evlvvval - Mathbox for Steven Nguyen

	Mathbox for Steven Nguyen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > evlvvval		Structured version Visualization version GIF version

Theorem evlvvval 42672

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 20170	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		eqid 2731	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
10	9	subrgid 20494	. . . . 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 17161	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s (Base‘𝑅)) = 𝑅)
14	7, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ↾_s (Base‘𝑅)) = 𝑅)
15	14	oveq2d 7368	. . . . . . . 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 6832	. . . . . 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 42670	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑅)‘(Base‘𝑅))‘𝐹) = (𝑄‘𝐹))
23	22	fveq1d 6830	. 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 42662	. 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 5174 ^◡ccnv 5618 “ cima 5622 ‘cfv 6487 (class class class)co 7352 ↑_m cmap 8756 Fincfn 8875 ℕcn 12131 ℕ₀cn0 12387 Basecbs 17126 ↾_s cress 17147 ._rcmulr 17168 Σ_g cgsu 17350 ._gcmg 18986 mulGrpcmgp 20064 Ringcrg 20157 CRingccrg 20158 SubRingcsubrg 20490 mPoly cmpl 21849 evalSub ces 22013 eval cevl 22014

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 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089

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 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-ofr 7617 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9252 df-sup 9332 df-oi 9402 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-fz 13414 df-fzo 13561 df-seq 13915 df-hash 14244 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-hom 17191 df-cco 17192 df-0g 17351 df-gsum 17352 df-prds 17357 df-pws 17359 df-mre 17494 df-mrc 17495 df-acs 17497 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-mhm 18697 df-submnd 18698 df-grp 18855 df-minusg 18856 df-sbg 18857 df-mulg 18987 df-subg 19042 df-ghm 19131 df-cntz 19235 df-cmn 19700 df-abl 19701 df-mgp 20065 df-rng 20077 df-ur 20106 df-srg 20111 df-ring 20159 df-cring 20160 df-rhm 20396 df-subrng 20467 df-subrg 20491 df-lmod 20801 df-lss 20871 df-lsp 20911 df-assa 21796 df-asp 21797 df-ascl 21798 df-psr 21852 df-mvr 21853 df-mpl 21854 df-evls 22015 df-evl 22016

This theorem is referenced by: selvvvval 42684 evlselv 42686

W3C validator