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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 20928 | . 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 1542 ∈ wcel 2113 {crab 3057 ⊆ wss 3841 ◡ccnv 5518 “ cima 5522 ‘cfv 6333 (class class class)co 7164 supp csupp 7849 ↑m cmap 8430 Fincfn 8548 ℕcn 11709 ℕ0cn0 11969 Basecbs 16579 ↾s cress 16580 0gc0g 16809 Σg cgsu 16810 ℂfldccnfld 20210 mPoly cmpl 20712 mHomP cmhp 20916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-1cn 10666 ax-addcl 10668 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-nn 11710 df-n0 11970 df-mhp 20920 |
This theorem is referenced by: mhp0cl 20933 mhpaddcl 20938 mhpinvcl 20939 mhpvscacl 20941 mhpind 39846 |
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