evlslem6 - Metamath Proof Explorer

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Theorem evlslem6 19909

Description: Lemma for evlseu 19912. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 26-Jul-2019.)

**Hypotheses**
Ref	Expression
evlslem1.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
evlslem1.b	⊢ 𝐵 = (Base‘𝑃)
evlslem1.c	⊢ 𝐶 = (Base‘𝑆)
evlslem1.k	⊢ 𝐾 = (Base‘𝑅)
evlslem1.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
evlslem1.t	⊢ 𝑇 = (mulGrp‘𝑆)
evlslem1.x	⊢ ↑ = (._g‘𝑇)
evlslem1.m	⊢ · = (._r‘𝑆)
evlslem1.v	⊢ 𝑉 = (𝐼 mVar 𝑅)
evlslem1.e	⊢ 𝐸 = (𝑝 ∈ 𝐵 ↦ (𝑆 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺))))))
evlslem1.i	⊢ (𝜑 → 𝐼 ∈ V)
evlslem1.r	⊢ (𝜑 → 𝑅 ∈ CRing)
evlslem1.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlslem1.f	⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
evlslem1.g	⊢ (𝜑 → 𝐺:𝐼⟶𝐶)
evlslem6.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
evlslem6	⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))):𝐷⟶𝐶 ∧ (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) finSupp (0_g‘𝑆)))

Distinct variable groups: 𝜑,𝑏 𝐶,𝑏 𝐷,𝑏 ℎ,𝐼 𝑅,𝑏 𝑆,𝑏 𝑌,𝑏 ℎ,𝑏 Allowed substitution hints: 𝜑(ℎ,𝑝) 𝐵(ℎ,𝑝,𝑏) 𝐶(ℎ,𝑝) 𝐷(ℎ,𝑝) 𝑃(ℎ,𝑝,𝑏) 𝑅(ℎ,𝑝) 𝑆(ℎ,𝑝) 𝑇(ℎ,𝑝,𝑏) · (ℎ,𝑝,𝑏) 𝐸(ℎ,𝑝,𝑏) ↑ (ℎ,𝑝,𝑏) 𝐹(ℎ,𝑝,𝑏) 𝐺(ℎ,𝑝,𝑏) 𝐼(𝑝,𝑏) 𝐾(ℎ,𝑝,𝑏) 𝑉(ℎ,𝑝,𝑏) 𝑌(ℎ,𝑝)

Proof of Theorem evlslem6 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlslem1.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
2		crngring 18945	. . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
4	3	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ Ring)
5		evlslem1.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
6		evlslem1.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑅)
7		evlslem1.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆)
8	6, 7	rhmf 19115	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐾⟶𝐶)
9	5, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐾⟶𝐶)
10	9	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐹:𝐾⟶𝐶)
11		evlslem1.p	. . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
12		evlslem1.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑃)
13		evlslem1.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
14		evlslem6.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
15	11, 6, 12, 13, 14	mplelf 19830	. . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶𝐾)
16	15	ffvelrnda 6623	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ 𝐾)
17	10, 16	ffvelrnd 6624	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ 𝐶)
18		evlslem1.t	. . . . . 6 ⊢ 𝑇 = (mulGrp‘𝑆)
19	18, 7	mgpbas 18882	. . . . 5 ⊢ 𝐶 = (Base‘𝑇)
20		evlslem1.x	. . . . 5 ⊢ ↑ = (._g‘𝑇)
21		eqid 2778	. . . . 5 ⊢ (0_g‘𝑇) = (0_g‘𝑇)
22	18	crngmgp 18942	. . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑇 ∈ CMnd)
23	1, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ CMnd)
24	23	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑇 ∈ CMnd)
25		simpr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷)
26		evlslem1.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)
27	26	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐺:𝐼⟶𝐶)
28		evlslem1.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
29	28	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ V)
30	13, 19, 20, 21, 24, 25, 27, 29	psrbagev2 19907	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)) ∈ 𝐶)
31		evlslem1.m	. . . . 5 ⊢ · = (._r‘𝑆)
32	7, 31	ringcl 18948	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘(𝑌‘𝑏)) ∈ 𝐶 ∧ (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)) ∈ 𝐶) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺))) ∈ 𝐶)
33	4, 17, 30, 32	syl3anc 1439	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺))) ∈ 𝐶)
34	33	fmpttd 6649	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))):𝐷⟶𝐶)
35		ovexd 6956	. . . . 5 ⊢ (𝜑 → (ℕ₀ ↑_𝑚 𝐼) ∈ V)
36	13, 35	rabexd 5050	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
37		mptexg 6756	. . . 4 ⊢ (𝐷 ∈ V → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) ∈ V)
38	36, 37	syl 17	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) ∈ V)
39		funmpt 6173	. . . 4 ⊢ Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺))))
40	39	a1i 11	. . 3 ⊢ (𝜑 → Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))))
41		fvexd 6461	. . 3 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
42		eqid 2778	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
43		evlslem1.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
44	11, 12, 42, 14, 43	mplelsfi 19887	. . . 4 ⊢ (𝜑 → 𝑌 finSupp (0_g‘𝑅))
45	44	fsuppimpd 8570	. . 3 ⊢ (𝜑 → (𝑌 supp (0_g‘𝑅)) ∈ Fin)
46	15	feqmptd 6509	. . . . . . 7 ⊢ (𝜑 → 𝑌 = (𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)))
47	46	oveq1d 6937	. . . . . 6 ⊢ (𝜑 → (𝑌 supp (0_g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0_g‘𝑅)))
48		eqimss2 3877	. . . . . 6 ⊢ ((𝑌 supp (0_g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0_g‘𝑅)) → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0_g‘𝑅)) ⊆ (𝑌 supp (0_g‘𝑅)))
49	47, 48	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0_g‘𝑅)) ⊆ (𝑌 supp (0_g‘𝑅)))
50		rhmghm 19114	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
51		eqid 2778	. . . . . . 7 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
52	42, 51	ghmid 18050	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
53	5, 50, 52	3syl 18	. . . . 5 ⊢ (𝜑 → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
54		fvexd 6461	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ V)
55		fvexd 6461	. . . . 5 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
56	49, 53, 54, 55	suppssfv 7613	. . . 4 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝐹‘(𝑌‘𝑏))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))
57	7, 31, 51	ringlz 18974	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝐶) → ((0_g‘𝑆) · 𝑥) = (0_g‘𝑆))
58	3, 57	sylan 575	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((0_g‘𝑆) · 𝑥) = (0_g‘𝑆))
59		fvexd 6461	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ V)
60	56, 58, 59, 30, 41	suppssov1 7609	. . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))
61		suppssfifsupp 8578	. . 3 ⊢ ((((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) ∈ V ∧ Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) ∧ (0_g‘𝑆) ∈ V) ∧ ((𝑌 supp (0_g‘𝑅)) ∈ Fin ∧ ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) finSupp (0_g‘𝑆))
62	38, 40, 41, 45, 60, 61	syl32anc 1446	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) finSupp (0_g‘𝑆))
63	34, 62	jca 507	1 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))):𝐷⟶𝐶 ∧ (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) finSupp (0_g‘𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {crab 3094 Vcvv 3398 ⊆ wss 3792 class class class wbr 4886 ↦ cmpt 4965 ^◡ccnv 5354 “ cima 5358 Fun wfun 6129 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ∘_𝑓 cof 7172 supp csupp 7576 ↑_𝑚 cmap 8140 Fincfn 8241 finSupp cfsupp 8563 ℕcn 11374 ℕ₀cn0 11642 Basecbs 16255 ._rcmulr 16339 0_gc0g 16486 Σ_g cgsu 16487 ._gcmg 17927 GrpHom cghm 18041 CMndccmn 18579 mulGrpcmgp 18876 Ringcrg 18934 CRingccrg 18935 RingHom crh 19101 mVar cmvr 19749 mPoly cmpl 19750

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-tset 16357 df-0g 16488 df-gsum 16489 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-minusg 17813 df-mulg 17928 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-rnghom 19104 df-psr 19753 df-mpl 19755

This theorem is referenced by: evlslem1 19911

W3C validator