evlslem6 - Metamath Proof Explorer

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Theorem evlslem6 19873

Description: Lemma for evlseu 19876. 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 18912	. . . . . 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 19082	. . . . . . 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 19794	. . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶𝐾)
16	15	ffvelrnda 6608	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ 𝐾)
17	10, 16	ffvelrnd 6609	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ 𝐶)
18		evlslem1.t	. . . . . 6 ⊢ 𝑇 = (mulGrp‘𝑆)
19	18, 7	mgpbas 18849	. . . . 5 ⊢ 𝐶 = (Base‘𝑇)
20		evlslem1.x	. . . . 5 ⊢ ↑ = (._g‘𝑇)
21		eqid 2825	. . . . 5 ⊢ (0_g‘𝑇) = (0_g‘𝑇)
22	18	crngmgp 18909	. . . . . . 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 19871	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)) ∈ 𝐶)
31		evlslem1.m	. . . . 5 ⊢ · = (._r‘𝑆)
32	7, 31	ringcl 18915	. . . 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 1496	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺))) ∈ 𝐶)
34	33	fmpttd 6634	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))):𝐷⟶𝐶)
35		ovexd 6939	. . . . 5 ⊢ (𝜑 → (ℕ₀ ↑_𝑚 𝐼) ∈ V)
36	13, 35	rabexd 5038	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
37		mptexg 6740	. . . 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 6161	. . . 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 6448	. . 3 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
42		eqid 2825	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
43		evlslem1.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
44	11, 12, 42, 14, 43	mplelsfi 19851	. . . 4 ⊢ (𝜑 → 𝑌 finSupp (0_g‘𝑅))
45	44	fsuppimpd 8551	. . 3 ⊢ (𝜑 → (𝑌 supp (0_g‘𝑅)) ∈ Fin)
46	15	feqmptd 6496	. . . . . . 7 ⊢ (𝜑 → 𝑌 = (𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)))
47	46	oveq1d 6920	. . . . . 6 ⊢ (𝜑 → (𝑌 supp (0_g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0_g‘𝑅)))
48		eqimss2 3883	. . . . . 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 19081	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
51		eqid 2825	. . . . . . 7 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
52	42, 51	ghmid 18017	. . . . . 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 6448	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ V)
55		fvexd 6448	. . . . 5 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
56	49, 53, 54, 55	suppssfv 7596	. . . 4 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝐹‘(𝑌‘𝑏))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))
57	7, 31, 51	ringlz 18941	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝐶) → ((0_g‘𝑆) · 𝑥) = (0_g‘𝑆))
58	3, 57	sylan 577	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((0_g‘𝑆) · 𝑥) = (0_g‘𝑆))
59		fvexd 6448	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ V)
60	56, 58, 59, 30, 41	suppssov1 7592	. . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))
61		suppssfifsupp 8559	. . 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 1503	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_𝑓 ↑ 𝐺)))) finSupp (0_g‘𝑆))
63	34, 62	jca 509	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 1658 ∈ wcel 2166 {crab 3121 Vcvv 3414 ⊆ wss 3798 class class class wbr 4873 ↦ cmpt 4952 ^◡ccnv 5341 “ cima 5345 Fun wfun 6117 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ∘_𝑓 cof 7155 supp csupp 7559 ↑_𝑚 cmap 8122 Fincfn 8222 finSupp cfsupp 8544 ℕcn 11350 ℕ₀cn0 11618 Basecbs 16222 ._rcmulr 16306 0_gc0g 16453 Σ_g cgsu 16454 ._gcmg 17894 GrpHom cghm 18008 CMndccmn 18546 mulGrpcmgp 18843 Ringcrg 18901 CRingccrg 18902 RingHom crh 19068 mVar cmvr 19713 mPoly cmpl 19714

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-hash 13411 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-tset 16324 df-0g 16455 df-gsum 16456 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-grp 17779 df-minusg 17780 df-mulg 17895 df-ghm 18009 df-cntz 18100 df-cmn 18548 df-mgp 18844 df-ur 18856 df-ring 18903 df-cring 18904 df-rnghom 19071 df-psr 19717 df-mpl 19719

This theorem is referenced by: evlslem1 19875

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