evlslem6 - Metamath Proof Explorer

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Theorem evlslem6 22383

Description: Lemma for evlseu 22385. 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 20465	. . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
3	1, 2	syl 18	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
4	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ Ring)
5		evlslem1.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
6		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		evlslem1.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆)
8	6, 7	rhmf 20708	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐶)
9	5, 8	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶𝐶)
10	9	adantr 486	. . . . 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 22298	. . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))
16	15	ffvelcdmda 7082	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ (Base‘𝑅))
17	10, 16	ffvelcdmd 7083	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ 𝐶)
18		evlslem1.t	. . . . . 6 ⊢ 𝑇 = (mulGrp‘𝑆)
19	18, 7	mgpbas 20358	. . . . 5 ⊢ 𝐶 = (Base‘𝑇)
20		evlslem1.x	. . . . 5 ⊢ ↑ = (._g‘𝑇)
21	18	crngmgp 20460	. . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑇 ∈ CMnd)
22	1, 21	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ CMnd)
23	22	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑇 ∈ CMnd)
24		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷)
25		evlslem1.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶)
26	25	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐺:𝐼⟶𝐶)
27	13, 19, 20, 23, 24, 26	psrbagev2 22380	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)) ∈ 𝐶)
28		evlslem1.m	. . . . 5 ⊢ · = (._r‘𝑆)
29	7, 28	ringcl 20470	. . . 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 1398	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺))) ∈ 𝐶)
31	30	fmpttd 7113	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))):𝐷⟶𝐶)
32		ovexd 7453	. . . . 5 ⊢ (𝜑 → (ℕ₀ ↑_m 𝐼) ∈ V)
33	13, 32	rabexd 5301	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
34	33	mptexd 7228	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))) ∈ V)
35		funmpt 6576	. . . 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 6898	. . 3 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
38		eqid 2761	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
39	11, 12, 38, 14	mplelsfi 22295	. . . 4 ⊢ (𝜑 → 𝑌 finSupp (0_g‘𝑅))
40	39	fsuppimpd 9354	. . 3 ⊢ (𝜑 → (𝑌 supp (0_g‘𝑅)) ∈ Fin)
41	15	feqmptd 6951	. . . . . . 7 ⊢ (𝜑 → 𝑌 = (𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)))
42	41	oveq1d 7433	. . . . . 6 ⊢ (𝜑 → (𝑌 supp (0_g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0_g‘𝑅)))
43		eqimss2 3990	. . . . . 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 18	. . . . 5 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0_g‘𝑅)) ⊆ (𝑌 supp (0_g‘𝑅)))
45		rhmghm 20707	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
46		eqid 2761	. . . . . . 7 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
47	38, 46	ghmid 19429	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
48	5, 45, 47	3syl 19	. . . . 5 ⊢ (𝜑 → (𝐹‘(0_g‘𝑅)) = (0_g‘𝑆))
49		fvexd 6898	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ V)
50		fvexd 6898	. . . . 5 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
51	44, 48, 49, 50	suppssfv 8212	. . . 4 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝐹‘(𝑌‘𝑏))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))
52	7, 28, 46	ringlz 20517	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝐶) → ((0_g‘𝑆) · 𝑥) = (0_g‘𝑆))
53	3, 52	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((0_g‘𝑆) · 𝑥) = (0_g‘𝑆))
54		fvexd 6898	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ V)
55	51, 53, 54, 27, 37	suppssov1 8207	. . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))
56		suppssfifsupp 9365	. . 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 1405	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))) finSupp (0_g‘𝑆))
58	31, 57	jca 521	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3413 Vcvv 3451 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 ^◡ccnv 5650 “ cima 5654 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ∘_f cof 7689 supp csupp 8170 ↑_m cmap 8840 Fincfn 8966 finSupp cfsupp 9346 ℕcn 12328 ℕ₀cn0 12599 Basecbs 17380 ._rcmulr 17422 0_gc0g 17603 Σ_g cgsu 17604 ._gcmg 19270 GrpHom cghm 19420 CMndccmn 19987 mulGrpcmgp 20353 Ringcrg 20452 CRingccrg 20453 RingHom crh 20692 mVar cmvr 22206 mPoly cmpl 22207

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-map 8842 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-uz 12959 df-fz 13633 df-fzo 13782 df-seq 14138 df-hash 14468 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-sca 17437 df-vsca 17438 df-tset 17440 df-0g 17605 df-gsum 17606 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-mhm 18971 df-grp 19140 df-minusg 19141 df-mulg 19271 df-ghm 19421 df-cntz 19524 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-cring 20455 df-rhm 20695 df-psr 22210 df-mpl 22212

This theorem is used by: evlslem1 22384

W3C validator