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Theorem dchrptlem3 25190

Description: Lemma for dchrpt 25191. (Contributed by Mario Carneiro, 28-Apr-2016.)

**Hypotheses**
Ref	Expression
dchrpt.g	⊢ 𝐺 = (DChr‘𝑁)
dchrpt.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrpt.d	⊢ 𝐷 = (Base‘𝐺)
dchrpt.b	⊢ 𝐵 = (Base‘𝑍)
dchrpt.1	⊢ 1 = (1_r‘𝑍)
dchrpt.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
dchrpt.n1	⊢ (𝜑 → 𝐴 ≠ 1 )
dchrpt.u	⊢ 𝑈 = (Unit‘𝑍)
dchrpt.h	⊢ 𝐻 = ((mulGrp‘𝑍) ↾_s 𝑈)
dchrpt.m	⊢ · = (._g‘𝐻)
dchrpt.s	⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))))
dchrpt.au	⊢ (𝜑 → 𝐴 ∈ 𝑈)
dchrpt.w	⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
dchrpt.2	⊢ (𝜑 → 𝐻dom DProd 𝑆)
dchrpt.3	⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)

**Assertion**
Ref	Expression
dchrptlem3	⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)

Distinct variable groups: 𝑘,𝑛,𝑥, 1 𝐴,𝑘,𝑛,𝑥 𝑥,𝐵 𝑥,𝐺 𝑘,𝐻,𝑛,𝑥 𝑥,𝑁 𝑘,𝑊,𝑛,𝑥 · ,𝑘,𝑛,𝑥 𝑆,𝑘,𝑛,𝑥 𝑘,𝑍,𝑛,𝑥 𝑥,𝐷 𝜑,𝑘,𝑛,𝑥 𝑥,𝑈 Allowed substitution hints: 𝐵(𝑘,𝑛) 𝐷(𝑘,𝑛) 𝑈(𝑘,𝑛) 𝐺(𝑘,𝑛) 𝑁(𝑘,𝑛)

Proof of Theorem dchrptlem3 Dummy variables 𝑎 ℎ 𝑚 𝑢 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrpt.n1	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 1 )
2		dchrpt.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	nnnn0d 11543	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4		dchrpt.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
5	4	zncrng 20095	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
6	3, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ CRing)
7		crngring 18758	. . . . . . . . . 10 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ Ring)
9		dchrpt.u	. . . . . . . . . 10 ⊢ 𝑈 = (Unit‘𝑍)
10		dchrpt.h	. . . . . . . . . 10 ⊢ 𝐻 = ((mulGrp‘𝑍) ↾_s 𝑈)
11	9, 10	unitgrp 18867	. . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
12	8, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp)
13		grpmnd 17630	. . . . . . . 8 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Mnd)
15		dchrpt.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
16		dmexg 7262	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑈 → dom 𝑊 ∈ V)
17	15, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝑊 ∈ V)
18		eqid 2760	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
19	18	gsumz 17575	. . . . . . 7 ⊢ ((𝐻 ∈ Mnd ∧ dom 𝑊 ∈ V) → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
20	14, 17, 19	syl2anc 696	. . . . . 6 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
21		dchrpt.1	. . . . . . . . . 10 ⊢ 1 = (1_r‘𝑍)
22	9, 10, 21	unitgrpid 18869	. . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 1 = (0_g‘𝐻))
23	8, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 1 = (0_g‘𝐻))
24	23	mpteq2dv 4897	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻)))
25	24	oveq2d 6829	. . . . . 6 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) = (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))))
26	20, 25, 23	3eqtr4d 2804	. . . . 5 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) = 1 )
27	1, 26	neeqtrrd 3006	. . . 4 ⊢ (𝜑 → 𝐴 ≠ (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )))
28		dchrpt.2	. . . . . 6 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
29		zex 11578	. . . . . . . . . 10 ⊢ ℤ ∈ V
30	29	mptex 6650	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
31	30	rnex 7265	. . . . . . . 8 ⊢ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
32		dchrpt.s	. . . . . . . 8 ⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))))
33	31, 32	dmmpti 6184	. . . . . . 7 ⊢ dom 𝑆 = dom 𝑊
34	33	a1i 11	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom 𝑊)
35		eqid 2760	. . . . . 6 ⊢ (𝐻dProj𝑆) = (𝐻dProj𝑆)
36		dchrpt.au	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
37		dchrpt.3	. . . . . . 7 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
38	36, 37	eleqtrrd 2842	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐻 DProd 𝑆))
39		eqid 2760	. . . . . 6 ⊢ {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐻)} = {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐻)}
40	23	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 = (0_g‘𝐻))
41	28, 34	dprdf2 18606	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆:dom 𝑊⟶(SubGrp‘𝐻))
42	41	ffvelrnda 6522	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (𝑆‘𝑎) ∈ (SubGrp‘𝐻))
43	18	subg0cl 17803	. . . . . . . . 9 ⊢ ((𝑆‘𝑎) ∈ (SubGrp‘𝐻) → (0_g‘𝐻) ∈ (𝑆‘𝑎))
44	42, 43	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (0_g‘𝐻) ∈ (𝑆‘𝑎))
45	40, 44	eqeltrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 ∈ (𝑆‘𝑎))
46		fvex 6362	. . . . . . . . . . 11 ⊢ (1_r‘𝑍) ∈ V
47	21, 46	eqeltri 2835	. . . . . . . . . 10 ⊢ 1 ∈ V
48	47	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ V)
49	17, 48	fczfsuppd 8458	. . . . . . . 8 ⊢ (𝜑 → (dom 𝑊 × { 1 }) finSupp 1 )
50		fconstmpt 5320	. . . . . . . . . 10 ⊢ (dom 𝑊 × { 1 }) = (𝑎 ∈ dom 𝑊 ↦ 1 )
51	50	eqcomi 2769	. . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 })
52	51	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 }))
53	23	eqcomd 2766	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝐻) = 1 )
54	49, 52, 53	3brtr4d 4836	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) finSupp (0_g‘𝐻))
55	39, 28, 34, 45, 54	dprdwd 18610	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) ∈ {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐻)})
56	28, 34, 35, 38, 18, 39, 55	dpjeq 18658	. . . . 5 ⊢ (𝜑 → (𝐴 = (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ))
57	56	necon3abid 2968	. . . 4 ⊢ (𝜑 → (𝐴 ≠ (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ))
58	27, 57	mpbid 222	. . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
59		rexnal 3133	. . 3 ⊢ (∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ↔ ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
60	58, 59	sylibr 224	. 2 ⊢ (𝜑 → ∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
61		df-ne 2933	. . . 4 ⊢ ((((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 ↔ ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
62		dchrpt.g	. . . . . 6 ⊢ 𝐺 = (DChr‘𝑁)
63		dchrpt.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐺)
64		dchrpt.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑍)
65	2	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑁 ∈ ℕ)
66	1	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ≠ 1 )
67		dchrpt.m	. . . . . 6 ⊢ · = (._g‘𝐻)
68	36	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ∈ 𝑈)
69	15	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑊 ∈ Word 𝑈)
70	28	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐻dom DProd 𝑆)
71	37	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (𝐻 DProd 𝑆) = 𝑈)
72		eqid 2760	. . . . . 6 ⊢ (od‘𝐻) = (od‘𝐻)
73		eqid 2760	. . . . . 6 ⊢ (-1↑_𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) = (-1↑_𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎))))
74		simprl 811	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑎 ∈ dom 𝑊)
75		simprr 813	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )
76		eqid 2760	. . . . . 6 ⊢ (𝑢 ∈ 𝑈 ↦ (℩ℎ∃𝑚 ∈ ℤ ((((𝐻dProj𝑆)‘𝑎)‘𝑢) = (𝑚 · (𝑊‘𝑎)) ∧ ℎ = ((-1↑_𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎))))↑𝑚)))) = (𝑢 ∈ 𝑈 ↦ (℩ℎ∃𝑚 ∈ ℤ ((((𝐻dProj𝑆)‘𝑎)‘𝑢) = (𝑚 · (𝑊‘𝑎)) ∧ ℎ = ((-1↑_𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎))))↑𝑚))))
77	62, 4, 63, 64, 21, 65, 66, 9, 10, 67, 32, 68, 69, 70, 71, 35, 72, 73, 74, 75, 76	dchrptlem2 25189	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)
78	77	expr 644	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → ((((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1))
79	61, 78	syl5bir 233	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1))
80	79	rexlimdva 3169	. 2 ⊢ (𝜑 → (∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1))
81	60, 80	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∀wral 3050 ∃wrex 3051 {crab 3054 Vcvv 3340 {csn 4321 class class class wbr 4804 ↦ cmpt 4881 × cxp 5264 dom cdm 5266 ran crn 5267 ℩cio 6010 ‘cfv 6049 (class class class)co 6813 Xcixp 8074 finSupp cfsupp 8440 1c1 10129 -cneg 10459 / cdiv 10876 ℕcn 11212 2c2 11262 ℕ₀cn0 11484 ℤcz 11569 ↑cexp 13054 Word cword 13477 Basecbs 16059 ↾_s cress 16060 0_gc0g 16302 Σ_g cgsu 16303 Mndcmnd 17495 Grpcgrp 17623 ._gcmg 17741 SubGrpcsubg 17789 odcod 18144 DProd cdprd 18592 dProjcdpj 18593 mulGrpcmgp 18689 1_rcur 18701 Ringcrg 18747 CRingccrg 18748 Unitcui 18839 ℤ/nℤczn 20053 ↑_𝑐ccxp 24501 DChrcdchr 25156

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-tpos 7521 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-omul 7734 df-er 7911 df-ec 7913 df-qs 7917 df-map 8025 df-pm 8026 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-fi 8482 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-acn 8958 df-cda 9182 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-ioo 12372 df-ioc 12373 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-fl 12787 df-mod 12863 df-seq 12996 df-exp 13055 df-fac 13255 df-bc 13284 df-hash 13312 df-word 13485 df-shft 14006 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-limsup 14401 df-clim 14418 df-rlim 14419 df-sum 14616 df-ef 14997 df-sin 14999 df-cos 15000 df-pi 15002 df-dvds 15183 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-hom 16168 df-cco 16169 df-rest 16285 df-topn 16286 df-0g 16304 df-gsum 16305 df-topgen 16306 df-pt 16307 df-prds 16310 df-xrs 16364 df-qtop 16369 df-imas 16370 df-qus 16371 df-xps 16372 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mhm 17536 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-mulg 17742 df-subg 17792 df-nsg 17793 df-eqg 17794 df-ghm 17859 df-gim 17902 df-cntz 17950 df-oppg 17976 df-od 18148 df-lsm 18251 df-pj1 18252 df-cmn 18395 df-abl 18396 df-dprd 18594 df-dpj 18595 df-mgp 18690 df-ur 18702 df-ring 18749 df-cring 18750 df-oppr 18823 df-dvdsr 18841 df-unit 18842 df-rnghom 18917 df-subrg 18980 df-lmod 19067 df-lss 19135 df-lsp 19174 df-sra 19374 df-rgmod 19375 df-lidl 19376 df-rsp 19377 df-2idl 19434 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-fbas 19945 df-fg 19946 df-cnfld 19949 df-zring 20021 df-zrh 20054 df-zn 20057 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cld 21025 df-ntr 21026 df-cls 21027 df-nei 21104 df-lp 21142 df-perf 21143 df-cn 21233 df-cnp 21234 df-haus 21321 df-tx 21567 df-hmeo 21760 df-fil 21851 df-fm 21943 df-flim 21944 df-flf 21945 df-xms 22326 df-ms 22327 df-tms 22328 df-cncf 22882 df-limc 23829 df-dv 23830 df-log 24502 df-cxp 24503 df-dchr 25157

This theorem is referenced by: dchrpt 25191

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