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Mirrors > Home > MPE Home > Th. List > mhpmpl | Structured version Visualization version GIF version |
Description: A homogeneous polynomial is a polynomial. (Contributed by Steven Nguyen, 25-Aug-2023.) |
Ref | Expression |
---|---|
mhpmpl.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
mhpmpl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mhpmpl.b | ⊢ 𝐵 = (Base‘𝑃) |
mhpmpl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mhpmpl.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
mhpmpl.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
mhpmpl.x | ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) |
Ref | Expression |
---|---|
mhpmpl | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhpmpl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) | |
2 | mhpmpl.h | . . . 4 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
3 | mhpmpl.p | . . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
4 | mhpmpl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
5 | eqid 2736 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | eqid 2736 | . . . 4 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
7 | mhpmpl.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
8 | mhpmpl.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
9 | mhpmpl.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | ismhp 21515 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))) |
11 | 10 | simprbda 499 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐻‘𝑁)) → 𝑋 ∈ 𝐵) |
12 | 1, 11 | mpdan 685 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3405 ⊆ wss 3908 ◡ccnv 5630 “ cima 5634 ‘cfv 6493 (class class class)co 7353 supp csupp 8088 ↑m cmap 8761 Fincfn 8879 ℕcn 12149 ℕ0cn0 12409 Basecbs 17075 ↾s cress 17104 0gc0g 17313 Σg cgsu 17314 ℂfldccnfld 20781 mPoly cmpl 21293 mHomP cmhp 21503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-1cn 11105 ax-addcl 11107 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 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 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-nn 12150 df-n0 12410 df-mhp 21507 |
This theorem is referenced by: mhpmulcl 21523 mhppwdeg 21524 mhpaddcl 21525 mhpinvcl 21526 mhpsubg 21527 mhpvscacl 21528 mhpind 40707 mhphf 40709 |
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