Theorem mhprcl 22058

Description: Reverse closure for homogeneous polynomials, use elfvov1 7388 and elfvov2 7389 with reldmmhp 22052 for the reverse closure of 𝐼 and 𝑅. (Contributed by SN, 4-Aug-2025.)

Hypotheses

Ref	Expression
mhprcl.h	⊢ 𝐻 = (𝐼 mHomP 𝑅)
mhprcl.x	⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))

Assertion

Ref	Expression
mhprcl	⊢ (𝜑 → 𝑁 ∈ ℕ0)

Proof of Theorem mhprcl

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

Step	Hyp	Ref	Expression
1		mhprcl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))
2		mhprcl.h	. . . . 5 ⊢ 𝐻 = (𝐼 mHomP 𝑅)
3		eqid 2731	. . . . 5 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
4		eqid 2731	. . . . 5 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅))
5		eqid 2731	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
6		eqid 2731	. . . . 5 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
7		reldmmhp 22052	. . . . . 6 ⊢ Rel dom mHomP
8	7, 2, 1	elfvov1 7388	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
9	7, 2, 1	elfvov2 7389	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ V)
10	2, 3, 4, 5, 6, 8, 9	mhpfval 22053	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ∣ (𝑓 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
11	10	fveq1d 6824	. . 3 ⊢ (𝜑 → (𝐻‘𝑁) = ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ∣ (𝑓 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})‘𝑁))
12	1, 11	eleqtrd 2833	. 2 ⊢ (𝜑 → 𝑋 ∈ ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ∣ (𝑓 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})‘𝑁))
13		eqid 2731	. . 3 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ∣ (𝑓 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ∣ (𝑓 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})
14	13	mptrcl 6938	. 2 ⊢ (𝑋 ∈ ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ∣ (𝑓 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})‘𝑁) → 𝑁 ∈ ℕ0)
15	12, 14	syl 17	1 ⊢ (𝜑 → 𝑁 ∈ ℕ0)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ⊆ wss 3897   ↦ cmpt 5170  ^◡ccnv 5613   “ cima 5617  ‘cfv 6481  (class class class)co 7346   supp csupp 8090   ↑_m cmap 8750  Fincfn 8869  ℕcn 12125  ℕ₀cn0 12381  Basecbs 17120   ↾_s cress 17141  0_gc0g 17343   Σ_g cgsu 17344  ℂ_fldccnfld 21291   mPoly cmpl 21843   mHomP cmhp 22044

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-1cn 11064  ax-addcl 11066

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-nn 12126  df-n0 12382  df-mhp 22051

This theorem is referenced by:  mhpmpl  22059  mhpdeg  22060  mhpmulcl  22064  mhppwdeg  22065  mhpaddcl  22066  mhpinvcl  22067  mhpvscacl  22069  mhpind  42686  mhphf  42689