evlvvval - Mathbox for Steven Nguyen

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Theorem evlvvval 42612

Description: Give a formula for the evaluation of a polynomial given assignments from variables to values. (Contributed by SN, 5-Mar-2025.)

**Hypotheses**
Ref	Expression
evlvvval.q	⊢ 𝑄 = (𝐼 eval 𝑅)
evlvvval.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
evlvvval.b	⊢ 𝐵 = (Base‘𝑃)
evlvvval.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
evlvvval.k	⊢ 𝐾 = (Base‘𝑅)
evlvvval.m	⊢ 𝑀 = (mulGrp‘𝑅)
evlvvval.w	⊢ ↑ = (._g‘𝑀)
evlvvval.x	⊢ · = (._r‘𝑅)
evlvvval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlvvval.r	⊢ (𝜑 → 𝑅 ∈ CRing)
evlvvval.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
evlvvval.a	⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))

**Assertion**
Ref	Expression
evlvvval	⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑅 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))

Distinct variable groups: 𝜑,𝑏,𝑖 𝑃,𝑏 𝐵,𝑏 𝐷,𝑏,𝑖 𝐾,𝑏,𝑖 𝐹,𝑏 𝑖,𝐼,𝑏 ℎ,𝐼,𝑏 𝑅,𝑏,ℎ 𝑅,𝑖 𝐴,𝑏,𝑖 Allowed substitution hints: 𝜑(ℎ) 𝐴(ℎ) 𝐵(ℎ,𝑖) 𝐷(ℎ) 𝑃(ℎ,𝑖) 𝑄(ℎ,𝑖,𝑏) · (ℎ,𝑖,𝑏) ↑ (ℎ,𝑖,𝑏) 𝐹(ℎ,𝑖) 𝐾(ℎ) 𝑀(ℎ,𝑖,𝑏) 𝑉(ℎ,𝑖,𝑏)

**Proof of Theorem evlvvval**
Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ ((𝐼 evalSub 𝑅)‘(Base‘𝑅)) = ((𝐼 evalSub 𝑅)‘(Base‘𝑅))
2		evlvvval.q	. . . 4 ⊢ 𝑄 = (𝐼 eval 𝑅)
3		eqid 2731	. . . 4 ⊢ (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))
4		eqid 2731	. . . 4 ⊢ (𝑅 ↾_s (Base‘𝑅)) = (𝑅 ↾_s (Base‘𝑅))
5		eqid 2731	. . . 4 ⊢ (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))) = (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))))
6		evlvvval.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
7		evlvvval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing)
8	7	crngringd 20165	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		eqid 2731	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
10	9	subrgid 20489	. . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
11	8, 10	syl 17	. . . 4 ⊢ (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
12		evlvvval.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
13	9	ressid 17155	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s (Base‘𝑅)) = 𝑅)
14	7, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ↾_s (Base‘𝑅)) = 𝑅)
15	14	oveq2d 7362	. . . . . . . 8 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = (𝐼 mPoly 𝑅))
16		evlvvval.p	. . . . . . . 8 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
17	15, 16	eqtr4di 2784	. . . . . . 7 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = 𝑃)
18	17	fveq2d 6826	. . . . . 6 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))) = (Base‘𝑃))
19		evlvvval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
20	18, 19	eqtr4di 2784	. . . . 5 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))) = 𝐵)
21	12, 20	eleqtrrd 2834	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))))
22	1, 2, 3, 4, 5, 6, 7, 11, 21	evlsevl 42610	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑅)‘(Base‘𝑅))‘𝐹) = (𝑄‘𝐹))
23	22	fveq1d 6824	. 2 ⊢ (𝜑 → ((((𝐼 evalSub 𝑅)‘(Base‘𝑅))‘𝐹)‘𝐴) = ((𝑄‘𝐹)‘𝐴))
24		evlvvval.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
25		evlvvval.k	. . 3 ⊢ 𝐾 = (Base‘𝑅)
26		evlvvval.m	. . 3 ⊢ 𝑀 = (mulGrp‘𝑅)
27		evlvvval.w	. . 3 ⊢ ↑ = (._g‘𝑀)
28		evlvvval.x	. . 3 ⊢ · = (._r‘𝑅)
29		evlvvval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
30	1, 3, 5, 4, 24, 25, 26, 27, 28, 6, 7, 11, 21, 29	evlsvvval 42602	. 2 ⊢ (𝜑 → ((((𝐼 evalSub 𝑅)‘(Base‘𝑅))‘𝐹)‘𝐴) = (𝑅 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
31	23, 30	eqtr3d 2768	1 ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑅 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 {crab 3395 ↦ cmpt 5172 ^◡ccnv 5615 “ cima 5619 ‘cfv 6481 (class class class)co 7346 ↑_m cmap 8750 Fincfn 8869 ℕcn 12125 ℕ₀cn0 12381 Basecbs 17120 ↾_s cress 17141 ._rcmulr 17162 Σ_g cgsu 17344 ._gcmg 18980 mulGrpcmgp 20059 Ringcrg 20152 CRingccrg 20153 SubRingcsubrg 20485 mPoly cmpl 21844 evalSub ces 22008 eval cevl 22009

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-fzo 13555 df-seq 13909 df-hash 14238 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19126 df-cntz 19230 df-cmn 19695 df-abl 19696 df-mgp 20060 df-rng 20072 df-ur 20101 df-srg 20106 df-ring 20154 df-cring 20155 df-rhm 20391 df-subrng 20462 df-subrg 20486 df-lmod 20796 df-lss 20866 df-lsp 20906 df-assa 21791 df-asp 21792 df-ascl 21793 df-psr 21847 df-mvr 21848 df-mpl 21849 df-evls 22010 df-evl 22011

This theorem is referenced by: selvvvval 42624 evlselv 42626

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