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Theorem evlvvvallem 43009

Description: Lemma for theorems using evlvvval 43008. Version of evlsvvvallem2 22070 using df-evl 22053. (Contributed by SN, 11-Mar-2025.)

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

**Assertion**
Ref	Expression
evlvvvallem	⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) finSupp (0_g‘𝑅))

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

**Proof of Theorem evlvvvallem**
Step	Hyp	Ref	Expression
1		evlvvvallem.d	. 2 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
2		eqid 2736	. 2 ⊢ (𝐼 mPoly (𝑅 ↾_s 𝐾)) = (𝐼 mPoly (𝑅 ↾_s 𝐾))
3		eqid 2736	. 2 ⊢ (𝑅 ↾_s 𝐾) = (𝑅 ↾_s 𝐾)
4		eqid 2736	. 2 ⊢ (Base‘(𝐼 mPoly (𝑅 ↾_s 𝐾))) = (Base‘(𝐼 mPoly (𝑅 ↾_s 𝐾)))
5		evlvvvallem.k	. 2 ⊢ 𝐾 = (Base‘𝑅)
6		evlvvvallem.m	. 2 ⊢ 𝑀 = (mulGrp‘𝑅)
7		evlvvvallem.w	. 2 ⊢ ↑ = (._g‘𝑀)
8		evlvvvallem.x	. 2 ⊢ · = (._r‘𝑅)
9		evlvvvallem.i	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
10		evlvvvallem.r	. 2 ⊢ (𝜑 → 𝑅 ∈ CRing)
11	10	crngringd 20227	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
12	5	subrgid 20550	. . 3 ⊢ (𝑅 ∈ Ring → 𝐾 ∈ (SubRing‘𝑅))
13	11, 12	syl 17	. 2 ⊢ (𝜑 → 𝐾 ∈ (SubRing‘𝑅))
14		evlvvvallem.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
15	5	ressid 17214	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s 𝐾) = 𝑅)
16	10, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑅 ↾_s 𝐾) = 𝑅)
17	16	oveq2d 7383	. . . . . 6 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s 𝐾)) = (𝐼 mPoly 𝑅))
18		evlvvvallem.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
19	17, 18	eqtr4di 2789	. . . . 5 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s 𝐾)) = 𝑃)
20	19	fveq2d 6844	. . . 4 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s 𝐾))) = (Base‘𝑃))
21		evlvvvallem.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
22	20, 21	eqtr4di 2789	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s 𝐾))) = 𝐵)
23	14, 22	eleqtrrd 2839	. 2 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPoly (𝑅 ↾_s 𝐾))))
24		evlvvvallem.a	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
25	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 23, 24	evlsvvvallem2 22070	1 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) finSupp (0_g‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {crab 3389 class class class wbr 5085 ↦ cmpt 5166 ^◡ccnv 5630 “ cima 5634 ‘cfv 6498 (class class class)co 7367 ↑_m cmap 8773 Fincfn 8893 finSupp cfsupp 9274 ℕcn 12174 ℕ₀cn0 12437 Basecbs 17179 ↾_s cress 17200 ._rcmulr 17221 0_gc0g 17402 Σ_g cgsu 17403 ._gcmg 19043 mulGrpcmgp 20121 Ringcrg 20214 CRingccrg 20215 SubRingcsubrg 20546 mPoly cmpl 21886

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-seq 13964 df-hash 14293 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-tset 17239 df-0g 17404 df-gsum 17405 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18912 df-minusg 18913 df-mulg 19044 df-subg 19099 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-subrg 20547 df-psr 21889 df-mpl 21891

This theorem is referenced by: evlselv 43020

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