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Theorem evlsvvvallem2 22083

Description: Lemma for theorems using evlsvvval 22084. (Contributed by SN, 8-Mar-2025.)

**Hypotheses**
Ref	Expression
evlsvvvallem2.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
evlsvvvallem2.p	⊢ 𝑃 = (𝐼 mPoly 𝑈)
evlsvvvallem2.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
evlsvvvallem2.b	⊢ 𝐵 = (Base‘𝑃)
evlsvvvallem2.k	⊢ 𝐾 = (Base‘𝑆)
evlsvvvallem2.m	⊢ 𝑀 = (mulGrp‘𝑆)
evlsvvvallem2.w	⊢ ↑ = (._g‘𝑀)
evlsvvvallem2.x	⊢ · = (._r‘𝑆)
evlsvvvallem2.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlsvvvallem2.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlsvvvallem2.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evlsvvvallem2.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
evlsvvvallem2.a	⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))

**Assertion**
Ref	Expression
evlsvvvallem2	⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) finSupp (0_g‘𝑆))

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

**Proof of Theorem evlsvvvallem2**
Step	Hyp	Ref	Expression
1		evlsvvvallem2.d	. . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
2		ovex 7394	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
3	1, 2	rabex2 5279	. . . 4 ⊢ 𝐷 ∈ V
4	3	mptex 7172	. . 3 ⊢ (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) ∈ V
5	4	a1i 11	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) ∈ V)
6		fvexd 6850	. 2 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
7		funmpt 6531	. . 3 ⊢ Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))))
8	7	a1i 11	. 2 ⊢ (𝜑 → Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))))
9		evlsvvvallem2.p	. . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑈)
10		evlsvvvallem2.b	. . 3 ⊢ 𝐵 = (Base‘𝑃)
11		eqid 2737	. . 3 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
12		evlsvvvallem2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
13	9, 10, 11, 12	mplelsfi 21986	. 2 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑈))
14		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
15	9, 14, 10, 1, 12	mplelf 21989	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑈))
16		ssidd 3946	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑈)) ⊆ (𝐹 supp (0_g‘𝑈)))
17		fvexd 6850	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑈) ∈ V)
18	15, 16, 12, 17	suppssrg 8140	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (𝐹‘𝑏) = (0_g‘𝑈))
19		evlsvvvallem2.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
20		evlsvvvallem2.u	. . . . . . . . . 10 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
21		eqid 2737	. . . . . . . . . 10 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
22	20, 21	subrg0 20550	. . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (0_g‘𝑆) = (0_g‘𝑈))
23	19, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝑆) = (0_g‘𝑈))
24	23	eqcomd 2743	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑈) = (0_g‘𝑆))
25	24	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (0_g‘𝑈) = (0_g‘𝑆))
26	18, 25	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (𝐹‘𝑏) = (0_g‘𝑆))
27	26	oveq1d 7376	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))) = ((0_g‘𝑆) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))))
28		evlsvvvallem2.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑆)
29		evlsvvvallem2.x	. . . . 5 ⊢ · = (._r‘𝑆)
30		evlsvvvallem2.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ CRing)
31	30	crngringd 20221	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
32	31	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → 𝑆 ∈ Ring)
33		eldifi 4072	. . . . . 6 ⊢ (𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈))) → 𝑏 ∈ 𝐷)
34		evlsvvvallem2.m	. . . . . . 7 ⊢ 𝑀 = (mulGrp‘𝑆)
35		evlsvvvallem2.w	. . . . . . 7 ⊢ ↑ = (._g‘𝑀)
36		evlsvvvallem2.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
37	36	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ 𝑉)
38	30	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ CRing)
39		evlsvvvallem2.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
40	39	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐴 ∈ (𝐾 ↑_m 𝐼))
41		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷)
42	1, 28, 34, 35, 37, 38, 40, 41	evlsvvvallem 22082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾)
43	33, 42	sylan2 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾)
44	28, 29, 21, 32, 43	ringlzd 20270	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → ((0_g‘𝑆) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))) = (0_g‘𝑆))
45	27, 44	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))) = (0_g‘𝑆))
46	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
47	45, 46	suppss2 8144	. 2 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) supp (0_g‘𝑆)) ⊆ (𝐹 supp (0_g‘𝑈)))
48	5, 6, 8, 13, 47	fsuppsssuppgd 9289	1 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) finSupp (0_g‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3390 Vcvv 3430 ∖ cdif 3887 class class class wbr 5086 ↦ cmpt 5167 ^◡ccnv 5624 “ cima 5628 Fun wfun 6487 ‘cfv 6493 (class class class)co 7361 supp csupp 8104 ↑_m cmap 8767 Fincfn 8887 finSupp cfsupp 9268 ℕcn 12168 ℕ₀cn0 12431 Basecbs 17173 ↾_s cress 17194 ._rcmulr 17215 0_gc0g 17396 Σ_g cgsu 17397 ._gcmg 19037 mulGrpcmgp 20115 Ringcrg 20208 CRingccrg 20209 SubRingcsubrg 20540 mPoly cmpl 21899

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 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-fzo 13603 df-seq 13958 df-hash 14287 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-tset 17233 df-0g 17398 df-gsum 17399 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-grp 18906 df-minusg 18907 df-mulg 19038 df-subg 19093 df-cntz 19286 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-cring 20211 df-subrg 20541 df-psr 21902 df-mpl 21904

This theorem is referenced by: evlsbagval 43019 evlvvvallem 43026 evlsmhpvvval 43045 mhphf 43047

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