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| Description: Property of being a homogeneous polynomial. (Contributed by Steven Nguyen, 25-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| ismhp.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) | 
| ismhp.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) | 
| ismhp.b | ⊢ 𝐵 = (Base‘𝑃) | 
| ismhp.0 | ⊢ 0 = (0g‘𝑅) | 
| ismhp.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | 
| ismhp.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) | 
| Ref | Expression | 
|---|---|
| ismhp | ⊢ (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | reldmmhp 22142 | . . . . 5 ⊢ Rel dom mHomP | |
| 2 | ismhp.h | . . . . 5 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
| 3 | id 22 | . . . . 5 ⊢ (𝑋 ∈ (𝐻‘𝑁) → 𝑋 ∈ (𝐻‘𝑁)) | |
| 4 | 1, 2, 3 | elfvov1 7474 | . . . 4 ⊢ (𝑋 ∈ (𝐻‘𝑁) → 𝐼 ∈ V) | 
| 5 | 1, 2, 3 | elfvov2 7475 | . . . 4 ⊢ (𝑋 ∈ (𝐻‘𝑁) → 𝑅 ∈ V) | 
| 6 | 4, 5 | jca 511 | . . 3 ⊢ (𝑋 ∈ (𝐻‘𝑁) → (𝐼 ∈ V ∧ 𝑅 ∈ V)) | 
| 7 | 6 | anim2i 617 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐻‘𝑁)) → (𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V))) | 
| 8 | reldmmpl 22009 | . . . . 5 ⊢ Rel dom mPoly | |
| 9 | ismhp.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
| 10 | ismhp.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
| 11 | 8, 9, 10 | elbasov 17255 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) | 
| 12 | 11 | adantr 480 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (𝐼 ∈ V ∧ 𝑅 ∈ V)) | 
| 13 | 12 | anim2i 617 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁})) → (𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V))) | 
| 14 | ismhp.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
| 15 | ismhp.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
| 16 | simprl 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐼 ∈ V) | |
| 17 | simprr 772 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑅 ∈ V) | |
| 18 | ismhp.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
| 19 | 18 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑁 ∈ ℕ0) | 
| 20 | 2, 9, 10, 14, 15, 16, 17, 19 | mhpval 22144 | . . . 4 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}}) | 
| 21 | 20 | eleq2d 2826 | . . 3 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝑋 ∈ (𝐻‘𝑁) ↔ 𝑋 ∈ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}})) | 
| 22 | oveq1 7439 | . . . . 5 ⊢ (𝑓 = 𝑋 → (𝑓 supp 0 ) = (𝑋 supp 0 )) | |
| 23 | 22 | sseq1d 4014 | . . . 4 ⊢ (𝑓 = 𝑋 → ((𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ↔ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁})) | 
| 24 | 23 | elrab 3691 | . . 3 ⊢ (𝑋 ∈ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}} ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁})) | 
| 25 | 21, 24 | bitrdi 287 | . 2 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝑋 ∈ (𝐻‘𝑁) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))) | 
| 26 | 7, 13, 25 | pm5.21nd 801 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 Vcvv 3479 ⊆ 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: ismhp2 22146 ismhp3 22147 mhpmpl 22149 mhpdeg 22150 | 
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