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Mirrors > Home > MPE Home > Th. List > ismhp | Structured version Visualization version GIF version |
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 22159 | . . . . 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 7473 | . . . 4 ⊢ (𝑋 ∈ (𝐻‘𝑁) → 𝐼 ∈ V) |
5 | 1, 2, 3 | elfvov2 7474 | . . . 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 22026 | . . . . 5 ⊢ Rel dom mPoly | |
9 | ismhp.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
10 | ismhp.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
11 | 8, 9, 10 | elbasov 17252 | . . . 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 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐼 ∈ V) | |
17 | simprr 773 | . . . . 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 22161 | . . . 4 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}}) |
21 | 20 | eleq2d 2825 | . . 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 7438 | . . . . 5 ⊢ (𝑓 = 𝑋 → (𝑓 supp 0 ) = (𝑋 supp 0 )) | |
23 | 22 | sseq1d 4027 | . . . 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 3695 | . . 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 802 | 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 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ⊆ wss 3963 ◡ccnv 5688 “ cima 5692 ‘cfv 6563 (class class class)co 7431 supp csupp 8184 ↑m cmap 8865 Fincfn 8984 ℕcn 12264 ℕ0cn0 12524 Basecbs 17245 ↾s cress 17274 0gc0g 17486 Σg cgsu 17487 ℂfldccnfld 21382 mPoly cmpl 21944 mHomP cmhp 22151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-n0 12525 df-slot 17216 df-ndx 17228 df-base 17246 df-mpl 21949 df-mhp 22158 |
This theorem is referenced by: ismhp2 22163 ismhp3 22164 mhpmpl 22166 mhpdeg 22167 |
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