evlsmhpvvval - Mathbox for Steven Nguyen

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Theorem evlsmhpvvval 41630

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.)

**Hypotheses**
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 ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 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))

**Assertion**
Ref	Expression
evlsmhpvvval	⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σ_g (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))

Distinct variable groups: 𝐴,𝑏,𝑖 𝐷,𝑏,𝑖 𝐷,𝑔 𝐹,𝑏 𝐺,𝑏 𝐼,𝑏,ℎ 𝑖,𝐼 𝐾,𝑏,𝑖 𝑔,𝑁 𝑅,𝑏 𝑆,𝑏,𝑖 𝑈,𝑏,ℎ 𝑈,𝑖 𝜑,𝑏,𝑖 𝑔,ℎ Allowed substitution hints: 𝜑(𝑔,ℎ) 𝐴(𝑔,ℎ) 𝐷(ℎ) 𝑄(𝑔,ℎ,𝑖,𝑏) 𝑅(𝑔,ℎ,𝑖) 𝑆(𝑔,ℎ) · (𝑔,ℎ,𝑖,𝑏) 𝑈(𝑔) ↑ (𝑔,ℎ,𝑖,𝑏) 𝐹(𝑔,ℎ,𝑖) 𝐺(𝑔,ℎ,𝑖) 𝐻(𝑔,ℎ,𝑖,𝑏) 𝐼(𝑔) 𝐾(𝑔,ℎ) 𝑀(𝑔,ℎ,𝑖,𝑏) 𝑁(ℎ,𝑖,𝑏) 𝑉(𝑔,ℎ,𝑖,𝑏)

**Proof of Theorem evlsmhpvvval**
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 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
21		eqid 2731	. . 3 ⊢ (0_g‘𝑆) = (0_g‘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 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))) ∈ 𝐾)
41	6, 9, 27, 35, 40	ringcld 20158	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) ∈ 𝐾)
42	41	fmpttd 7116	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))):𝐷⟶𝐾)
43	4, 21	subrg0 20477	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (0_g‘𝑆) = (0_g‘𝑈))
44	12, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0_g‘𝑆) = (0_g‘𝑈))
45	44	oveq2d 7428	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑆)) = (𝐹 supp (0_g‘𝑈)))
46		eqid 2731	. . . . . . . . . 10 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
47	13, 46, 5, 10, 15, 16, 17	mhpdeg 21997	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑈)) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁})
48		evlsmhpvvval.g	. . . . . . . . 9 ⊢ 𝐺 = {𝑔 ∈ 𝐷 ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}
49	47, 48	sseqtrrdi 4033	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑈)) ⊆ 𝐺)
50	45, 49	eqsstrd 4020	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑆)) ⊆ 𝐺)
51		fvexd 6906	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
52	34, 50, 26, 51	suppssr 8186	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → (𝐹‘𝑏) = (0_g‘𝑆))
53	52	oveq1d 7427	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) = ((0_g‘𝑆) · (𝑀 Σ_g (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 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))) ∈ 𝐾)
57	6, 9, 21, 54, 56	ringlzd 20190	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → ((0_g‘𝑆) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) = (0_g‘𝑆))
58	53, 57	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ 𝐺)) → ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) = (0_g‘𝑆))
59	58, 26	suppss2 8191	. . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) supp (0_g‘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 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) finSupp (0_g‘𝑆))
61	6, 21, 23, 26, 42, 59, 60	gsumres 19829	. 2 ⊢ (𝜑 → (𝑆 Σ_g ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) ↾ 𝐺)) = (𝑆 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (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 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) ↾ 𝐺) = (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))
65	64	oveq2d 7428	. 2 ⊢ (𝜑 → (𝑆 Σ_g ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) ↾ 𝐺)) = (𝑆 Σ_g (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
66	20, 61, 65	3eqtr2d 2777	1 ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σ_g (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (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 0_gc0g 17392 Σ_g cgsu 17393 ._gcmg 18993 mulGrpcmgp 20035 Ringcrg 20134 CRingccrg 20135 SubRingcsubrg 20465 ℂ_fldccnfld 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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