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Theorem evlvvval 22497

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 2763	. . . 4 ⊢ ((𝐼 evalSub 𝑅)‘(Base‘𝑅)) = ((𝐼 evalSub 𝑅)‘(Base‘𝑅))
2		evlvvval.q	. . . 4 ⊢ 𝑄 = (𝐼 eval 𝑅)
3		eqid 2763	. . . 4 ⊢ (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))
4		eqid 2763	. . . 4 ⊢ (𝑅 ↾_s (Base‘𝑅)) = (𝑅 ↾_s (Base‘𝑅))
5		eqid 2763	. . . 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 20524	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		eqid 2763	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
10	9	subrgid 20876	. . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅))
11	8, 10	syl 18	. . . 4 ⊢ (𝜑 → (Base‘𝑅) ∈ (SubRing‘𝑅))
12		evlvvval.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
13	9	ressid 17466	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s (Base‘𝑅)) = 𝑅)
14	7, 13	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ↾_s (Base‘𝑅)) = 𝑅)
15	14	oveq2d 7467	. . . . . . . 8 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = (𝐼 mPoly 𝑅))
16		evlvvval.p	. . . . . . . 8 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
17	15, 16	eqtr4di 2816	. . . . . . 7 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = 𝑃)
18	17	fveq2d 6920	. . . . . 6 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))) = (Base‘𝑃))
19		evlvvval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
20	18, 19	eqtr4di 2816	. . . . 5 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))) = 𝐵)
21	12, 20	eleqtrrd 2867	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))))
22	1, 2, 3, 4, 5, 6, 7, 11, 21	evlsevl 22496	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑅)‘(Base‘𝑅))‘𝐹) = (𝑄‘𝐹))
23	22	fveq1d 6918	. 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 22457	. 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 2800	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

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 {crab 3416 ↦ cmpt 5192 ^◡ccnv 5657 “ cima 5661 ‘cfv 6566 (class class class)co 7451 ↑_m cmap 8885 Fincfn 9011 ℕcn 12375 ℕ₀cn0 12646 Basecbs 17431 ↾_s cress 17452 ._rcmulr 17473 Σ_g cgsu 17661 ._gcmg 19327 mulGrpcmgp 20411 Ringcrg 20510 CRingccrg 20511 SubRingcsubrg 20872 mPoly cmpl 22269 evalSub ces 22436 eval cevl 22437

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-ofr 7726 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-sup 9471 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-fz 13680 df-fzo 13829 df-seq 14185 df-hash 14516 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-hom 17497 df-cco 17498 df-0g 17662 df-gsum 17663 df-prds 17668 df-pws 17670 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-mhm 19028 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-mulg 19328 df-subg 19383 df-ghm 19478 df-cntz 19582 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-srg 20464 df-ring 20512 df-cring 20513 df-rhm 20753 df-subrng 20849 df-subrg 20873 df-lmod 21188 df-lss 21258 df-lsp 21298 df-assa 22216 df-asp 22217 df-ascl 22218 df-psr 22272 df-mvr 22273 df-mpl 22274 df-evls 22438 df-evl 22439

This theorem is used by: selvvvval 22506 evlselv 43883

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