evlvvval - Mathbox for Steven Nguyen

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Theorem evlvvval 41600

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 ℎ,𝑏 𝐹,𝑏 𝑖,𝐼,𝑏 ℎ,𝐼 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 2724	. . . 4 ⊢ ((𝐼 evalSub 𝑅)‘(Base‘𝑅)) = ((𝐼 evalSub 𝑅)‘(Base‘𝑅))
2		evlvvval.q	. . . 4 ⊢ 𝑄 = (𝐼 eval 𝑅)
3		eqid 2724	. . . 4 ⊢ (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))
4		eqid 2724	. . . 4 ⊢ (𝑅 ↾_s (Base‘𝑅)) = (𝑅 ↾_s (Base‘𝑅))
5		eqid 2724	. . . 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 20136	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		eqid 2724	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
10	9	subrgid 20460	. . . . 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 17185	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s (Base‘𝑅)) = 𝑅)
14	7, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ↾_s (Base‘𝑅)) = 𝑅)
15	14	oveq2d 7417	. . . . . . . 8 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = (𝐼 mPoly 𝑅))
16		evlvvval.p	. . . . . . . 8 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
17	15, 16	eqtr4di 2782	. . . . . . 7 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = 𝑃)
18	17	fveq2d 6885	. . . . . 6 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))) = (Base‘𝑃))
19		evlvvval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
20	18, 19	eqtr4di 2782	. . . . 5 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))) = 𝐵)
21	12, 20	eleqtrrd 2828	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))))
22	1, 2, 3, 4, 5, 6, 7, 11, 21	evlsevl 41598	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑅)‘(Base‘𝑅))‘𝐹) = (𝑄‘𝐹))
23	22	fveq1d 6883	. 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 41590	. 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 2766	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 1533 ∈ wcel 2098 {crab 3424 ↦ cmpt 5221 ^◡ccnv 5665 “ cima 5669 ‘cfv 6533 (class class class)co 7401 ↑_m cmap 8815 Fincfn 8934 ℕcn 12208 ℕ₀cn0 12468 Basecbs 17140 ↾_s cress 17169 ._rcmulr 17194 Σ_g cgsu 17382 ._gcmg 18982 mulGrpcmgp 20024 Ringcrg 20123 CRingccrg 20124 SubRingcsubrg 20454 mPoly cmpl 21759 evalSub ces 21934 eval cevl 21935

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-sup 9432 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-mhm 18700 df-submnd 18701 df-grp 18853 df-minusg 18854 df-sbg 18855 df-mulg 18983 df-subg 19035 df-ghm 19124 df-cntz 19218 df-cmn 19687 df-abl 19688 df-mgp 20025 df-rng 20043 df-ur 20072 df-srg 20077 df-ring 20125 df-cring 20126 df-rhm 20359 df-subrng 20431 df-subrg 20456 df-lmod 20693 df-lss 20764 df-lsp 20804 df-assa 21708 df-asp 21709 df-ascl 21710 df-psr 21762 df-mvr 21763 df-mpl 21764 df-evls 21936 df-evl 21937

This theorem is referenced by: selvvvval 41612 evlselv 41614

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