dchrptlem3 - Metamath Proof Explorer

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Theorem dchrptlem3 27197

Description: Lemma for dchrpt 27198. (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 12434	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4		dchrpt.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
5	4	zncrng 21474	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
6	3, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ CRing)
7		crngring 20156	. . . . . . . . . 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 20294	. . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
12	8, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp)
13	12	grpmndd 18851	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Mnd)
14		dchrpt.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
15	14	dmexd 7828	. . . . . . 7 ⊢ (𝜑 → dom 𝑊 ∈ V)
16		eqid 2730	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
17	16	gsumz 18736	. . . . . . 7 ⊢ ((𝐻 ∈ Mnd ∧ dom 𝑊 ∈ V) → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
18	13, 15, 17	syl2anc 584	. . . . . 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 20296	. . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 1 = (0_g‘𝐻))
21	8, 20	syl 17	. . . . . . . 8 ⊢ (𝜑 → 1 = (0_g‘𝐻))
22	21	mpteq2dv 5183	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻)))
23	22	oveq2d 7357	. . . . . 6 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) = (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))))
24	18, 23, 21	3eqtr4d 2775	. . . . 5 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) = 1 )
25	1, 24	neeqtrrd 3000	. . . 4 ⊢ (𝜑 → 𝐴 ≠ (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )))
26		dchrpt.2	. . . . . 6 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
27		zex 12469	. . . . . . . . . 10 ⊢ ℤ ∈ V
28	27	mptex 7152	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
29	28	rnex 7835	. . . . . . . 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 6621	. . . . . . 7 ⊢ dom 𝑆 = dom 𝑊
32	31	a1i 11	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom 𝑊)
33		eqid 2730	. . . . . 6 ⊢ (𝐻dProj𝑆) = (𝐻dProj𝑆)
34		dchrpt.au	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
35		dchrpt.3	. . . . . . 7 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
36	34, 35	eleqtrrd 2832	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐻 DProd 𝑆))
37		eqid 2730	. . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 = (0_g‘𝐻))
39	26, 32	dprdf2 19914	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆:dom 𝑊⟶(SubGrp‘𝐻))
40	39	ffvelcdmda 7012	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (𝑆‘𝑎) ∈ (SubGrp‘𝐻))
41	16	subg0cl 19039	. . . . . . . . 9 ⊢ ((𝑆‘𝑎) ∈ (SubGrp‘𝐻) → (0_g‘𝐻) ∈ (𝑆‘𝑎))
42	40, 41	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (0_g‘𝐻) ∈ (𝑆‘𝑎))
43	38, 42	eqeltrd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 ∈ (𝑆‘𝑎))
44	19	fvexi 6831	. . . . . . . . . 10 ⊢ 1 ∈ V
45	44	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ V)
46	15, 45	fczfsuppd 9265	. . . . . . . 8 ⊢ (𝜑 → (dom 𝑊 × { 1 }) finSupp 1 )
47		fconstmpt 5676	. . . . . . . . . 10 ⊢ (dom 𝑊 × { 1 }) = (𝑎 ∈ dom 𝑊 ↦ 1 )
48	47	eqcomi 2739	. . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 })
49	48	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 }))
50	21	eqcomd 2736	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝐻) = 1 )
51	46, 49, 50	3brtr4d 5121	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) finSupp (0_g‘𝐻))
52	37, 26, 32, 43, 51	dprdwd 19918	. . . . . 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 19966	. . . . 5 ⊢ (𝜑 → (𝐴 = (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ))
54	53	necon3abid 2962	. . . 4 ⊢ (𝜑 → (𝐴 ≠ (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ))
55	25, 54	mpbid 232	. . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
56		rexnal 3082	. . 3 ⊢ (∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ↔ ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
57	55, 56	sylibr 234	. 2 ⊢ (𝜑 → ∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )
58		df-ne 2927	. . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑁 ∈ ℕ)
63	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ≠ 1 )
64		dchrpt.m	. . . . . 6 ⊢ · = (._g‘𝐻)
65	34	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ∈ 𝑈)
66	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑊 ∈ Word 𝑈)
67	26	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐻dom DProd 𝑆)
68	35	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (𝐻 DProd 𝑆) = 𝑈)
69		eqid 2730	. . . . . 6 ⊢ (od‘𝐻) = (od‘𝐻)
70		eqid 2730	. . . . . 6 ⊢ (-1↑_𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) = (-1↑_𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎))))
71		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑎 ∈ dom 𝑊)
72		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )
73		eqid 2730	. . . . . 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 27196	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)
75	74	expr 456	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → ((((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1))
76	58, 75	biimtrrid 243	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1))
77	76	rexlimdva 3131	. 2 ⊢ (𝜑 → (∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1))
78	57, 77	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 {crab 3393 Vcvv 3434 {csn 4574 class class class wbr 5089 ↦ cmpt 5170 × cxp 5612 dom cdm 5614 ran crn 5615 ℩cio 6431 ‘cfv 6477 (class class class)co 7341 Xcixp 8816 finSupp cfsupp 9240 1c1 10999 -cneg 11337 / cdiv 11766 ℕcn 12117 2c2 12172 ℕ₀cn0 12373 ℤcz 12460 ↑cexp 13960 Word cword 14412 Basecbs 17112 ↾_s cress 17133 0_gc0g 17335 Σ_g cgsu 17336 Mndcmnd 18634 Grpcgrp 18838 ._gcmg 18972 SubGrpcsubg 19025 odcod 19429 DProd cdprd 19900 dProjcdpj 19901 mulGrpcmgp 20051 1_rcur 20092 Ringcrg 20144 CRingccrg 20145 Unitcui 20266 ℤ/nℤczn 21432 ↑_𝑐ccxp 26484 DChrcdchr 27163

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 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077 ax-mulf 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 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 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-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8617 df-ec 8619 df-qs 8623 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9824 df-acn 9827 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ioc 13242 df-ico 13243 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-fac 14173 df-bc 14202 df-hash 14230 df-word 14413 df-shft 14966 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-limsup 15370 df-clim 15387 df-rlim 15388 df-sum 15586 df-ef 15966 df-sin 15968 df-cos 15969 df-pi 15971 df-dvds 16156 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-qus 17405 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-mhm 18683 df-submnd 18684 df-grp 18841 df-minusg 18842 df-sbg 18843 df-mulg 18973 df-subg 19028 df-nsg 19029 df-eqg 19030 df-ghm 19118 df-gim 19164 df-cntz 19222 df-oppg 19251 df-od 19433 df-lsm 19541 df-pj1 19542 df-cmn 19687 df-abl 19688 df-dprd 19902 df-dpj 19903 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-cring 20147 df-oppr 20248 df-dvdsr 20268 df-unit 20269 df-rhm 20383 df-subrng 20454 df-subrg 20478 df-lmod 20788 df-lss 20858 df-lsp 20898 df-sra 21100 df-rgmod 21101 df-lidl 21138 df-rsp 21139 df-2idl 21180 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-zring 21377 df-zrh 21433 df-zn 21436 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-haus 23223 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-xms 24228 df-ms 24229 df-tms 24230 df-cncf 24791 df-limc 25787 df-dv 25788 df-log 26485 df-cxp 26486 df-dchr 27164

This theorem is referenced by: dchrpt 27198

W3C validator