Theorem selvval2lem4 39185

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		difexg 5231	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (𝐼 ∖ 𝐽) ∈ V)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V)
8		selvval2lem4.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
9	5, 8	ssexd 5228	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ V)
10		selvval2lem4.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing)
11	1, 2, 3, 4, 7, 9, 10	selvval2lem2 39182	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (𝑅 RingHom 𝑇))
12		eqid 2821	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
13		eqid 2821	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
14	12, 13	rhmf 19478	. . . . . . 7 ⊢ (𝐷 ∈ (𝑅 RingHom 𝑇) → 𝐷:(Base‘𝑅)⟶(Base‘𝑇))
15		ffrn 6526	. . . . . . 7 ⊢ (𝐷:(Base‘𝑅)⟶(Base‘𝑇) → 𝐷:(Base‘𝑅)⟶ran 𝐷)
16	11, 14, 15	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐷:(Base‘𝑅)⟶ran 𝐷)
17	1, 2, 3, 4, 7, 9, 10	selvval2lem3 39183	. . . . . . . 8 ⊢ (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇))
18	13	subrgss 19536	. . . . . . . 8 ⊢ (ran 𝐷 ∈ (SubRing‘𝑇) → ran 𝐷 ⊆ (Base‘𝑇))
19		selvval2lem4.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑇 ↾s ran 𝐷)
20	19, 13	ressbas2 16555	. . . . . . . 8 ⊢ (ran 𝐷 ⊆ (Base‘𝑇) → ran 𝐷 = (Base‘𝑆))
21	17, 18, 20	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐷 = (Base‘𝑆))
22	21	feq3d 6501	. . . . . 6 ⊢ (𝜑 → (𝐷:(Base‘𝑅)⟶ran 𝐷 ↔ 𝐷:(Base‘𝑅)⟶(Base‘𝑆)))
23	16, 22	mpbid 234	. . . . 5 ⊢ (𝜑 → 𝐷:(Base‘𝑅)⟶(Base‘𝑆))
24		selvval2lem4.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
25		selvval2lem4.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
26		eqid 2821	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
27		selvval2lem4.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
28	24, 12, 25, 26, 27	mplelf 20213	. . . . 5 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
29	23, 28	fcod 6532	. . . 4 ⊢ (𝜑 → (𝐷 ∘ 𝐹):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑆))
30		fvexd 6685	. . . . 5 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
31		ovex 7189	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
32	31	rabex 5235	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V
33	32	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
34	30, 33	elmapd 8420	. . . 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)))
35	29, 34	mpbird 259	. . 3 ⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
36		eqid 2821	. . . 4 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆)
37		eqid 2821	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
38		eqid 2821	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆))
39	36, 37, 26, 38, 5	psrbas 20158	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑆)) = ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
40	35, 39	eleqtrrd 2916	. 2 ⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)))
41		fvexd 6685	. . 3 ⊢ (𝜑 → (0g‘𝑆) ∈ V)
42		crngring 19308	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
43		eqid 2821	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
44	12, 43	ring0cl 19319	. . . 4 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
45	10, 42, 44	3syl 18	. . 3 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
46		ssidd 3990	. . 3 ⊢ (𝜑 → (Base‘𝑅) ⊆ (Base‘𝑅))
47		fvexd 6685	. . 3 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
48	24, 25, 43, 27, 10	mplelsfi 20271	. . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅))
49	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐷 = (𝐶 ∘ (algSc‘𝑈)))
50	49	fveq1d 6672	. . . 4 ⊢ (𝜑 → (𝐷‘(0g‘𝑅)) = ((𝐶 ∘ (algSc‘𝑈))‘(0g‘𝑅)))
51		eqid 2821	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
52		eqid 2821	. . . . . 6 ⊢ (algSc‘𝑈) = (algSc‘𝑈)
53	10, 42	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
54	1, 51, 12, 52, 7, 53	mplasclf 20277	. . . . 5 ⊢ (𝜑 → (algSc‘𝑈):(Base‘𝑅)⟶(Base‘𝑈))
55	54, 45	fvco3d 6761	. . . 4 ⊢ (𝜑 → ((𝐶 ∘ (algSc‘𝑈))‘(0g‘𝑅)) = (𝐶‘((algSc‘𝑈)‘(0g‘𝑅))))
56	1, 7, 10	mplsca 20225	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑈))
57	56	fveq2d 6674	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘𝑈)))
58	57	fveq2d 6674	. . . . . . 7 ⊢ (𝜑 → ((algSc‘𝑈)‘(0g‘𝑅)) = ((algSc‘𝑈)‘(0g‘(Scalar‘𝑈))))
59		eqid 2821	. . . . . . . 8 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
60	1	mplassa 20235	. . . . . . . . . 10 ⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ CRing) → 𝑈 ∈ AssAlg)
61	7, 10, 60	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ AssAlg)
62		assalmod 20092	. . . . . . . . 9 ⊢ (𝑈 ∈ AssAlg → 𝑈 ∈ LMod)
63	61, 62	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
64		assaring 20093	. . . . . . . . 9 ⊢ (𝑈 ∈ AssAlg → 𝑈 ∈ Ring)
65	61, 64	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ Ring)
66	52, 59, 63, 65	ascl0 20113	. . . . . . 7 ⊢ (𝜑 → ((algSc‘𝑈)‘(0g‘(Scalar‘𝑈))) = (0g‘𝑈))
67	58, 66	eqtrd 2856	. . . . . 6 ⊢ (𝜑 → ((algSc‘𝑈)‘(0g‘𝑅)) = (0g‘𝑈))
68	67	fveq2d 6674	. . . . 5 ⊢ (𝜑 → (𝐶‘((algSc‘𝑈)‘(0g‘𝑅))) = (𝐶‘(0g‘𝑈)))
69	2, 9, 61	mplsca 20225	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Scalar‘𝑇))
70	69	fveq2d 6674	. . . . . 6 ⊢ (𝜑 → (0g‘𝑈) = (0g‘(Scalar‘𝑇)))
71	70	fveq2d 6674	. . . . 5 ⊢ (𝜑 → (𝐶‘(0g‘𝑈)) = (𝐶‘(0g‘(Scalar‘𝑇))))
72		eqid 2821	. . . . . . 7 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
73	1, 2, 7, 9, 10	selvval2lem1 39181	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ AssAlg)
74		assalmod 20092	. . . . . . . 8 ⊢ (𝑇 ∈ AssAlg → 𝑇 ∈ LMod)
75	73, 74	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ LMod)
76		assaring 20093	. . . . . . . 8 ⊢ (𝑇 ∈ AssAlg → 𝑇 ∈ Ring)
77	73, 76	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ Ring)
78	3, 72, 75, 77	ascl0 20113	. . . . . 6 ⊢ (𝜑 → (𝐶‘(0g‘(Scalar‘𝑇))) = (0g‘𝑇))
79		subrgsubg 19541	. . . . . . 7 ⊢ (ran 𝐷 ∈ (SubRing‘𝑇) → ran 𝐷 ∈ (SubGrp‘𝑇))
80		eqid 2821	. . . . . . . 8 ⊢ (0g‘𝑇) = (0g‘𝑇)
81	19, 80	subg0 18285	. . . . . . 7 ⊢ (ran 𝐷 ∈ (SubGrp‘𝑇) → (0g‘𝑇) = (0g‘𝑆))
82	17, 79, 81	3syl 18	. . . . . 6 ⊢ (𝜑 → (0g‘𝑇) = (0g‘𝑆))
83	78, 82	eqtrd 2856	. . . . 5 ⊢ (𝜑 → (𝐶‘(0g‘(Scalar‘𝑇))) = (0g‘𝑆))
84	68, 71, 83	3eqtrd 2860	. . . 4 ⊢ (𝜑 → (𝐶‘((algSc‘𝑈)‘(0g‘𝑅))) = (0g‘𝑆))
85	50, 55, 84	3eqtrd 2860	. . 3 ⊢ (𝜑 → (𝐷‘(0g‘𝑅)) = (0g‘𝑆))
86	41, 45, 28, 16, 46, 33, 47, 48, 85	fsuppcor 8867	. 2 ⊢ (𝜑 → (𝐷 ∘ 𝐹) finSupp (0g‘𝑆))
87		selvval2lem4.w	. . 3 ⊢ 𝑊 = (𝐼 mPoly 𝑆)
88		eqid 2821	. . 3 ⊢ (0g‘𝑆) = (0g‘𝑆)
89		selvval2lem4.x	. . 3 ⊢ 𝑋 = (Base‘𝑊)
90	87, 36, 38, 88, 89	mplelbas 20210	. 2 ⊢ ((𝐷 ∘ 𝐹) ∈ 𝑋 ↔ ((𝐷 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)) ∧ (𝐷 ∘ 𝐹) finSupp (0g‘𝑆)))
91	40, 86, 90	sylanbrc 585	1 ⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  {crab 3142  Vcvv 3494   ∖ cdif 3933   ⊆ wss 3936   class class class wbr 5066  ^◡ccnv 5554  ran crn 5556   “ cima 5558   ∘ ccom 5559  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ↑_m cmap 8406  Fincfn 8509   finSupp cfsupp 8833  ℕcn 11638  ℕ₀cn0 11898  Basecbs 16483   ↾_s cress 16484  Scalarcsca 16568  0_gc0g 16713  SubGrpcsubg 18273  Ringcrg 19297  CRingccrg 19298   RingHom crh 19464  SubRingcsubrg 19531  LModclmod 19634  AssAlgcasa 20082  algSccascl 20084   mPwSer cmps 20131   mPoly cmpl 20133

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-ofr 7410  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-oi 8974  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-fzo 13035  df-seq 13371  df-hash 13692  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-tset 16584  df-0g 16715  df-gsum 16716  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-subg 18276  df-ghm 18356  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-cring 19300  df-rnghom 19467  df-subrg 19533  df-lmod 19636  df-lss 19704  df-assa 20085  df-ascl 20087  df-psr 20136  df-mpl 20138

This theorem is referenced by:  selvcl  39187