Theorem mhpinvcl 20339

Description: Homogeneous polynomials are closed under taking the opposite. (Contributed by SN, 12-Sep-2023.)

Hypotheses

Ref	Expression
mhpinvcl.h	⊢ 𝐻 = (𝐼 mHomP 𝑅)
mhpinvcl.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mhpinvcl.m	⊢ 𝑀 = (invg‘𝑃)
mhpinvcl.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mhpinvcl.r	⊢ (𝜑 → 𝑅 ∈ Grp)
mhpinvcl.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
mhpinvcl.x	⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))

Assertion

Ref	Expression
mhpinvcl	⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐻‘𝑁))

Proof of Theorem mhpinvcl

Dummy variables 𝑔 ℎ 𝑗 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhpinvcl.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
2		mhpinvcl.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
3		mhpinvcl.p	. . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
4	3	mplgrp 20230	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp)
5	1, 2, 4	syl2anc 586	. . 3 ⊢ (𝜑 → 𝑃 ∈ Grp)
6		mhpinvcl.h	. . . 4 ⊢ 𝐻 = (𝐼 mHomP 𝑅)
7		eqid 2821	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
8		mhpinvcl.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
9		mhpinvcl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))
10	6, 3, 7, 1, 2, 8, 9	mhpmpl 20335	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃))
11		mhpinvcl.m	. . . 4 ⊢ 𝑀 = (invg‘𝑃)
12	7, 11	grpinvcl 18151	. . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑃)) → (𝑀‘𝑋) ∈ (Base‘𝑃))
13	5, 10, 12	syl2anc 586	. 2 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (Base‘𝑃))
14		eqid 2821	. . . . . . . 8 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
15		eqid 2821	. . . . . . . . . 10 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
16		eqid 2821	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
17	3, 14, 15, 16, 7	mplbas 20209	. . . . . . . . 9 ⊢ (Base‘𝑃) = {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)}
18	17	eqcomi 2830	. . . . . . . 8 ⊢ {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)} = (Base‘𝑃)
19	14, 3, 18, 1, 2	mplsubg 20217	. . . . . . 7 ⊢ (𝜑 → {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)} ∈ (SubGrp‘(𝐼 mPwSer 𝑅)))
20	10, 17	eleqtrdi 2923	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)})
21		eqid 2821	. . . . . . . . 9 ⊢ {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)} = {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)}
22	3, 14, 15, 16, 21	mplval 20208	. . . . . . . 8 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)})
23		eqid 2821	. . . . . . . 8 ⊢ (invg‘(𝐼 mPwSer 𝑅)) = (invg‘(𝐼 mPwSer 𝑅))
24	22, 23, 11	subginv 18286	. . . . . . 7 ⊢ (({𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)} ∈ (SubGrp‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 ∈ {𝑓 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑓 finSupp (0g‘𝑅)}) → ((invg‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋))
25	19, 20, 24	syl2anc 586	. . . . . 6 ⊢ (𝜑 → ((invg‘(𝐼 mPwSer 𝑅))‘𝑋) = (𝑀‘𝑋))
26		eqid 2821	. . . . . . 7 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
27		eqid 2821	. . . . . . 7 ⊢ (invg‘𝑅) = (invg‘𝑅)
28	3, 14, 15, 16, 7	mplelbas 20210	. . . . . . . . 9 ⊢ (𝑋 ∈ (Base‘𝑃) ↔ (𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 finSupp (0g‘𝑅)))
29	10, 28	sylib 220	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 finSupp (0g‘𝑅)))
30	29	simpld 497	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅)))
31	14, 1, 2, 26, 27, 15, 23, 30	psrneg 20180	. . . . . 6 ⊢ (𝜑 → ((invg‘(𝐼 mPwSer 𝑅))‘𝑋) = ((invg‘𝑅) ∘ 𝑋))
32	25, 31	eqtr3d 2858	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑋) = ((invg‘𝑅) ∘ 𝑋))
33	32	oveq1d 7171	. . . 4 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) = (((invg‘𝑅) ∘ 𝑋) supp (0g‘𝑅)))
34		fvexd 6685	. . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
35		eqid 2821	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
36	35, 27	grpinvf 18150	. . . . . . 7 ⊢ (𝑅 ∈ Grp → (invg‘𝑅):(Base‘𝑅)⟶(Base‘𝑅))
37	2, 36	syl 17	. . . . . 6 ⊢ (𝜑 → (invg‘𝑅):(Base‘𝑅)⟶(Base‘𝑅))
38	37	ffnd 6515	. . . . 5 ⊢ (𝜑 → (invg‘𝑅) Fn (Base‘𝑅))
39		fvexd 6685	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
40	3, 35, 7, 26, 10	mplelf 20213	. . . . . 6 ⊢ (𝜑 → 𝑋:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
41	40	ffnd 6515	. . . . 5 ⊢ (𝜑 → 𝑋 Fn {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
42		ovexd 7191	. . . . . 6 ⊢ (𝜑 → (ℕ0 ↑m 𝐼) ∈ V)
43	26, 42	rabexd 5236	. . . . 5 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
44	34, 38, 39, 41, 43	suppcofnd 7871	. . . 4 ⊢ (𝜑 → (((invg‘𝑅) ∘ 𝑋) supp (0g‘𝑅)) = {𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ ((𝑋‘𝑥) ∈ (Base‘𝑅) ∧ ((invg‘𝑅)‘(𝑋‘𝑥)) ≠ (0g‘𝑅))})
45		suppvalfn 7837	. . . . . 6 ⊢ ((𝑋 Fn {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V ∧ (0g‘𝑅) ∈ V) → (𝑋 supp (0g‘𝑅)) = {𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ (𝑋‘𝑥) ≠ (0g‘𝑅)})
46	41, 43, 34, 45	syl3anc 1367	. . . . 5 ⊢ (𝜑 → (𝑋 supp (0g‘𝑅)) = {𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ (𝑋‘𝑥) ≠ (0g‘𝑅)})
47	2	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Grp)
48	40	ffvelrnda 6851	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑋‘𝑥) ∈ (Base‘𝑅))
49	35, 16	grpidcl 18131	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ (Base‘𝑅))
50	47, 49	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (0g‘𝑅) ∈ (Base‘𝑅))
51		eqid 2821	. . . . . . . . . . 11 ⊢ (+g‘𝑅) = (+g‘𝑅)
52	35, 51, 16, 27	grpinvid1 18154	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅) ∧ (0g‘𝑅) ∈ (Base‘𝑅)) → (((invg‘𝑅)‘(𝑋‘𝑥)) = (0g‘𝑅) ↔ ((𝑋‘𝑥)(+g‘𝑅)(0g‘𝑅)) = (0g‘𝑅)))
53	47, 48, 50, 52	syl3anc 1367	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((invg‘𝑅)‘(𝑋‘𝑥)) = (0g‘𝑅) ↔ ((𝑋‘𝑥)(+g‘𝑅)(0g‘𝑅)) = (0g‘𝑅)))
54	35, 51, 16	grprid 18134	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑥) ∈ (Base‘𝑅)) → ((𝑋‘𝑥)(+g‘𝑅)(0g‘𝑅)) = (𝑋‘𝑥))
55	47, 48, 54	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑋‘𝑥)(+g‘𝑅)(0g‘𝑅)) = (𝑋‘𝑥))
56	55	eqeq1d 2823	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((𝑋‘𝑥)(+g‘𝑅)(0g‘𝑅)) = (0g‘𝑅) ↔ (𝑋‘𝑥) = (0g‘𝑅)))
57	53, 56	bitr2d 282	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑋‘𝑥) = (0g‘𝑅) ↔ ((invg‘𝑅)‘(𝑋‘𝑥)) = (0g‘𝑅)))
58	57	necon3bid 3060	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑋‘𝑥) ≠ (0g‘𝑅) ↔ ((invg‘𝑅)‘(𝑋‘𝑥)) ≠ (0g‘𝑅)))
59	48	biantrurd 535	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (((invg‘𝑅)‘(𝑋‘𝑥)) ≠ (0g‘𝑅) ↔ ((𝑋‘𝑥) ∈ (Base‘𝑅) ∧ ((invg‘𝑅)‘(𝑋‘𝑥)) ≠ (0g‘𝑅))))
60	58, 59	bitrd 281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑋‘𝑥) ≠ (0g‘𝑅) ↔ ((𝑋‘𝑥) ∈ (Base‘𝑅) ∧ ((invg‘𝑅)‘(𝑋‘𝑥)) ≠ (0g‘𝑅))))
61	60	rabbidva 3478	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ (𝑋‘𝑥) ≠ (0g‘𝑅)} = {𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ ((𝑋‘𝑥) ∈ (Base‘𝑅) ∧ ((invg‘𝑅)‘(𝑋‘𝑥)) ≠ (0g‘𝑅))})
62	46, 61	eqtr2d 2857	. . . 4 ⊢ (𝜑 → {𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ ((𝑋‘𝑥) ∈ (Base‘𝑅) ∧ ((invg‘𝑅)‘(𝑋‘𝑥)) ≠ (0g‘𝑅))} = (𝑋 supp (0g‘𝑅)))
63	33, 44, 62	3eqtrd 2860	. . 3 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) = (𝑋 supp (0g‘𝑅)))
64		eqid 2821	. . . 4 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
65	6, 16, 64, 1, 2, 8, 9	mhpdeg 20336	. . 3 ⊢ (𝜑 → (𝑋 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁})
66	63, 65	eqsstrd 4005	. 2 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁})
67	6, 3, 7, 16, 64, 1, 2, 8	ismhp 20334	. 2 ⊢ (𝜑 → ((𝑀‘𝑋) ∈ (𝐻‘𝑁) ↔ ((𝑀‘𝑋) ∈ (Base‘𝑃) ∧ ((𝑀‘𝑋) supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁})))
68	13, 66, 67	mpbir2and 711	1 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐻‘𝑁))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  {crab 3142  Vcvv 3494   ⊆ wss 3936   class class class wbr 5066  ^◡ccnv 5554   “ cima 5558   ∘ ccom 5559   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   supp csupp 7830   ↑_m cmap 8406  Fincfn 8509   finSupp cfsupp 8833  ℕcn 11638  ℕ₀cn0 11898  Σcsu 15042  Basecbs 16483  +_gcplusg 16565  0_gc0g 16713  Grpcgrp 18103  inv_gcminusg 18104  SubGrpcsubg 18273   mPwSer cmps 20131   mPoly cmpl 20133   mHomP cmhp 20322

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-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-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-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  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-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  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-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-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-subg 18276  df-psr 20136  df-mpl 20138  df-mhp 20326

This theorem is referenced by:  mhpsubg  20340