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Mirrors > Home > MPE Home > Th. List > mhpval | Structured version Visualization version GIF version |
Description: Value of the "homogeneous polynomial" function. (Contributed by Steven Nguyen, 25-Aug-2023.) |
Ref | Expression |
---|---|
mhpfval.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
mhpfval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mhpfval.b | ⊢ 𝐵 = (Base‘𝑃) |
mhpfval.0 | ⊢ 0 = (0g‘𝑅) |
mhpfval.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
mhpfval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mhpfval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
mhpval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
mhpval | ⊢ (𝜑 → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhpfval.h | . . 3 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
2 | mhpfval.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | mhpfval.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
4 | mhpfval.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
5 | mhpfval.d | . . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
6 | mhpfval.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
7 | mhpfval.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | mhpfval 21033 | . 2 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})) |
9 | eqeq2 2748 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛 ↔ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁)) | |
10 | 9 | rabbidv 3380 | . . . . 5 ⊢ (𝑛 = 𝑁 → {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛} = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
11 | 10 | sseq2d 3919 | . . . 4 ⊢ (𝑛 = 𝑁 → ((𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛} ↔ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁})) |
12 | 11 | rabbidv 3380 | . . 3 ⊢ (𝑛 = 𝑁 → {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}} = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}}) |
13 | 12 | adantl 485 | . 2 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}} = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}}) |
14 | mhpval.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
15 | 3 | fvexi 6709 | . . . 4 ⊢ 𝐵 ∈ V |
16 | 15 | rabex 5210 | . . 3 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}} ∈ V |
17 | 16 | a1i 11 | . 2 ⊢ (𝜑 → {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}} ∈ V) |
18 | 8, 13, 14, 17 | fvmptd 6803 | 1 ⊢ (𝜑 → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 {crab 3055 Vcvv 3398 ⊆ wss 3853 ◡ccnv 5535 “ cima 5539 ‘cfv 6358 (class class class)co 7191 supp csupp 7881 ↑m cmap 8486 Fincfn 8604 ℕcn 11795 ℕ0cn0 12055 Basecbs 16666 ↾s cress 16667 0gc0g 16898 Σg cgsu 16899 ℂfldccnfld 20317 mPoly cmpl 20819 mHomP cmhp 21023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-1cn 10752 ax-addcl 10754 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-nn 11796 df-n0 12056 df-mhp 21027 |
This theorem is referenced by: ismhp 21035 |
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