Theorem mhprcl 22063

Description: Reverse closure for homogeneous polynomials, use elfvov1 7411 and elfvov2 7412 with reldmmhp 22057 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 2729	. . . . 5 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
4		eqid 2729	. . . . 5 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅))
5		eqid 2729	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
6		eqid 2729	. . . . 5 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
7		reldmmhp 22057	. . . . . 6 ⊢ Rel dom mHomP
8	7, 2, 1	elfvov1 7411	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
9	7, 2, 1	elfvov2 7412	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ V)
10	2, 3, 4, 5, 6, 8, 9	mhpfval 22058	. . . 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 6842	. . 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 2830	. 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 2729	. . 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 6959	. 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 1540   ∈ wcel 2109  {crab 3402  Vcvv 3444   ⊆ wss 3911   ↦ cmpt 5183  ^◡ccnv 5630   “ cima 5634  ‘cfv 6499  (class class class)co 7369   supp csupp 8116   ↑_m cmap 8776  Fincfn 8895  ℕcn 12162  ℕ₀cn0 12418  Basecbs 17155   ↾_s cress 17176  0_gc0g 17378   Σ_g cgsu 17379  ℂ_fldccnfld 21296   mPoly cmpl 21848   mHomP cmhp 22049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-1cn 11102  ax-addcl 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-nn 12163  df-n0 12419  df-mhp 22056

This theorem is referenced by:  mhpmpl  22064  mhpdeg  22065  mhpmulcl  22069  mhppwdeg  22070  mhpaddcl  22071  mhpinvcl  22072  mhpvscacl  22074  mhpind  42575  mhphf  42578