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Theorem evlsvvvallem2 22038

Description: Lemma for theorems using evlsvvval 22039. (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 7388	. . . . 5 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
3	1, 2	rabex2 5283	. . . 4 ⊢ 𝐷 ∈ V
4	3	mptex 7166	. . 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 6846	. 2 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
7		funmpt 6527	. . 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 2733	. . 3 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
12		evlsvvvallem2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
13	9, 10, 11, 12	mplelsfi 21941	. 2 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑈))
14		eqid 2733	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
15	9, 14, 10, 1, 12	mplelf 21944	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑈))
16		ssidd 3954	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑈)) ⊆ (𝐹 supp (0_g‘𝑈)))
17		fvexd 6846	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑈) ∈ V)
18	15, 16, 12, 17	suppssrg 8135	. . . . . 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 2733	. . . . . . . . . 10 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
22	20, 21	subrg0 20503	. . . . . . . . 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 2739	. . . . . . 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 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (𝐹‘𝑏) = (0_g‘𝑆))
27	26	oveq1d 7370	. . . 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 20172	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
32	31	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → 𝑆 ∈ Ring)
33		eldifi 4080	. . . . . 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 22037	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾)
43	33, 42	sylan2 593	. . . . 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 20221	. . . 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 2768	. . 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 8139	. 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 9277	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 1541 ∈ wcel 2113 {crab 3396 Vcvv 3437 ∖ cdif 3895 class class class wbr 5095 ↦ cmpt 5176 ^◡ccnv 5620 “ cima 5624 Fun wfun 6483 ‘cfv 6489 (class class class)co 7355 supp csupp 8099 ↑_m cmap 8759 Fincfn 8879 finSupp cfsupp 9256 ℕcn 12136 ℕ₀cn0 12392 Basecbs 17127 ↾_s cress 17148 ._rcmulr 17169 0_gc0g 17350 Σ_g cgsu 17351 ._gcmg 18988 mulGrpcmgp 20066 Ringcrg 20159 CRingccrg 20160 SubRingcsubrg 20493 mPoly cmpl 21853

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-uz 12743 df-fz 13415 df-fzo 13562 df-seq 13916 df-hash 14245 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-sca 17184 df-vsca 17185 df-tset 17187 df-0g 17352 df-gsum 17353 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-grp 18857 df-minusg 18858 df-mulg 18989 df-subg 19044 df-cntz 19237 df-cmn 19702 df-abl 19703 df-mgp 20067 df-rng 20079 df-ur 20108 df-ring 20161 df-cring 20162 df-subrg 20494 df-psr 21856 df-mpl 21858

This theorem is referenced by: evlsbagval 42727 evlvvvallem 42734 evlsmhpvvval 42753 mhphf 42755

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