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Theorem evlsvvvallem2 22142

Description: Lemma for theorems using evlsvvval 22143. (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 7429	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
3	1, 2	rabex2 5297	. . . 4 ⊢ 𝐷 ∈ V
4	3	mptex 7207	. . 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 6882	. 2 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
7		funmpt 6559	. . 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 2762	. . 3 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
12		evlsvvvallem2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
13	9, 10, 11, 12	mplelsfi 22043	. 2 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑈))
14		eqid 2762	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
15	9, 14, 10, 1, 12	mplelf 22046	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑈))
16		ssidd 3959	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑈)) ⊆ (𝐹 supp (0_g‘𝑈)))
17		fvexd 6882	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑈) ∈ V)
18	15, 16, 12, 17	suppssrg 8176	. . . . . 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 2762	. . . . . . . . . 10 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
22	20, 21	subrg0 20625	. . . . . . . . 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 2768	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑈) = (0_g‘𝑆))
25	24	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (0_g‘𝑈) = (0_g‘𝑆))
26	18, 25	eqtrd 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (𝐹‘𝑏) = (0_g‘𝑆))
27	26	oveq1d 7411	. . . 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 20292	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
32	31	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → 𝑆 ∈ Ring)
33		eldifi 4084	. . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ 𝑉)
38	30	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ CRing)
39		evlsvvvallem2.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
40	39	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐴 ∈ (𝐾 ↑_m 𝐼))
41		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷)
42	1, 28, 34, 35, 37, 38, 40, 41	evlsvvvallem 22141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾)
43	33, 42	sylan2 602	. . . . 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 20341	. . . 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 2797	. . 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 8180	. 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 9328	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 399 = wceq 1560 ∈ wcel 2142 {crab 3414 Vcvv 3454 ∖ cdif 3901 class class class wbr 5100 ↦ cmpt 5181 ^◡ccnv 5646 “ cima 5650 Fun wfun 6515 ‘cfv 6521 (class class class)co 7396 supp csupp 8140 ↑_m cmap 8808 Fincfn 8927 finSupp cfsupp 9307 ℕcn 12210 ℕ₀cn0 12481 Basecbs 17245 ↾_s cress 17266 ._rcmulr 17287 0_gc0g 17468 Σ_g cgsu 17469 ._gcmg 19109 mulGrpcmgp 20186 Ringcrg 20279 CRingccrg 20280 SubRingcsubrg 20615 mPoly cmpl 21955

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-uz 12840 df-fz 13513 df-fzo 13660 df-seq 14015 df-hash 14344 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-sca 17302 df-vsca 17303 df-tset 17305 df-0g 17470 df-gsum 17471 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-grp 18978 df-minusg 18979 df-mulg 19110 df-subg 19165 df-cntz 19357 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20228 df-ring 20281 df-cring 20282 df-subrg 20616 df-psr 21958 df-mpl 21960

This theorem is referenced by: evlsbagval 43165 evlvvvallem 43166 evlsmhpvvval 43174 mhphf 43176

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