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Theorem evlvvvallem 43037

Description: Lemma for theorems using evlvvval 22109. Version of evlsvvvallem2 22068 using df-evl 22051. (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 2739	. 2 ⊢ (𝐼 mPoly (𝑅 ↾_s 𝐾)) = (𝐼 mPoly (𝑅 ↾_s 𝐾))
3		eqid 2739	. 2 ⊢ (𝑅 ↾_s 𝐾) = (𝑅 ↾_s 𝐾)
4		eqid 2739	. 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 20218	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
12	5	subrgid 20545	. . 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 17205	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s 𝐾) = 𝑅)
16	10, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑅 ↾_s 𝐾) = 𝑅)
17	16	oveq2d 7372	. . . . . 6 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s 𝐾)) = (𝐼 mPoly 𝑅))
18		evlvvvallem.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
19	17, 18	eqtr4di 2792	. . . . 5 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾_s 𝐾)) = 𝑃)
20	19	fveq2d 6831	. . . 4 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s 𝐾))) = (Base‘𝑃))
21		evlvvvallem.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
22	20, 21	eqtr4di 2792	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾_s 𝐾))) = 𝐵)
23	14, 22	eleqtrrd 2842	. 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 22068	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 1547 ∈ wcel 2119 {crab 3391 class class class wbr 5072 ↦ cmpt 5153 ^◡ccnv 5617 “ cima 5621 ‘cfv 6485 (class class class)co 7356 ↑_m cmap 8763 Fincfn 8883 finSupp cfsupp 9264 ℕcn 12165 ℕ₀cn0 12428 Basecbs 17170 ↾_s cress 17191 ._rcmulr 17212 0_gc0g 17393 Σ_g cgsu 17394 ._gcmg 19034 mulGrpcmgp 20112 Ringcrg 20205 CRingccrg 20206 SubRingcsubrg 20541 mPoly cmpl 21881

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-tset 17230 df-0g 17395 df-gsum 17396 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904 df-mulg 19035 df-subg 19090 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-subrg 20542 df-psr 21884 df-mpl 21886

This theorem is referenced by: evlselv 43039

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