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Theorem dchrptlem3 27441

Description: Lemma for dchrpt 27442. (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 12571	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4		dchrpt.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
5	4	zncrng 21705	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
6	3, 5	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ CRing)
7		crngring 20333	. . . . . . . . . 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 20472	. . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
12	8, 11	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp)
13	12	grpmndd 19019	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Mnd)
14		dchrpt.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
15	14	dmexd 7898	. . . . . . 7 ⊢ (𝜑 → dom 𝑊 ∈ V)
16		eqid 2762	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
17	16	gsumz 18901	. . . . . . 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 20474	. . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 1 = (0_g‘𝐻))
21	8, 20	syl 18	. . . . . . . 8 ⊢ (𝜑 → 1 = (0_g‘𝐻))
22	21	mpteq2dv 5204	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻)))
23	22	oveq2d 7428	. . . . . 6 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) = (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))))
24	18, 23, 21	3eqtr4d 2807	. . . . 5 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) = 1 )
25	1, 24	neeqtrrd 3031	. . . 4 ⊢ (𝜑 → 𝐴 ≠ (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )))
26		dchrpt.2	. . . . . 6 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
27		zex 12606	. . . . . . . . . 10 ⊢ ℤ ∈ V
28	27	mptex 7221	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
29	28	rnex 7905	. . . . . . . 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 6679	. . . . . . 7 ⊢ dom 𝑆 = dom 𝑊
32	31	a1i 11	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom 𝑊)
33		eqid 2762	. . . . . 6 ⊢ (𝐻dProj𝑆) = (𝐻dProj𝑆)
34		dchrpt.au	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
35		dchrpt.3	. . . . . . 7 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
36	34, 35	eleqtrrd 2865	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐻 DProd 𝑆))
37		eqid 2762	. . . . . 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 20085	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆:dom 𝑊⟶(SubGrp‘𝐻))
40	39	ffvelcdmda 7079	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (𝑆‘𝑎) ∈ (SubGrp‘𝐻))
41	16	subg0cl 19206	. . . . . . . . 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 2862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 ∈ (𝑆‘𝑎))
44	19	fvexi 6895	. . . . . . . . . 10 ⊢ 1 ∈ V
45	44	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ V)
46	15, 45	fczfsuppd 9344	. . . . . . . 8 ⊢ (𝜑 → (dom 𝑊 × { 1 }) finSupp 1 )
47		fconstmpt 5722	. . . . . . . . . 10 ⊢ (dom 𝑊 × { 1 }) = (𝑎 ∈ dom 𝑊 ↦ 1 )
48	47	eqcomi 2771	. . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 })
49	48	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 }))
50	21	eqcomd 2768	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝐻) = 1 )
51	46, 49, 50	3brtr4d 5142	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) finSupp (0_g‘𝐻))
52	37, 26, 32, 43, 51	dprdwd 20089	. . . . . 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 20137	. . . . 5 ⊢ (𝜑 → (𝐴 = (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ))
54	53	necon3abid 2993	. . . 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 3116	. . 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 2958	. . . 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 2762	. . . . . 6 ⊢ (od‘𝐻) = (od‘𝐻)
70		eqid 2762	. . . . . 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 2762	. . . . . 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 27440	. . . . 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 3165	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 {crab 3415 Vcvv 3454 {csn 4588 class class class wbr 5108 ↦ cmpt 5191 × cxp 5658 dom cdm 5660 ran crn 5661 ℩cio 6490 ‘cfv 6536 (class class class)co 7412 Xcixp 8893 finSupp cfsupp 9319 1c1 11107 -cneg 11448 / cdiv 11877 ℕcn 12239 2c2 12301 ℕ₀cn0 12510 ℤcz 12597 ↑cexp 14104 Word cword 14557 Basecbs 17275 ↾_s cress 17296 0_gc0g 17498 Σ_g cgsu 17499 Mndcmnd 18798 Grpcgrp 19006 ._gcmg 19139 SubGrpcsubg 19192 odcod 19600 DProd cdprd 20071 dProjcdpj 20072 mulGrpcmgp 20222 1_rcur 20269 Ringcrg 20321 CRingccrg 20322 Unitcui 20444 ℤ/nℤczn 21663 ↑_𝑐ccxp 26731 DChrcdchr 27407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-omul 8456 df-er 8692 df-ec 8694 df-qs 8698 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-acn 9935 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-word 14558 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-pi 16132 df-dvds 16317 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-qus 17569 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-submnd 18848 df-grp 19009 df-minusg 19010 df-sbg 19011 df-mulg 19140 df-subg 19195 df-nsg 19196 df-eqg 19197 df-ghm 19290 df-gim 19335 df-cntz 19393 df-oppg 19422 df-od 19604 df-lsm 19712 df-pj1 19713 df-cmn 19858 df-abl 19859 df-dprd 20073 df-dpj 20074 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-oppr 20426 df-dvdsr 20446 df-unit 20447 df-rhm 20561 df-subrng 20656 df-subrg 20680 df-lmod 20994 df-lss 21064 df-lsp 21104 df-sra 21305 df-rgmod 21306 df-lidl 21343 df-rsp 21344 df-2idl 21400 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-zring 21608 df-zrh 21664 df-zn 21667 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-limc 26036 df-dv 26037 df-log 26732 df-cxp 26733 df-dchr 27408

This theorem is used by: dchrpt 27442

W3C validator