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| Mirrors > Home > MPE Home > Th. List > mhprcl | Structured version Visualization version GIF version | ||
| Description: Reverse closure for homogeneous polynomials, use elfvov1 7473 and elfvov2 7474 with reldmmhp 22141 for the reverse closure of 𝐼 and 𝑅. (Contributed by SN, 4-Aug-2025.) |
| Ref | Expression |
|---|---|
| mhprcl.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
| mhprcl.x | ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) |
| Ref | Expression |
|---|---|
| mhprcl | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mhprcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) | |
| 2 | mhprcl.h | . . . . 5 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
| 4 | eqid 2737 | . . . . 5 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
| 5 | eqid 2737 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 6 | eqid 2737 | . . . . 5 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
| 7 | reldmmhp 22141 | . . . . . 6 ⊢ Rel dom mHomP | |
| 8 | 7, 2, 1 | elfvov1 7473 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
| 9 | 7, 2, 1 | elfvov2 7474 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) |
| 10 | 2, 3, 4, 5, 6, 8, 9 | mhpfval 22142 | . . . 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 6908 | . . 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 2843 | . 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 2737 | . . 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 7025 | . 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) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 ↑m cmap 8866 Fincfn 8985 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 ↾s cress 17274 0gc0g 17484 Σg cgsu 17485 ℂfldccnfld 21364 mPoly cmpl 21926 mHomP cmhp 22133 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-n0 12527 df-mhp 22140 |
| This theorem is referenced by: mhpmpl 22148 mhpdeg 22149 mhpmulcl 22153 mhppwdeg 22154 mhpaddcl 22155 mhpinvcl 22156 mhpvscacl 22158 mhpind 42604 mhphf 42607 |
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