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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 42578 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.s | ⊢ (𝜑 → 𝑆 ∈ CRing) | 
| evlsmhpvvval.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | 
| 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 2736 | . . 3 ⊢ (𝐼 mPoly 𝑈) = (𝐼 mPoly 𝑈) | |
| 3 | eqid 2736 | . . 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 | reldmmhp 22142 | . . . 4 ⊢ Rel dom mHomP | |
| 11 | evlsmhpvvval.p | . . . 4 ⊢ 𝐻 = (𝐼 mHomP 𝑈) | |
| 12 | evlsmhpvvval.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐻‘𝑁)) | |
| 13 | 10, 11, 12 | elfvov1 7474 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) | 
| 14 | evlsmhpvvval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
| 15 | evlsmhpvvval.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
| 16 | 11, 2, 3, 12 | mhpmpl 22149 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPoly 𝑈))) | 
| 17 | evlsmhpvvval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
| 18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 17 | evlsvvval 42578 | . 2 ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))) | 
| 19 | eqid 2736 | . . 3 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
| 20 | 14 | crngringd 20244 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring) | 
| 21 | 20 | ringcmnd 20282 | . . 3 ⊢ (𝜑 → 𝑆 ∈ CMnd) | 
| 22 | ovex 7465 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
| 23 | 5, 22 | rabex2 5340 | . . . 4 ⊢ 𝐷 ∈ V | 
| 24 | 23 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) | 
| 25 | 20 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ Ring) | 
| 26 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
| 27 | 2, 26, 3, 5, 16 | mplelf 22019 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑈)) | 
| 28 | 4 | subrgbas 20582 | . . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 = (Base‘𝑈)) | 
| 29 | 6 | subrgss 20573 | . . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾) | 
| 30 | 28, 29 | eqsstrrd 4018 | . . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (Base‘𝑈) ⊆ 𝐾) | 
| 31 | 15, 30 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑈) ⊆ 𝐾) | 
| 32 | 27, 31 | fssd 6752 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐷⟶𝐾) | 
| 33 | 32 | ffvelcdmda 7103 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘𝑏) ∈ 𝐾) | 
| 34 | 13 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ V) | 
| 35 | 14 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ CRing) | 
| 36 | 17 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐴 ∈ (𝐾 ↑m 𝐼)) | 
| 37 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷) | |
| 38 | 5, 6, 7, 8, 34, 35, 36, 37 | evlsvvvallem 42576 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))) ∈ 𝐾) | 
| 39 | 6, 9, 25, 33, 38 | ringcld 20258 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) ∈ 𝐾) | 
| 40 | 39 | fmpttd 7134 | . . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))):𝐷⟶𝐾) | 
| 41 | 4, 19 | subrg0 20580 | . . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (0g‘𝑆) = (0g‘𝑈)) | 
| 42 | 15, 41 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑈)) | 
| 43 | 42 | oveq2d 7448 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp (0g‘𝑆)) = (𝐹 supp (0g‘𝑈))) | 
| 44 | eqid 2736 | . . . . . . . . . 10 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
| 45 | 11, 44, 5, 12 | mhpdeg 22150 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp (0g‘𝑈)) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) | 
| 46 | evlsmhpvvval.g | . . . . . . . . 9 ⊢ 𝐺 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} | |
| 47 | 45, 46 | sseqtrrdi 4024 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp (0g‘𝑈)) ⊆ 𝐺) | 
| 48 | 43, 47 | eqsstrd 4017 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp (0g‘𝑆)) ⊆ 𝐺) | 
| 49 | fvexd 6920 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
| 50 | 32, 48, 24, 49 | suppssr 8221 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → (𝐹‘𝑏) = (0g‘𝑆)) | 
| 51 | 50 | oveq1d 7447 | . . . . 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)))))) | 
| 52 | 20 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → 𝑆 ∈ Ring) | 
| 53 | eldifi 4130 | . . . . . . 7 ⊢ (𝑏 ∈ (𝐷 ∖ 𝐺) → 𝑏 ∈ 𝐷) | |
| 54 | 53, 38 | sylan2 593 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))) ∈ 𝐾) | 
| 55 | 6, 9, 19, 52, 54 | ringlzd 20293 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → ((0g‘𝑆) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) = (0g‘𝑆)) | 
| 56 | 51, 55 | eqtrd 2776 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) = (0g‘𝑆)) | 
| 57 | 56, 24 | suppss2 8226 | . . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) supp (0g‘𝑆)) ⊆ 𝐺) | 
| 58 | 5, 2, 4, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17 | evlsvvvallem2 42577 | . . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) finSupp (0g‘𝑆)) | 
| 59 | 6, 19, 21, 24, 40, 57, 58 | gsumres 19932 | . 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)))))))) | 
| 60 | 46 | ssrab3 4081 | . . . . 5 ⊢ 𝐺 ⊆ 𝐷 | 
| 61 | 60 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐺 ⊆ 𝐷) | 
| 62 | 61 | resmptd 6057 | . . 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))))))) | 
| 63 | 62 | oveq2d 7448 | . 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)))))))) | 
| 64 | 18, 59, 63 | 3eqtr2d 2782 | 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 1539 ∈ wcel 2107 {crab 3435 Vcvv 3479 ∖ cdif 3947 ⊆ wss 3950 ↦ cmpt 5224 ◡ccnv 5683 ↾ cres 5686 “ 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 .rcmulr 17299 0gc0g 17485 Σg cgsu 17486 .gcmg 19086 mulGrpcmgp 20138 Ringcrg 20231 CRingccrg 20232 SubRingcsubrg 20570 ℂfldccnfld 21365 mPoly cmpl 21927 evalSub ces 22097 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-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| 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-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 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-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 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-se 5637 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-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-srg 20185 df-ring 20233 df-cring 20234 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-lmod 20861 df-lss 20931 df-lsp 20971 df-assa 21874 df-asp 21875 df-ascl 21876 df-psr 21930 df-mvr 21931 df-mpl 21932 df-evls 22099 df-mhp 22141 | 
| This theorem is referenced by: mhphf 42612 | 
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