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| 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.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 2736 | . . . 4 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
| 4 | eqid 2736 | . . . 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 | 2, 1 | mhprcl 22148 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | 
| 8 | 2, 3, 4, 5, 6, 7 | ismhp 22145 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ (𝑋 ∈ (Base‘(𝐼 mPoly 𝑅)) ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))) | 
| 9 | 8 | simplbda 499 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐻‘𝑁)) → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) | 
| 10 | 1, 9 | mpdan 687 | 1 ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 {crab 3435 ⊆ wss 3950 ◡ccnv 5683 “ cima 5687 ‘cfv 6560 (class class class)co 7432 supp csupp 8186 ↑m cmap 8867 Fincfn 8986 ℕcn 12267 ℕ0cn0 12528 Basecbs 17248 ↾s cress 17275 0gc0g 17485 Σg cgsu 17486 ℂfldccnfld 21365 mPoly cmpl 21927 mHomP cmhp 22134 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-1cn 11214 ax-addcl 11216 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-nn 12268 df-n0 12529 df-slot 17220 df-ndx 17232 df-base 17249 df-mpl 21932 df-mhp 22141 | 
| This theorem is referenced by: mhpmulcl 22154 mhpaddcl 22156 mhpinvcl 22157 mhpvscacl 22159 mhpind 42609 evlsmhpvvval 42610 | 
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