evlslem6 - Metamath Proof Explorer

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Theorem evlslem6 22011

Description: Lemma for evlseu 22013. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 26-Jul-2019.) (Revised by AV, 11-Apr-2024.)

**Hypotheses**
Ref	Expression
evlslem1.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
evlslem1.b	⊢ 𝐵 = (Base‘𝑃)
evlslem1.c	⊢ 𝐶 = (Base‘𝑆)
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 ∈ 𝑊)
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) 𝑉(ℎ,𝑝,𝑏) 𝑊(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 20158	. . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ Ring)
5		evlslem1.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
6		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		evlslem1.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆)
8	6, 7	rhmf 20397	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐶)
9	5, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶𝐶)
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐹:(Base‘𝑅)⟶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 21930	. . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))
16	15	ffvelcdmda 7012	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ (Base‘𝑅))
17	10, 16	ffvelcdmd 7013	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ 𝐶)
18		evlslem1.t	. . . . . 6 ⊢ 𝑇 = (mulGrp‘𝑆)
19	18, 7	mgpbas 20058	. . . . 5 ⊢ 𝐶 = (Base‘𝑇)
20		evlslem1.x	. . . . 5 ⊢ ↑ = (._g‘𝑇)
21	18	crngmgp 20154	. . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑇 ∈ CMnd)
22	1, 21	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ CMnd)
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑇 ∈ CMnd)
24		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷)
25		evlslem1.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)
26	25	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐺:𝐼⟶𝐶)
27	13, 19, 20, 23, 24, 26	psrbagev2 22008	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)) ∈ 𝐶)
28		evlslem1.m	. . . . 5 ⊢ · = (._r‘𝑆)
29	7, 28	ringcl 20163	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘(𝑌‘𝑏)) ∈ 𝐶 ∧ (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)) ∈ 𝐶) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺))) ∈ 𝐶)
30	4, 17, 27, 29	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺))) ∈ 𝐶)
31	30	fmpttd 7043	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))):𝐷⟶𝐶)
32		ovexd 7376	. . . . 5 ⊢ (𝜑 → (ℕ₀ ↑_m 𝐼) ∈ V)
33	13, 32	rabexd 5273	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
34	33	mptexd 7153	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))) ∈ V)
35		funmpt 6514	. . . 4 ⊢ Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺))))
36	35	a1i 11	. . 3 ⊢ (𝜑 → Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))))
37		fvexd 6832	. . 3 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
38		eqid 2731	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
39	11, 12, 38, 14	mplelsfi 21927	. . . 4 ⊢ (𝜑 → 𝑌 finSupp (0_g‘𝑅))
40	39	fsuppimpd 9248	. . 3 ⊢ (𝜑 → (𝑌 supp (0_g‘𝑅)) ∈ Fin)
41	15	feqmptd 6885	. . . . . . 7 ⊢ (𝜑 → 𝑌 = (𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)))
42	41	oveq1d 7356	. . . . . 6 ⊢ (𝜑 → (𝑌 supp (0_g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0_g‘𝑅)))
43		eqimss2 3989	. . . . . 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)))
44	42, 43	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0_g‘𝑅)) ⊆ (𝑌 supp (0_g‘𝑅)))
45		rhmghm 20396	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
46		eqid 2731	. . . . . . 7 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
47	38, 46	ghmid 19129	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
48	5, 45, 47	3syl 18	. . . . 5 ⊢ (𝜑 → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
49		fvexd 6832	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ V)
50		fvexd 6832	. . . . 5 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
51	44, 48, 49, 50	suppssfv 8127	. . . 4 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝐹‘(𝑌‘𝑏))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))
52	7, 28, 46	ringlz 20206	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝐶) → ((0_g‘𝑆) · 𝑥) = (0_g‘𝑆))
53	3, 52	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((0_g‘𝑆) · 𝑥) = (0_g‘𝑆))
54		fvexd 6832	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ V)
55	51, 53, 54, 27, 37	suppssov1 8122	. . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))
56		suppssfifsupp 9259	. . 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))
57	34, 36, 37, 40, 55, 56	syl32anc 1380	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))) finSupp (0_g‘𝑆))
58	31, 57	jca 511	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 395 = wceq 1541 ∈ wcel 2111 {crab 3395 Vcvv 3436 ⊆ wss 3897 class class class wbr 5086 ↦ cmpt 5167 ^◡ccnv 5610 “ cima 5614 Fun wfun 6470 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 ∘_f cof 7603 supp csupp 8085 ↑_m cmap 8745 Fincfn 8864 finSupp cfsupp 9240 ℕcn 12120 ℕ₀cn0 12376 Basecbs 17115 ._rcmulr 17157 0_gc0g 17338 Σ_g cgsu 17339 ._gcmg 18975 GrpHom cghm 19119 CMndccmn 19687 mulGrpcmgp 20053 Ringcrg 20146 CRingccrg 20147 RingHom crh 20382 mVar cmvr 21837 mPoly cmpl 21838

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-oi 9391 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-uz 12728 df-fz 13403 df-fzo 13550 df-seq 13904 df-hash 14233 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-sca 17172 df-vsca 17173 df-tset 17175 df-0g 17340 df-gsum 17341 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-mhm 18686 df-grp 18844 df-minusg 18845 df-mulg 18976 df-ghm 19120 df-cntz 19224 df-cmn 19689 df-abl 19690 df-mgp 20054 df-rng 20066 df-ur 20095 df-ring 20148 df-cring 20149 df-rhm 20385 df-psr 21841 df-mpl 21843

This theorem is referenced by: evlslem1 22012

W3C validator