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Mirrors > Home > MPE Home > Th. List > mhpdeg | Structured version Visualization version GIF version |
Description: All nonzero terms of a homogeneous polynomial have degree 𝑁. (Contributed by Steven Nguyen, 25-Aug-2023.) |
Ref | Expression |
---|---|
mhpdeg.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
mhpdeg.0 | ⊢ 0 = (0g‘𝑅) |
mhpdeg.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
mhpdeg.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mhpdeg.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
mhpdeg.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
mhpdeg.x | ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) |
Ref | Expression |
---|---|
mhpdeg | ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhpdeg.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) | |
2 | mhpdeg.h | . . . 4 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
3 | eqid 2737 | . . . 4 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
4 | eqid 2737 | . . . 4 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
5 | mhpdeg.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
6 | mhpdeg.d | . . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
7 | mhpdeg.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
8 | mhpdeg.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
9 | mhpdeg.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | ismhp 21547 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ (𝑋 ∈ (Base‘(𝐼 mPoly 𝑅)) ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))) |
11 | 10 | simplbda 501 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐻‘𝑁)) → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
12 | 1, 11 | mpdan 686 | 1 ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3410 ⊆ wss 3915 ◡ccnv 5637 “ cima 5641 ‘cfv 6501 (class class class)co 7362 supp csupp 8097 ↑m cmap 8772 Fincfn 8890 ℕcn 12160 ℕ0cn0 12420 Basecbs 17090 ↾s cress 17119 0gc0g 17328 Σg cgsu 17329 ℂfldccnfld 20812 mPoly cmpl 21324 mHomP cmhp 21535 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-1cn 11116 ax-addcl 11118 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-nn 12161 df-n0 12421 df-mhp 21539 |
This theorem is referenced by: mhpmulcl 21555 mhpaddcl 21557 mhpinvcl 21558 mhpvscacl 21560 mhpind 40798 |
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