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Mirrors > Home > MPE Home > Th. List > dchrptlem3 | Structured version Visualization version GIF version |
Description: Lemma for dchrpt 27291. (Contributed by Mario Carneiro, 28-Apr-2016.) |
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 = (1r‘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) |
Ref | Expression |
---|---|
dchrptlem3 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrpt.n1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 1 ) | |
2 | dchrpt.n | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
3 | 2 | nnnn0d 12576 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
4 | dchrpt.z | . . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
5 | 4 | zncrng 21536 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
6 | 3, 5 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ CRing) |
7 | crngring 20222 | . . . . . . . . . 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 20359 | . . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp) |
12 | 8, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp) |
13 | 12 | grpmndd 18934 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
14 | dchrpt.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈) | |
15 | 14 | dmexd 7906 | . . . . . . 7 ⊢ (𝜑 → dom 𝑊 ∈ V) |
16 | eqid 2726 | . . . . . . . 8 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
17 | 16 | gsumz 18819 | . . . . . . 7 ⊢ ((𝐻 ∈ Mnd ∧ dom 𝑊 ∈ V) → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
18 | 13, 15, 17 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
19 | dchrpt.1 | . . . . . . . . . 10 ⊢ 1 = (1r‘𝑍) | |
20 | 9, 10, 19 | unitgrpid 20361 | . . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 1 = (0g‘𝐻)) |
21 | 8, 20 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 = (0g‘𝐻)) |
22 | 21 | mpteq2dv 5246 | . . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻))) |
23 | 22 | oveq2d 7430 | . . . . . 6 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) = (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻)))) |
24 | 18, 23, 21 | 3eqtr4d 2776 | . . . . 5 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) = 1 ) |
25 | 1, 24 | neeqtrrd 3005 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 ))) |
26 | dchrpt.2 | . . . . . 6 ⊢ (𝜑 → 𝐻dom DProd 𝑆) | |
27 | zex 12611 | . . . . . . . . . 10 ⊢ ℤ ∈ V | |
28 | 27 | mptex 7230 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V |
29 | 28 | rnex 7913 | . . . . . . . 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 6695 | . . . . . . 7 ⊢ dom 𝑆 = dom 𝑊 |
32 | 31 | a1i 11 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom 𝑊) |
33 | eqid 2726 | . . . . . 6 ⊢ (𝐻dProj𝑆) = (𝐻dProj𝑆) | |
34 | dchrpt.au | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
35 | dchrpt.3 | . . . . . . 7 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈) | |
36 | 34, 35 | eleqtrrd 2829 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐻 DProd 𝑆)) |
37 | eqid 2726 | . . . . . 6 ⊢ {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐻)} = {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐻)} | |
38 | 21 | adantr 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 = (0g‘𝐻)) |
39 | 26, 32 | dprdf2 20001 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆:dom 𝑊⟶(SubGrp‘𝐻)) |
40 | 39 | ffvelcdmda 7088 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (𝑆‘𝑎) ∈ (SubGrp‘𝐻)) |
41 | 16 | subg0cl 19122 | . . . . . . . . 9 ⊢ ((𝑆‘𝑎) ∈ (SubGrp‘𝐻) → (0g‘𝐻) ∈ (𝑆‘𝑎)) |
42 | 40, 41 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (0g‘𝐻) ∈ (𝑆‘𝑎)) |
43 | 38, 42 | eqeltrd 2826 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 ∈ (𝑆‘𝑎)) |
44 | 19 | fvexi 6905 | . . . . . . . . . 10 ⊢ 1 ∈ V |
45 | 44 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ V) |
46 | 15, 45 | fczfsuppd 9420 | . . . . . . . 8 ⊢ (𝜑 → (dom 𝑊 × { 1 }) finSupp 1 ) |
47 | fconstmpt 5735 | . . . . . . . . . 10 ⊢ (dom 𝑊 × { 1 }) = (𝑎 ∈ dom 𝑊 ↦ 1 ) | |
48 | 47 | eqcomi 2735 | . . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 }) |
49 | 48 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 })) |
50 | 21 | eqcomd 2732 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐻) = 1 ) |
51 | 46, 49, 50 | 3brtr4d 5176 | . . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) finSupp (0g‘𝐻)) |
52 | 37, 26, 32, 43, 51 | dprdwd 20005 | . . . . . 6 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) ∈ {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐻)}) |
53 | 26, 32, 33, 36, 16, 37, 52 | dpjeq 20053 | . . . . 5 ⊢ (𝜑 → (𝐴 = (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )) |
54 | 53 | necon3abid 2967 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )) |
55 | 25, 54 | mpbid 231 | . . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ) |
56 | rexnal 3090 | . . 3 ⊢ (∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ↔ ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ) | |
57 | 55, 56 | sylibr 233 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ) |
58 | df-ne 2931 | . . . 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 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑁 ∈ ℕ) |
63 | 1 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ≠ 1 ) |
64 | dchrpt.m | . . . . . 6 ⊢ · = (.g‘𝐻) | |
65 | 34 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ∈ 𝑈) |
66 | 14 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑊 ∈ Word 𝑈) |
67 | 26 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐻dom DProd 𝑆) |
68 | 35 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (𝐻 DProd 𝑆) = 𝑈) |
69 | eqid 2726 | . . . . . 6 ⊢ (od‘𝐻) = (od‘𝐻) | |
70 | eqid 2726 | . . . . . 6 ⊢ (-1↑𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) = (-1↑𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) | |
71 | simprl 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑎 ∈ dom 𝑊) | |
72 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 ) | |
73 | eqid 2726 | . . . . . 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 27289 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1) |
75 | 74 | expr 455 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → ((((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)) |
76 | 58, 75 | biimtrrid 242 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)) |
77 | 76 | rexlimdva 3145 | . 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 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 {crab 3420 Vcvv 3463 {csn 4624 class class class wbr 5144 ↦ cmpt 5227 × cxp 5671 dom cdm 5673 ran crn 5674 ℩cio 6494 ‘cfv 6544 (class class class)co 7414 Xcixp 8916 finSupp cfsupp 9396 1c1 11148 -cneg 11484 / cdiv 11910 ℕcn 12256 2c2 12311 ℕ0cn0 12516 ℤcz 12602 ↑cexp 14073 Word cword 14515 Basecbs 17206 ↾s cress 17235 0gc0g 17447 Σg cgsu 17448 Mndcmnd 18720 Grpcgrp 18921 .gcmg 19055 SubGrpcsubg 19108 odcod 19516 DProd cdprd 19987 dProjcdpj 19988 mulGrpcmgp 20111 1rcur 20158 Ringcrg 20210 CRingccrg 20211 Unitcui 20331 ℤ/nℤczn 21486 ↑𝑐ccxp 26577 DChrcdchr 27256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-inf2 9675 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 ax-addf 11226 ax-mulf 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-oadd 8490 df-omul 8491 df-er 8724 df-ec 8726 df-qs 8730 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9397 df-fi 9445 df-sup 9476 df-inf 9477 df-oi 9544 df-card 9973 df-acn 9976 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-z 12603 df-dec 12722 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13374 df-ioc 13375 df-ico 13376 df-icc 13377 df-fz 13531 df-fzo 13674 df-fl 13804 df-mod 13882 df-seq 14014 df-exp 14074 df-fac 14284 df-bc 14313 df-hash 14341 df-word 14516 df-shft 15065 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-limsup 15466 df-clim 15483 df-rlim 15484 df-sum 15684 df-ef 16062 df-sin 16064 df-cos 16065 df-pi 16067 df-dvds 16250 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-mulr 17273 df-starv 17274 df-sca 17275 df-vsca 17276 df-ip 17277 df-tset 17278 df-ple 17279 df-ds 17281 df-unif 17282 df-hom 17283 df-cco 17284 df-rest 17430 df-topn 17431 df-0g 17449 df-gsum 17450 df-topgen 17451 df-pt 17452 df-prds 17455 df-xrs 17510 df-qtop 17515 df-imas 17516 df-qus 17517 df-xps 17518 df-mre 17592 df-mrc 17593 df-acs 17595 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-nsg 19112 df-eqg 19113 df-ghm 19201 df-gim 19247 df-cntz 19305 df-oppg 19334 df-od 19520 df-lsm 19628 df-pj1 19629 df-cmn 19774 df-abl 19775 df-dprd 19989 df-dpj 19990 df-mgp 20112 df-rng 20130 df-ur 20159 df-ring 20212 df-cring 20213 df-oppr 20310 df-dvdsr 20333 df-unit 20334 df-rhm 20448 df-subrng 20522 df-subrg 20547 df-lmod 20832 df-lss 20903 df-lsp 20943 df-sra 21145 df-rgmod 21146 df-lidl 21191 df-rsp 21192 df-2idl 21233 df-psmet 21329 df-xmet 21330 df-met 21331 df-bl 21332 df-mopn 21333 df-fbas 21334 df-fg 21335 df-cnfld 21338 df-zring 21431 df-zrh 21487 df-zn 21490 df-top 22882 df-topon 22899 df-topsp 22921 df-bases 22935 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24312 df-ms 24313 df-tms 24314 df-cncf 24884 df-limc 25881 df-dv 25882 df-log 26578 df-cxp 26579 df-dchr 27257 |
This theorem is referenced by: dchrpt 27291 |
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