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Theorem dchrptlem3 27388

Description: Lemma for dchrpt 27389. (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 12556	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4		dchrpt.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
5	4	zncrng 21654	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
6	3, 5	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ CRing)
7		crngring 20318	. . . . . . . . . 10 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
8	6, 7	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ Ring)
9		dchrpt.u	. . . . . . . . . 10 ⊢ 𝑈 = (Unit‘𝑍)
10		dchrpt.h	. . . . . . . . . 10 ⊢ 𝐻 = ((mulGrp‘𝑍) ↾_s 𝑈)
11	9, 10	unitgrp 20456	. . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
12	8, 11	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp)
13	12	grpmndd 19003	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Mnd)
14		dchrpt.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
15	14	dmexd 7888	. . . . . . 7 ⊢ (𝜑 → dom 𝑊 ∈ V)
16		eqid 2765	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
17	16	gsumz 18885	. . . . . . 7 ⊢ ((𝐻 ∈ Mnd ∧ dom 𝑊 ∈ V) → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
18	13, 15, 17	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
19		dchrpt.1	. . . . . . . . . 10 ⊢ 1 = (1_r‘𝑍)
20	9, 10, 19	unitgrpid 20458	. . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 1 = (0_g‘𝐻))
21	8, 20	syl 18	. . . . . . . 8 ⊢ (𝜑 → 1 = (0_g‘𝐻))
22	21	mpteq2dv 5199	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻)))
23	22	oveq2d 7416	. . . . . 6 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) = (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))))
24	18, 23, 21	3eqtr4d 2810	. . . . 5 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) = 1 )
25	1, 24	neeqtrrd 3034	. . . 4 ⊢ (𝜑 → 𝐴 ≠ (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )))
26		dchrpt.2	. . . . . 6 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
27		zex 12591	. . . . . . . . . 10 ⊢ ℤ ∈ V
28	27	mptex 7211	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
29	28	rnex 7895	. . . . . . . 8 ⊢ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
30		dchrpt.s	. . . . . . . 8 ⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))))
31	29, 30	dmmpti 6669	. . . . . . 7 ⊢ dom 𝑆 = dom 𝑊
32	31	a1i 11	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom 𝑊)
33		eqid 2765	. . . . . 6 ⊢ (𝐻dProj𝑆) = (𝐻dProj𝑆)
34		dchrpt.au	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
35		dchrpt.3	. . . . . . 7 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
36	34, 35	eleqtrrd 2868	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐻 DProd 𝑆))
37		eqid 2765	. . . . . 6 ⊢ {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐻)} = {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐻)}
38	21	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 = (0_g‘𝐻))
39	26, 32	dprdf2 20070	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆:dom 𝑊⟶(SubGrp‘𝐻))
40	39	ffvelcdmda 7069	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (𝑆‘𝑎) ∈ (SubGrp‘𝐻))
41	16	subg0cl 19191	. . . . . . . . 9 ⊢ ((𝑆‘𝑎) ∈ (SubGrp‘𝐻) → (0_g‘𝐻) ∈ (𝑆‘𝑎))
42	40, 41	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (0_g‘𝐻) ∈ (𝑆‘𝑎))
43	38, 42	eqeltrd 2865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 ∈ (𝑆‘𝑎))
44	19	fvexi 6885	. . . . . . . . . 10 ⊢ 1 ∈ V
45	44	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ V)
46	15, 45	fczfsuppd 9334	. . . . . . . 8 ⊢ (𝜑 → (dom 𝑊 × { 1 }) finSupp 1 )
47		fconstmpt 5714	. . . . . . . . . 10 ⊢ (dom 𝑊 × { 1 }) = (𝑎 ∈ dom 𝑊 ↦ 1 )
48	47	eqcomi 2774	. . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 })
49	48	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 }))
50	21	eqcomd 2771	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝐻) = 1 )
51	46, 49, 50	3brtr4d 5137	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) finSupp (0_g‘𝐻))
52	37, 26, 32, 43, 51	dprdwd 20074	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) ∈ {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0_g‘𝐻)})
53	26, 32, 33, 36, 16, 37, 52	dpjeq 20122	. . . . 5 ⊢ (𝜑 → (𝐴 = (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ))
54	53	necon3abid 2996	. . . 4 ⊢ (𝜑 → (𝐴 ≠ (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ))
55	25, 54	mpbid 235	. . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
56		rexnal 3117	. . 3 ⊢ (∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ↔ ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
57	55, 56	sylibr 237	. 2 ⊢ (𝜑 → ∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
58		df-ne 2961	. . . 4 ⊢ ((((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 ↔ ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
59		dchrpt.g	. . . . . 6 ⊢ 𝐺 = (DChr‘𝑁)
60		dchrpt.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐺)
61		dchrpt.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑍)
62	2	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑁 ∈ ℕ)
63	1	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ≠ 1 )
64		dchrpt.m	. . . . . 6 ⊢ · = (._g‘𝐻)
65	34	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ∈ 𝑈)
66	14	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑊 ∈ Word 𝑈)
67	26	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐻dom DProd 𝑆)
68	35	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (𝐻 DProd 𝑆) = 𝑈)
69		eqid 2765	. . . . . 6 ⊢ (od‘𝐻) = (od‘𝐻)
70		eqid 2765	. . . . . 6 ⊢ (-1↑_𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) = (-1↑_𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎))))
71		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑎 ∈ dom 𝑊)
72		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )
73		eqid 2765	. . . . . 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))))
74	59, 4, 60, 61, 19, 62, 63, 9, 10, 64, 30, 65, 66, 67, 68, 33, 69, 70, 71, 72, 73	dchrptlem2 27387	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)
75	74	expr 461	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → ((((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1))
76	58, 75	biimtrrid 246	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1))
77	76	rexlimdva 3166	. 2 ⊢ (𝜑 → (∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1))
78	57, 77	mpd 16	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∀wral 3079 ∃wrex 3089 {crab 3417 Vcvv 3457 {csn 4585 class class class wbr 5105 ↦ cmpt 5186 × cxp 5650 dom cdm 5652 ran crn 5653 ℩cio 6479 ‘cfv 6525 (class class class)co 7400 Xcixp 8883 finSupp cfsupp 9309 1c1 11089 -cneg 11430 / cdiv 11859 ℕcn 12224 2c2 12286 ℕ₀cn0 12495 ℤcz 12582 ↑cexp 14088 Word cword 14540 Basecbs 17259 ↾_s cress 17280 0_gc0g 17482 Σ_g cgsu 17483 Mndcmnd 18782 Grpcgrp 18990 ._gcmg 19124 SubGrpcsubg 19177 odcod 19585 DProd cdprd 20056 dProjcdpj 20057 mulGrpcmgp 20207 1_rcur 20254 Ringcrg 20306 CRingccrg 20307 Unitcui 20428 ℤ/nℤczn 21612 ↑_𝑐ccxp 26678 DChrcdchr 27354

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167 ax-mulf 11168

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-tpos 8210 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-oadd 8445 df-omul 8446 df-er 8682 df-ec 8684 df-qs 8688 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-acn 9916 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-fac 14301 df-bc 14330 df-hash 14358 df-word 14541 df-shft 15094 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-limsup 15512 df-clim 15529 df-rlim 15530 df-sum 15728 df-ef 16111 df-sin 16113 df-cos 16114 df-pi 16116 df-dvds 16301 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-qus 17553 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-mhm 18831 df-submnd 18832 df-grp 18993 df-minusg 18994 df-sbg 18995 df-mulg 19125 df-subg 19180 df-nsg 19181 df-eqg 19182 df-ghm 19275 df-gim 19320 df-cntz 19378 df-oppg 19407 df-od 19589 df-lsm 19697 df-pj1 19698 df-cmn 19843 df-abl 19844 df-dprd 20058 df-dpj 20059 df-mgp 20208 df-rng 20222 df-ur 20255 df-ring 20308 df-cring 20309 df-oppr 20410 df-dvdsr 20430 df-unit 20431 df-rhm 20545 df-subrng 20622 df-subrg 20646 df-lmod 20952 df-lss 21022 df-lsp 21062 df-sra 21263 df-rgmod 21264 df-lidl 21301 df-rsp 21302 df-2idl 21351 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-fbas 21479 df-fg 21480 df-cnfld 21483 df-zring 21557 df-zrh 21613 df-zn 21616 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-nei 23216 df-lp 23254 df-perf 23255 df-cn 23345 df-cnp 23346 df-haus 23433 df-tx 23680 df-hmeo 23873 df-fil 23964 df-fm 24056 df-flim 24057 df-flf 24058 df-xms 24438 df-ms 24439 df-tms 24440 df-cncf 24998 df-limc 25986 df-dv 25987 df-log 26679 df-cxp 26680 df-dchr 27355

This theorem is referenced by: dchrpt 27389

W3C validator