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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 7472 and elfvov2 7473 with reldmmhp 22158 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 2734 | . . . . 5 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
4 | eqid 2734 | . . . . 5 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
5 | eqid 2734 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | eqid 2734 | . . . . 5 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
7 | reldmmhp 22158 | . . . . . 6 ⊢ Rel dom mHomP | |
8 | 7, 2, 1 | elfvov1 7472 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
9 | 7, 2, 1 | elfvov2 7473 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) |
10 | 2, 3, 4, 5, 6, 8, 9 | mhpfval 22159 | . . . 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 2840 | . 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 2734 | . . 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 7024 | . 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 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 ⊆ wss 3962 ↦ cmpt 5230 ◡ccnv 5687 “ cima 5691 ‘cfv 6562 (class class class)co 7430 supp csupp 8183 ↑m cmap 8864 Fincfn 8983 ℕcn 12263 ℕ0cn0 12523 Basecbs 17244 ↾s cress 17273 0gc0g 17485 Σg cgsu 17486 ℂfldccnfld 21381 mPoly cmpl 21943 mHomP cmhp 22150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-1cn 11210 ax-addcl 11212 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-nn 12264 df-n0 12524 df-mhp 22157 |
This theorem is referenced by: mhpmpl 22165 mhpdeg 22166 mhpmulcl 22170 mhppwdeg 22171 mhpaddcl 22172 mhpinvcl 22173 mhpvscacl 22175 mhpind 42580 mhphf 42583 |
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