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Mirrors > Home > MPE Home > Th. List > ismhp2 | Structured version Visualization version GIF version |
Description: Deduce a homogeneous polynomial from its properties. (Contributed by SN, 25-May-2024.) |
Ref | Expression |
---|---|
mhpfval.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
mhpfval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mhpfval.b | ⊢ 𝐵 = (Base‘𝑃) |
mhpfval.0 | ⊢ 0 = (0g‘𝑅) |
mhpfval.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
mhpfval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mhpfval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
mhpval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
ismhp2.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ismhp2.2 | ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
Ref | Expression |
---|---|
ismhp2 | ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismhp2.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | ismhp2.2 | . 2 ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) | |
3 | mhpfval.h | . . 3 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
4 | mhpfval.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
5 | mhpfval.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
6 | mhpfval.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
7 | mhpfval.d | . . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
8 | mhpfval.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
9 | mhpfval.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
10 | mhpval.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | ismhp 20869 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))) |
12 | 1, 2, 11 | mpbir2and 713 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 {crab 3072 ⊆ wss 3854 ◡ccnv 5516 “ cima 5520 ‘cfv 6328 (class class class)co 7143 supp csupp 7828 ↑m cmap 8409 Fincfn 8520 ℕcn 11659 ℕ0cn0 11919 Basecbs 16526 ↾s cress 16527 0gc0g 16756 Σg cgsu 16757 ℂfldccnfld 20151 mPoly cmpl 20653 mHomP cmhp 20857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-1cn 10618 ax-addcl 10620 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-nn 11660 df-n0 11920 df-mhp 20861 |
This theorem is referenced by: mhp0cl 20874 mhpaddcl 20879 mhpinvcl 20880 mhpvscacl 20882 mhpind 39773 |
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