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Theorem evlvvval 22350

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 2760	. . . 4 ⊢ ((𝐼 evalSub 𝑅)‘(Base‘𝑅)) = ((𝐼 evalSub 𝑅)‘(Base‘𝑅))
2		evlvvval.q	. . . 4 ⊢ 𝑄 = (𝐼 eval 𝑅)
3		eqid 2760	. . . 4 ⊢ (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))
4		eqid 2760	. . . 4 ⊢ (𝑅 ↾_s (Base‘𝑅)) = (𝑅 ↾_s (Base‘𝑅))
5		eqid 2760	. . . 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 20386	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		eqid 2760	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
10	9	subrgid 20736	. . . . 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 17337	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s (Base‘𝑅)) = 𝑅)
14	7, 13	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ↾_s (Base‘𝑅)) = 𝑅)
15	14	oveq2d 7430	. . . . . . . 8 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = (𝐼 mPoly 𝑅))
16		evlvvval.p	. . . . . . . 8 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
17	15, 16	eqtr4di 2813	. . . . . . 7 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s (Base‘𝑅))) = 𝑃)
18	17	fveq2d 6883	. . . . . 6 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))) = (Base‘𝑃))
19		evlvvval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
20	18, 19	eqtr4di 2813	. . . . 5 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))) = 𝐵)
21	12, 20	eleqtrrd 2863	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPoly (𝑅 ↾_s (Base‘𝑅)))))
22	1, 2, 3, 4, 5, 6, 7, 11, 21	evlsevl 22349	. . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑅)‘(Base‘𝑅))‘𝐹) = (𝑄‘𝐹))
23	22	fveq1d 6881	. 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 22310	. 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 2797	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 2145 {crab 3412 ↦ cmpt 5186 ^◡ccnv 5654 “ cima 5658 ‘cfv 6533 (class class class)co 7414 ↑_m cmap 8827 Fincfn 8953 ℕcn 12258 ℕ₀cn0 12529 Basecbs 17302 ↾_s cress 17323 ._rcmulr 17344 Σ_g cgsu 17526 ._gcmg 19191 mulGrpcmgp 20274 Ringcrg 20373 CRingccrg 20374 SubRingcsubrg 20732 mPoly cmpl 22122 evalSub ces 22289 eval cevl 22290

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 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 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9333 df-sup 9413 df-oi 9483 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-fz 13563 df-fzo 13711 df-seq 14067 df-hash 14396 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-sca 17359 df-vsca 17360 df-ip 17361 df-tset 17362 df-ple 17363 df-ds 17365 df-hom 17367 df-cco 17368 df-0g 17527 df-gsum 17528 df-prds 17533 df-pws 17535 df-mre 17671 df-mrc 17672 df-acs 17674 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-mhm 18892 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-mulg 19192 df-subg 19247 df-ghm 19342 df-cntz 19445 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-srg 20327 df-ring 20375 df-cring 20376 df-rhm 20614 df-subrng 20709 df-subrg 20733 df-lmod 21047 df-lss 21117 df-lsp 21157 df-assa 22069 df-asp 22070 df-ascl 22071 df-psr 22125 df-mvr 22126 df-mpl 22127 df-evls 22291 df-evl 22292

This theorem is used by: selvvvval 22359 evlselv 43436

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