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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 2728 | . . . 4 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
4 | eqid 2728 | . . . 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 22065 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ (𝑋 ∈ (Base‘(𝐼 mPoly 𝑅)) ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))) |
11 | 10 | simplbda 499 | . 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 1534 ∈ wcel 2099 {crab 3429 ⊆ wss 3947 ◡ccnv 5677 “ cima 5681 ‘cfv 6548 (class class class)co 7420 supp csupp 8165 ↑m cmap 8845 Fincfn 8964 ℕcn 12243 ℕ0cn0 12503 Basecbs 17180 ↾s cress 17209 0gc0g 17421 Σg cgsu 17422 ℂfldccnfld 21279 mPoly cmpl 21839 mHomP cmhp 22055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-1cn 11197 ax-addcl 11199 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-nn 12244 df-n0 12504 df-mhp 22062 |
This theorem is referenced by: mhpmulcl 22073 mhpaddcl 22075 mhpinvcl 22076 mhpvscacl 22078 mhpind 41827 evlsmhpvvval 41828 |
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