheses

Ref	Expression
selvval2lem4.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
selvval2lem4.b	⊢ 𝐵 = (Base‘𝑃)
selvval2lem4.u	⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
selvval2lem4.t	⊢ 𝑇 = (𝐽 mPoly 𝑈)
selvval2lem4.c	⊢ 𝐶 = (algSc‘𝑇)
selvval2lem4.d	⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈))
selvval2lem4.s	⊢ 𝑆 = (𝑇 ↾s ran 𝐷)
selvval2lem4.w	⊢ 𝑊 = (𝐼 mPoly 𝑆)
selvval2lem4.x	⊢ 𝑋 = (Base‘𝑊)
selvval2lem4.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
selvval2lem4.r	⊢ (𝜑 → 𝑅 ∈ CRing)
selvval2lem4.j	⊢ (𝜑 → 𝐽 ⊆ 𝐼)
selvval2lem4.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)

Assertion

Ref	Expression
selvval2lem4	⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ 𝑋)

Proof of Theorem selvval2lem4

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		selvval2lem4.u	. . . . . . . 8 ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
2		selvval2lem4.t	. . . . . . . 8 ⊢ 𝑇 = (𝐽 mPoly 𝑈)
3		selvval2lem4.c	. . . . . . . 8 ⊢ 𝐶 = (algSc‘𝑇)
4		selvval2lem4.d	. . . . . . . 8 ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈))
5		selvval2lem4.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6	5	difexd 5257	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V)
7		selvval2lem4.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
8	5, 7	ssexd 5252	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ V)
9		selvval2lem4.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing)
10	1, 2, 3, 4, 6, 8, 9	selvval2lem2 40220	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (𝑅 RingHom 𝑇))
11		eqid 2740	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
12		eqid 2740	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
13	11, 12	rhmf 19966	. . . . . . 7 ⊢ (𝐷 ∈ (𝑅 RingHom 𝑇) → 𝐷:(Base‘𝑅)⟶(Base‘𝑇))
14		ffrn 6611	. . . . . . 7 ⊢ (𝐷:(Base‘𝑅)⟶(Base‘𝑇) → 𝐷:(Base‘𝑅)⟶ran 𝐷)
15	10, 13, 14	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐷:(Base‘𝑅)⟶ran 𝐷)
16	1, 2, 3, 4, 6, 8, 9	selvval2lem3 40221	. . . . . . . 8 ⊢ (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇))
17	12	subrgss 20021	. . . . . . . 8 ⊢ (ran 𝐷 ∈ (SubRing‘𝑇) → ran 𝐷 ⊆ (Base‘𝑇))
18		selvval2lem4.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑇 ↾s ran 𝐷)
19	18, 12	ressbas2 16945	. . . . . . . 8 ⊢ (ran 𝐷 ⊆ (Base‘𝑇) → ran 𝐷 = (Base‘𝑆))
20	16, 17, 19	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐷 = (Base‘𝑆))
21	20	feq3d 6584	. . . . . 6 ⊢ (𝜑 → (𝐷:(Base‘𝑅)⟶ran 𝐷 ↔ 𝐷:(Base‘𝑅)⟶(Base‘𝑆)))
22	15, 21	mpbid 231	. . . . 5 ⊢ (𝜑 → 𝐷:(Base‘𝑅)⟶(Base‘𝑆))
23		selvval2lem4.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
24		selvval2lem4.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
25		eqid 2740	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
26		selvval2lem4.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
27	23, 11, 24, 25, 26	mplelf 21200	. . . . 5 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
28	22, 27	fcod 6623	. . . 4 ⊢ (𝜑 → (𝐷 ∘ 𝐹):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑆))
29		fvexd 6784	. . . . 5 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
30		ovex 7302	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
31	30	rabex 5260	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V
32	31	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
33	29, 32	elmapd 8610	. . . 4 ⊢ (𝜑 → ((𝐷 ∘ 𝐹) ∈ ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ↔ (𝐷 ∘ 𝐹):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑆)))
34	28, 33	mpbird 256	. . 3 ⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
35		eqid 2740	. . . 4 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆)
36		eqid 2740	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
37		eqid 2740	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆))
38	35, 36, 25, 37, 5	psrbas 21143	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑆)) = ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
39	34, 38	eleqtrrd 2844	. 2 ⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)))
40		fvexd 6784	. . 3 ⊢ (𝜑 → (0g‘𝑆) ∈ V)
41		crngring 19791	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
42		eqid 2740	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
43	11, 42	ring0cl 19804	. . . 4 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
44	9, 41, 43	3syl 18	. . 3 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
45		ssidd 3949	. . 3 ⊢ (𝜑 → (Base‘𝑅) ⊆ (Base‘𝑅))
46		fvexd 6784	. . 3 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
47	23, 24, 42, 26, 9	mplelsfi 21197	. . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅))
48	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐷 = (𝐶 ∘ (algSc‘𝑈)))
49	48	fveq1d 6771	. . . 4 ⊢ (𝜑 → (𝐷‘(0g‘𝑅)) = ((𝐶 ∘ (algSc‘𝑈))‘(0g‘𝑅)))
50		eqid 2740	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
51		eqid 2740	. . . . . 6 ⊢ (algSc‘𝑈) = (algSc‘𝑈)
52	9, 41	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
53	1, 50, 11, 51, 6, 52	mplasclf 21269	. . . . 5 ⊢ (𝜑 → (algSc‘𝑈):(Base‘𝑅)⟶(Base‘𝑈))
54	53, 44	fvco3d 6863	. . . 4 ⊢ (𝜑 → ((𝐶 ∘ (algSc‘𝑈))‘(0g‘𝑅)) = (𝐶‘((algSc‘𝑈)‘(0g‘𝑅))))
55	1, 6, 9	mplsca 21213	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑈))
56	55	fveq2d 6773	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑈)))
57	56	fveq2d 6773	. . . . . . 7 ⊢ (𝜑 → ((algSc‘𝑈)‘(0g‘𝑅)) = ((algSc‘𝑈)‘(0g‘(Scalar‘𝑈))))
58		eqid 2740	. . . . . . . 8 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
59	1	mplassa 21223	. . . . . . . . . 10 ⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ CRing) → 𝑈 ∈ AssAlg)
60	6, 9, 59	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ AssAlg)
61		assalmod 21063	. . . . . . . . 9 ⊢ (𝑈 ∈ AssAlg → 𝑈 ∈ LMod)
62	60, 61	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
63		assaring 21064	. . . . . . . . 9 ⊢ (𝑈 ∈ AssAlg → 𝑈 ∈ Ring)
64	60, 63	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ Ring)
65	51, 58, 62, 64	ascl0 21084	. . . . . . 7 ⊢ (𝜑 → ((algSc‘𝑈)‘(0g‘(Scalar‘𝑈))) = (0g‘𝑈))
66	57, 65	eqtrd 2780	. . . . . 6 ⊢ (𝜑 → ((algSc‘𝑈)‘(0g‘𝑅)) = (0g‘𝑈))
67	66	fveq2d 6773	. . . . 5 ⊢ (𝜑 → (𝐶‘((algSc‘𝑈)‘(0g‘𝑅))) = (𝐶‘(0g‘𝑈)))
68	2, 8, 60	mplsca 21213	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Scalar‘𝑇))
69	68	fveq2d 6773	. . . . . 6 ⊢ (𝜑 → (0g‘𝑈) = (0g‘(Scalar‘𝑇)))
70	69	fveq2d 6773	. . . . 5 ⊢ (𝜑 → (𝐶‘(0g‘𝑈)) = (𝐶‘(0g‘(Scalar‘𝑇))))
71		eqid 2740	. . . . . . 7 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
72	1, 2, 6, 8, 9	selvval2lem1 40219	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ AssAlg)
73		assalmod 21063	. . . . . . . 8 ⊢ (𝑇 ∈ AssAlg → 𝑇 ∈ LMod)
74	72, 73	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ LMod)
75		assaring 21064	. . . . . . . 8 ⊢ (𝑇 ∈ AssAlg → 𝑇 ∈ Ring)
76	72, 75	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ Ring)
77	3, 71, 74, 76	ascl0 21084	. . . . . 6 ⊢ (𝜑 → (𝐶‘(0g‘(Scalar‘𝑇))) = (0g‘𝑇))
78		subrgsubg 20026	. . . . . . 7 ⊢ (ran 𝐷 ∈ (SubRing‘𝑇) → ran 𝐷 ∈ (SubGrp‘𝑇))
79		eqid 2740	. . . . . . . 8 ⊢ (0g‘𝑇) = (0g‘𝑇)
80	18, 79	subg0 18757	. . . . . . 7 ⊢ (ran 𝐷 ∈ (SubGrp‘𝑇) → (0g‘𝑇) = (0g‘𝑆))
81	16, 78, 80	3syl 18	. . . . . 6 ⊢ (𝜑 → (0g‘𝑇) = (0g‘𝑆))
82	77, 81	eqtrd 2780	. . . . 5 ⊢ (𝜑 → (𝐶‘(0g‘(Scalar‘𝑇))) = (0g‘𝑆))
83	67, 70, 82	3eqtrd 2784	. . . 4 ⊢ (𝜑 → (𝐶‘((algSc‘𝑈)‘(0g‘𝑅))) = (0g‘𝑆))
84	49, 54, 83	3eqtrd 2784	. . 3 ⊢ (𝜑 → (𝐷‘(0g‘𝑅)) = (0g‘𝑆))
85	40, 44, 27, 15, 45, 32, 46, 47, 84	fsuppcor 9139	. 2 ⊢ (𝜑 → (𝐷 ∘ 𝐹) finSupp (0g‘𝑆))
86		selvval2lem4.w	. . 3 ⊢ 𝑊 = (𝐼 mPoly 𝑆)
87		eqid 2740	. . 3 ⊢ (0g‘𝑆) = (0g‘𝑆)
88		selvval2lem4.x	. . 3 ⊢ 𝑋 = (Base‘𝑊)
89	86, 35, 37, 87, 88	mplelbas 21195	. 2 ⊢ ((𝐷 ∘ 𝐹) ∈ 𝑋 ↔ ((𝐷 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)) ∧ (𝐷 ∘ 𝐹) finSupp (0g‘𝑆)))
90	39, 85, 89	sylanbrc 583	1 ⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2110  {crab 3070  Vcvv 3431   ∖ cdif 3889   ⊆ wss 3892   class class class wbr 5079  ^◡ccnv 5588  ran crn 5590   “ cima 5592   ∘ ccom 5593  ⟶wf 6427  ‘cfv 6431  (class class class)co 7269   ↑_m cmap 8596  Fincfn 8714   finSupp cfsupp 9104  ℕcn 11971  ℕ₀cn0 12231  Basecbs 16908   ↾_s cress 16937  Scalarcsca 16961  0_gc0g 17146  SubGrpcsubg 18745  Ringcrg 19779  CRingccrg 19780   RingHom crh 19952  SubRingcsubrg 20016  LModclmod 20119  AssAlgcasa 21053  algSccascl 21055   mPwSer cmps 21103   mPoly cmpl 21105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-cnex 10926  ax-resscn 10927  ax-1cn 10928  ax-icn 10929  ax-addcl 10930  ax-addrcl 10931  ax-mulcl 10932  ax-mulrcl 10933  ax-mulcom 10934  ax-addass 10935  ax-mulass 10936  ax-distr 10937  ax-i2m1 10938  ax-1ne0 10939  ax-1rid 10940  ax-rnegex 10941  ax-rrecex 10942  ax-cnre 10943  ax-pre-lttri 10944  ax-pre-lttrn 10945  ax-pre-ltadd 10946  ax-pre-mulgt0 10947

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-isom 6440  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-of 7525  df-ofr 7526  df-om 7705  df-1st 7822  df-2nd 7823  df-supp 7967  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-1o 8286  df-er 8479  df-map 8598  df-pm 8599  df-ixp 8667  df-en 8715  df-dom 8716  df-sdom 8717  df-fin 8718  df-fsupp 9105  df-oi 9245  df-card 9696  df-pnf 11010  df-mnf 11011  df-xr 11012  df-ltxr 11013  df-le 11014  df-sub 11205  df-neg 11206  df-nn 11972  df-2 12034  df-3 12035  df-4 12036  df-5 12037  df-6 12038  df-7 12039  df-8 12040  df-9 12041  df-n0 12232  df-z 12318  df-uz 12580  df-fz 13237  df-fzo 13380  df-seq 13718  df-hash 14041  df-struct 16844  df-sets 16861  df-slot 16879  df-ndx 16891  df-base 16909  df-ress 16938  df-plusg 16971  df-mulr 16972  df-sca 16974  df-vsca 16975  df-tset 16977  df-0g 17148  df-gsum 17149  df-mre 17291  df-mrc 17292  df-acs 17294  df-mgm 18322  df-sgrp 18371  df-mnd 18382  df-mhm 18426  df-submnd 18427  df-grp 18576  df-minusg 18577  df-sbg 18578  df-mulg 18697  df-subg 18748  df-ghm 18828  df-cntz 18919  df-cmn 19384  df-abl 19385  df-mgp 19717  df-ur 19734  df-ring 19781  df-cring 19782  df-rnghom 19955  df-subrg 20018  df-lmod 20121  df-lss 20190  df-assa 21056  df-ascl 21058  df-psr 21108  df-mpl 21110

This theorem is referenced by:  selvcl  40225