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Mirrors > Home > MPE Home > Th. List > Mathboxes > evlsmhpvvval | Structured version Visualization version GIF version |
Description: Give a formula for the evaluation of a homogeneous polynomial given assignments from variables to values. The difference between this and evlsvvval 41598 is that 𝑏 ∈ 𝐷 is restricted to 𝑏 ∈ 𝐺, that is, we can evaluate an 𝑁-th degree homogeneous polynomial over just the terms where the sum of all variable degrees is 𝑁. (Contributed by SN, 5-Mar-2025.) |
Ref | Expression |
---|---|
evlsmhpvvval.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsmhpvvval.p | ⊢ 𝐻 = (𝐼 mHomP 𝑈) |
evlsmhpvvval.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsmhpvvval.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
evlsmhpvvval.g | ⊢ 𝐺 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} |
evlsmhpvvval.k | ⊢ 𝐾 = (Base‘𝑆) |
evlsmhpvvval.m | ⊢ 𝑀 = (mulGrp‘𝑆) |
evlsmhpvvval.w | ⊢ ↑ = (.g‘𝑀) |
evlsmhpvvval.x | ⊢ · = (.r‘𝑆) |
evlsmhpvvval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlsmhpvvval.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsmhpvvval.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsmhpvvval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
evlsmhpvvval.f | ⊢ (𝜑 → 𝐹 ∈ (𝐻‘𝑁)) |
evlsmhpvvval.a | ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
Ref | Expression |
---|---|
evlsmhpvvval | ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σg (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsmhpvvval.q | . . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
2 | eqid 2731 | . . 3 ⊢ (𝐼 mPoly 𝑈) = (𝐼 mPoly 𝑈) | |
3 | eqid 2731 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑈)) = (Base‘(𝐼 mPoly 𝑈)) | |
4 | evlsmhpvvval.u | . . 3 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
5 | evlsmhpvvval.d | . . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
6 | evlsmhpvvval.k | . . 3 ⊢ 𝐾 = (Base‘𝑆) | |
7 | evlsmhpvvval.m | . . 3 ⊢ 𝑀 = (mulGrp‘𝑆) | |
8 | evlsmhpvvval.w | . . 3 ⊢ ↑ = (.g‘𝑀) | |
9 | evlsmhpvvval.x | . . 3 ⊢ · = (.r‘𝑆) | |
10 | evlsmhpvvval.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
11 | evlsmhpvvval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
12 | evlsmhpvvval.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
13 | evlsmhpvvval.p | . . . 4 ⊢ 𝐻 = (𝐼 mHomP 𝑈) | |
14 | 4 | ovexi 7446 | . . . . 5 ⊢ 𝑈 ∈ V |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ V) |
16 | evlsmhpvvval.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
17 | evlsmhpvvval.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐻‘𝑁)) | |
18 | 13, 2, 3, 10, 15, 16, 17 | mhpmpl 21996 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPoly 𝑈))) |
19 | evlsmhpvvval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18, 19 | evlsvvval 41598 | . 2 ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))) |
21 | eqid 2731 | . . 3 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
22 | 11 | crngringd 20147 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring) |
23 | 22 | ringcmnd 20179 | . . 3 ⊢ (𝜑 → 𝑆 ∈ CMnd) |
24 | ovex 7445 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
25 | 5, 24 | rabex2 5334 | . . . 4 ⊢ 𝐷 ∈ V |
26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
27 | 22 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ Ring) |
28 | eqid 2731 | . . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
29 | 2, 28, 3, 5, 18 | mplelf 21868 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑈)) |
30 | 4 | subrgbas 20479 | . . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 = (Base‘𝑈)) |
31 | 6 | subrgss 20470 | . . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾) |
32 | 30, 31 | eqsstrrd 4021 | . . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (Base‘𝑈) ⊆ 𝐾) |
33 | 12, 32 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑈) ⊆ 𝐾) |
34 | 29, 33 | fssd 6735 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐷⟶𝐾) |
35 | 34 | ffvelcdmda 7086 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘𝑏) ∈ 𝐾) |
36 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ 𝑉) |
37 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ CRing) |
38 | 19 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
39 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷) | |
40 | 5, 6, 7, 8, 36, 37, 38, 39 | evlsvvvallem 41596 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))) ∈ 𝐾) |
41 | 6, 9, 27, 35, 40 | ringcld 20158 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) ∈ 𝐾) |
42 | 41 | fmpttd 7116 | . . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))):𝐷⟶𝐾) |
43 | 4, 21 | subrg0 20477 | . . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (0g‘𝑆) = (0g‘𝑈)) |
44 | 12, 43 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑈)) |
45 | 44 | oveq2d 7428 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp (0g‘𝑆)) = (𝐹 supp (0g‘𝑈))) |
46 | eqid 2731 | . . . . . . . . . 10 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
47 | 13, 46, 5, 10, 15, 16, 17 | mhpdeg 21997 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp (0g‘𝑈)) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
48 | evlsmhpvvval.g | . . . . . . . . 9 ⊢ 𝐺 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} | |
49 | 47, 48 | sseqtrrdi 4033 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp (0g‘𝑈)) ⊆ 𝐺) |
50 | 45, 49 | eqsstrd 4020 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp (0g‘𝑆)) ⊆ 𝐺) |
51 | fvexd 6906 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
52 | 34, 50, 26, 51 | suppssr 8186 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → (𝐹‘𝑏) = (0g‘𝑆)) |
53 | 52 | oveq1d 7427 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) = ((0g‘𝑆) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) |
54 | 22 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → 𝑆 ∈ Ring) |
55 | eldifi 4126 | . . . . . . 7 ⊢ (𝑏 ∈ (𝐷 ∖ 𝐺) → 𝑏 ∈ 𝐷) | |
56 | 55, 40 | sylan2 592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))) ∈ 𝐾) |
57 | 6, 9, 21, 54, 56 | ringlzd 20190 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → ((0g‘𝑆) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) = (0g‘𝑆)) |
58 | 53, 57 | eqtrd 2771 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) = (0g‘𝑆)) |
59 | 58, 26 | suppss2 8191 | . . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) supp (0g‘𝑆)) ⊆ 𝐺) |
60 | 5, 2, 4, 3, 6, 7, 8, 9, 10, 11, 12, 18, 19 | evlsvvvallem2 41597 | . . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) finSupp (0g‘𝑆)) |
61 | 6, 21, 23, 26, 42, 59, 60 | gsumres 19829 | . 2 ⊢ (𝜑 → (𝑆 Σg ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) ↾ 𝐺)) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))) |
62 | 48 | ssrab3 4080 | . . . . 5 ⊢ 𝐺 ⊆ 𝐷 |
63 | 62 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐺 ⊆ 𝐷) |
64 | 63 | resmptd 6040 | . . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) ↾ 𝐺) = (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))) |
65 | 64 | oveq2d 7428 | . 2 ⊢ (𝜑 → (𝑆 Σg ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) ↾ 𝐺)) = (𝑆 Σg (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))) |
66 | 20, 61, 65 | 3eqtr2d 2777 | 1 ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σg (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 ∖ cdif 3945 ⊆ wss 3948 ↦ cmpt 5231 ◡ccnv 5675 ↾ cres 5678 “ cima 5679 ‘cfv 6543 (class class class)co 7412 supp csupp 8151 ↑m cmap 8826 Fincfn 8945 ℕcn 12219 ℕ0cn0 12479 Basecbs 17151 ↾s cress 17180 .rcmulr 17205 0gc0g 17392 Σg cgsu 17393 .gcmg 18993 mulGrpcmgp 20035 Ringcrg 20134 CRingccrg 20135 SubRingcsubrg 20465 ℂfldccnfld 21233 mPoly cmpl 21769 evalSub ces 21944 mHomP cmhp 21983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-srg 20088 df-ring 20136 df-cring 20137 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-assa 21718 df-asp 21719 df-ascl 21720 df-psr 21772 df-mvr 21773 df-mpl 21774 df-evls 21946 df-mhp 21990 |
This theorem is referenced by: mhphf 41632 |
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