evlslem6 - Metamath Proof Explorer

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Theorem evlslem6 22036

Description: Lemma for evlseu 22038. 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 20180	. . . . . 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 2736	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		evlslem1.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆)
8	6, 7	rhmf 20420	. . . . . . 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 21953	. . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))
16	15	ffvelcdmda 7029	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ (Base‘𝑅))
17	10, 16	ffvelcdmd 7030	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ 𝐶)
18		evlslem1.t	. . . . . 6 ⊢ 𝑇 = (mulGrp‘𝑆)
19	18, 7	mgpbas 20080	. . . . 5 ⊢ 𝐶 = (Base‘𝑇)
20		evlslem1.x	. . . . 5 ⊢ ↑ = (._g‘𝑇)
21	18	crngmgp 20176	. . . . . . 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 22033	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)) ∈ 𝐶)
28		evlslem1.m	. . . . 5 ⊢ · = (._r‘𝑆)
29	7, 28	ringcl 20185	. . . 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 7060	. 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))):𝐷⟶𝐶)
32		ovexd 7393	. . . . 5 ⊢ (𝜑 → (ℕ₀ ↑_m 𝐼) ∈ V)
33	13, 32	rabexd 5285	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
34	33	mptexd 7170	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))) ∈ V)
35		funmpt 6530	. . . 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 6849	. . 3 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
38		eqid 2736	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
39	11, 12, 38, 14	mplelsfi 21950	. . . 4 ⊢ (𝜑 → 𝑌 finSupp (0_g‘𝑅))
40	39	fsuppimpd 9272	. . 3 ⊢ (𝜑 → (𝑌 supp (0_g‘𝑅)) ∈ Fin)
41	15	feqmptd 6902	. . . . . . 7 ⊢ (𝜑 → 𝑌 = (𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)))
42	41	oveq1d 7373	. . . . . 6 ⊢ (𝜑 → (𝑌 supp (0_g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0_g‘𝑅)))
43		eqimss2 3993	. . . . . 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 20419	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
46		eqid 2736	. . . . . . 7 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
47	38, 46	ghmid 19151	. . . . . 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 6849	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ V)
50		fvexd 6849	. . . . 5 ⊢ (𝜑 → (0_g‘𝑅) ∈ V)
51	44, 48, 49, 50	suppssfv 8144	. . . 4 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝐹‘(𝑌‘𝑏))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))
52	7, 28, 46	ringlz 20228	. . . . 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 6849	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ V)
55	51, 53, 54, 27, 37	suppssov1 8139	. . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σ_g (𝑏 ∘_f ↑ 𝐺)))) supp (0_g‘𝑆)) ⊆ (𝑌 supp (0_g‘𝑅)))
56		suppssfifsupp 9283	. . 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 2113 {crab 3399 Vcvv 3440 ⊆ wss 3901 class class class wbr 5098 ↦ cmpt 5179 ^◡ccnv 5623 “ cima 5627 Fun wfun 6486 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ∘_f cof 7620 supp csupp 8102 ↑_m cmap 8763 Fincfn 8883 finSupp cfsupp 9264 ℕcn 12145 ℕ₀cn0 12401 Basecbs 17136 ._rcmulr 17178 0_gc0g 17359 Σ_g cgsu 17360 ._gcmg 18997 GrpHom cghm 19141 CMndccmn 19709 mulGrpcmgp 20075 Ringcrg 20168 CRingccrg 20169 RingHom crh 20405 mVar cmvr 21861 mPoly cmpl 21862

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-fzo 13571 df-seq 13925 df-hash 14254 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-tset 17196 df-0g 17361 df-gsum 17362 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-grp 18866 df-minusg 18867 df-mulg 18998 df-ghm 19142 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-rhm 20408 df-psr 21865 df-mpl 21867

This theorem is referenced by: evlslem1 22037

W3C validator