dchrptlem3 - Metamath Proof Explorer

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Theorem dchrptlem3 27623

Description: Lemma for dchrpt 27624. (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 12694	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4		dchrpt.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
5	4	zncrng 21880	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
6	3, 5	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ CRing)
7		crngring 20502	. . . . . . . . . 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 20643	. . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
12	8, 11	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp)
13	12	grpmndd 19186	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Mnd)
14		dchrpt.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
15	14	dmexd 7939	. . . . . . 7 ⊢ (𝜑 → dom 𝑊 ∈ V)
16		eqid 2763	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
17	16	gsumz 19061	. . . . . . 7 ⊢ ((𝐻 ∈ Mnd ∧ dom 𝑊 ∈ V) → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
18	13, 15, 17	syl2anc 596	. . . . . 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 20645	. . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 1 = (0_g‘𝐻))
21	8, 20	syl 18	. . . . . . . 8 ⊢ (𝜑 → 1 = (0_g‘𝐻))
22	21	mpteq2dv 5202	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻)))
23	22	oveq2d 7459	. . . . . 6 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) = (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ (0_g‘𝐻))))
24	18, 23, 21	3eqtr4d 2808	. . . . 5 ⊢ (𝜑 → (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) = 1 )
25	1, 24	neeqtrrd 3033	. . . 4 ⊢ (𝜑 → 𝐴 ≠ (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )))
26		dchrpt.2	. . . . . 6 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
27		zex 12729	. . . . . . . . . 10 ⊢ ℤ ∈ V
28	27	mptex 7252	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
29	28	rnex 7946	. . . . . . . 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 6705	. . . . . . 7 ⊢ dom 𝑆 = dom 𝑊
32	31	a1i 11	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom 𝑊)
33		eqid 2763	. . . . . 6 ⊢ (𝐻dProj𝑆) = (𝐻dProj𝑆)
34		dchrpt.au	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
35		dchrpt.3	. . . . . . 7 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
36	34, 35	eleqtrrd 2867	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐻 DProd 𝑆))
37		eqid 2763	. . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 = (0_g‘𝐻))
39	26, 32	dprdf2 20253	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆:dom 𝑊⟶(SubGrp‘𝐻))
40	39	ffvelcdmda 7108	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (𝑆‘𝑎) ∈ (SubGrp‘𝐻))
41	16	subg0cl 19373	. . . . . . . . 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 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 ∈ (𝑆‘𝑎))
44	19	fvexi 6922	. . . . . . . . . 10 ⊢ 1 ∈ V
45	44	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ V)
46	15, 45	fczfsuppd 9405	. . . . . . . 8 ⊢ (𝜑 → (dom 𝑊 × { 1 }) finSupp 1 )
47		fconstmpt 5717	. . . . . . . . . 10 ⊢ (dom 𝑊 × { 1 }) = (𝑎 ∈ dom 𝑊 ↦ 1 )
48	47	eqcomi 2772	. . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 })
49	48	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 }))
50	21	eqcomd 2769	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝐻) = 1 )
51	46, 49, 50	3brtr4d 5140	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) finSupp (0_g‘𝐻))
52	37, 26, 32, 43, 51	dprdwd 20257	. . . . . 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 20305	. . . . 5 ⊢ (𝜑 → (𝐴 = (𝐻 Σ_g (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ))
54	53	necon3abid 2995	. . . 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 3118	. . 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 2960	. . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑁 ∈ ℕ)
63	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ≠ 1 )
64		dchrpt.m	. . . . . 6 ⊢ · = (._g‘𝐻)
65	34	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ∈ 𝑈)
66	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑊 ∈ Word 𝑈)
67	26	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐻dom DProd 𝑆)
68	35	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (𝐻 DProd 𝑆) = 𝑈)
69		eqid 2763	. . . . . 6 ⊢ (od‘𝐻) = (od‘𝐻)
70		eqid 2763	. . . . . 6 ⊢ ( -1↑_𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) = ( -1↑_𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎))))
71		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑎 ∈ dom 𝑊)
72		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )
73		eqid 2763	. . . . . 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 27622	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)
75	74	expr 462	. . . 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 3167	. 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 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 {crab 3416 Vcvv 3454 {csn 4588 class class class wbr 5106 ↦ cmpt 5189 × cxp 5653 dom cdm 5655 ran crn 5656 ℩cio 6510 ‘cfv 6558 (class class class)co 7443 Xcixp 8953 finSupp cfsupp 9381 1c1 11228 -cneg 11569 / cdiv 12000 ℕcn 12362 2c2 12424 ℕ₀cn0 12633 ℤcz 12720 ↑cexp 14231 Word cword 14685 Basecbs 17416 ↾_s cress 17437 0_gc0g 17639 Σ_g cgsu 17640 Mndcmnd 18952 Grpcgrp 19173 ._gcmg 19306 SubGrpcsubg 19359 odcod 19768 DProd cdprd 20239 dProjcdpj 20240 mulGrpcmgp 20390 1_rcur 20437 Ringcrg 20489 CRingccrg 20490 Unitcui 20615 ℤ/nℤczn 21838 ↑_𝑐ccxp 26913 DChrcdchr 27589

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306 ax-mulf 11307

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-tpos 8271 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8745 df-ec 8747 df-qs 8751 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-fi 9430 df-sup 9461 df-inf 9462 df-oi 9531 df-card 10047 df-acn 10050 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-ioo 13507 df-ioc 13508 df-ico 13509 df-icc 13510 df-fz 13667 df-fzo 13816 df-fl 13959 df-mod 14037 df-seq 14172 df-exp 14232 df-fac 14445 df-bc 14474 df-hash 14502 df-word 14686 df-shft 15249 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-limsup 15667 df-clim 15684 df-rlim 15685 df-sum 15883 df-ef 16262 df-sin 16264 df-cos 16265 df-pi 16267 df-dvds 16452 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-hom 17481 df-cco 17482 df-rest 17622 df-topn 17623 df-0g 17641 df-gsum 17642 df-topgen 17643 df-pt 17644 df-prds 17647 df-xrs 17703 df-qtop 17708 df-imas 17709 df-qus 17710 df-xps 17711 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-mhm 19007 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-mulg 19307 df-subg 19362 df-nsg 19363 df-eqg 19364 df-ghm 19457 df-gim 19502 df-cntz 19561 df-oppg 19590 df-od 19772 df-lsm 19880 df-pj1 19881 df-cmn 20026 df-abl 20027 df-dprd 20241 df-dpj 20242 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-cring 20492 df-oppr 20597 df-dvdsr 20617 df-unit 20618 df-rhm 20732 df-subrng 20828 df-subrg 20852 df-lmod 21167 df-lss 21237 df-lsp 21277 df-sra 21478 df-rgmod 21479 df-lidl 21516 df-rsp 21517 df-2idl 21573 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-fbas 21705 df-fg 21706 df-cnfld 21709 df-zring 21783 df-zrh 21839 df-zn 21842 df-top 23242 df-topon 23259 df-topsp 23281 df-bases 23294 df-cld 23367 df-ntr 23368 df-cls 23369 df-nei 23446 df-lp 23484 df-perf 23485 df-cn 23575 df-cnp 23576 df-haus 23663 df-tx 23911 df-hmeo 24104 df-fil 24195 df-fm 24287 df-flim 24288 df-flf 24289 df-xms 24669 df-ms 24670 df-tms 24671 df-cncf 25229 df-limc 26216 df-dv 26217 df-log 26914 df-cxp 26915 df-dchr 27590

This theorem is used by: dchrpt 27624

W3C validator