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Mirrors > Home > MPE Home > Th. List > dchrptlem3 | Structured version Visualization version GIF version |
Description: Lemma for dchrpt 27326. (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 12585 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
4 | dchrpt.z | . . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
5 | 4 | zncrng 21581 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
6 | 3, 5 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ CRing) |
7 | crngring 20263 | . . . . . . . . . 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 20400 | . . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp) |
12 | 8, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp) |
13 | 12 | grpmndd 18977 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
14 | dchrpt.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈) | |
15 | 14 | dmexd 7926 | . . . . . . 7 ⊢ (𝜑 → dom 𝑊 ∈ V) |
16 | eqid 2735 | . . . . . . . 8 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
17 | 16 | gsumz 18862 | . . . . . . 7 ⊢ ((𝐻 ∈ Mnd ∧ dom 𝑊 ∈ V) → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
18 | 13, 15, 17 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
19 | dchrpt.1 | . . . . . . . . . 10 ⊢ 1 = (1r‘𝑍) | |
20 | 9, 10, 19 | unitgrpid 20402 | . . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 1 = (0g‘𝐻)) |
21 | 8, 20 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 = (0g‘𝐻)) |
22 | 21 | mpteq2dv 5250 | . . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻))) |
23 | 22 | oveq2d 7447 | . . . . . 6 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) = (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻)))) |
24 | 18, 23, 21 | 3eqtr4d 2785 | . . . . 5 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) = 1 ) |
25 | 1, 24 | neeqtrrd 3013 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 ))) |
26 | dchrpt.2 | . . . . . 6 ⊢ (𝜑 → 𝐻dom DProd 𝑆) | |
27 | zex 12620 | . . . . . . . . . 10 ⊢ ℤ ∈ V | |
28 | 27 | mptex 7243 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V |
29 | 28 | rnex 7933 | . . . . . . . 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 6713 | . . . . . . 7 ⊢ dom 𝑆 = dom 𝑊 |
32 | 31 | a1i 11 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom 𝑊) |
33 | eqid 2735 | . . . . . 6 ⊢ (𝐻dProj𝑆) = (𝐻dProj𝑆) | |
34 | dchrpt.au | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
35 | dchrpt.3 | . . . . . . 7 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈) | |
36 | 34, 35 | eleqtrrd 2842 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐻 DProd 𝑆)) |
37 | eqid 2735 | . . . . . 6 ⊢ {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐻)} = {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐻)} | |
38 | 21 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 = (0g‘𝐻)) |
39 | 26, 32 | dprdf2 20042 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆:dom 𝑊⟶(SubGrp‘𝐻)) |
40 | 39 | ffvelcdmda 7104 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (𝑆‘𝑎) ∈ (SubGrp‘𝐻)) |
41 | 16 | subg0cl 19165 | . . . . . . . . 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 2839 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 ∈ (𝑆‘𝑎)) |
44 | 19 | fvexi 6921 | . . . . . . . . . 10 ⊢ 1 ∈ V |
45 | 44 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ V) |
46 | 15, 45 | fczfsuppd 9424 | . . . . . . . 8 ⊢ (𝜑 → (dom 𝑊 × { 1 }) finSupp 1 ) |
47 | fconstmpt 5751 | . . . . . . . . . 10 ⊢ (dom 𝑊 × { 1 }) = (𝑎 ∈ dom 𝑊 ↦ 1 ) | |
48 | 47 | eqcomi 2744 | . . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 }) |
49 | 48 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 })) |
50 | 21 | eqcomd 2741 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐻) = 1 ) |
51 | 46, 49, 50 | 3brtr4d 5180 | . . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) finSupp (0g‘𝐻)) |
52 | 37, 26, 32, 43, 51 | dprdwd 20046 | . . . . . 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 20094 | . . . . 5 ⊢ (𝜑 → (𝐴 = (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )) |
54 | 53 | necon3abid 2975 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ (𝐻 Σg (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 3098 | . . 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 2939 | . . . 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 2735 | . . . . . 6 ⊢ (od‘𝐻) = (od‘𝐻) | |
70 | eqid 2735 | . . . . . 6 ⊢ (-1↑𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) = (-1↑𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) | |
71 | simprl 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑎 ∈ dom 𝑊) | |
72 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 ) | |
73 | eqid 2735 | . . . . . 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 27324 | . . . . 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 3153 | . 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 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 {crab 3433 Vcvv 3478 {csn 4631 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 dom cdm 5689 ran crn 5690 ℩cio 6514 ‘cfv 6563 (class class class)co 7431 Xcixp 8936 finSupp cfsupp 9399 1c1 11154 -cneg 11491 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ℤcz 12611 ↑cexp 14099 Word cword 14549 Basecbs 17245 ↾s cress 17274 0gc0g 17486 Σg cgsu 17487 Mndcmnd 18760 Grpcgrp 18964 .gcmg 19098 SubGrpcsubg 19151 odcod 19557 DProd cdprd 20028 dProjcdpj 20029 mulGrpcmgp 20152 1rcur 20199 Ringcrg 20251 CRingccrg 20252 Unitcui 20372 ℤ/nℤczn 21531 ↑𝑐ccxp 26612 DChrcdchr 27291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-word 14550 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-dvds 16288 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-qus 17556 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-gim 19290 df-cntz 19348 df-oppg 19377 df-od 19561 df-lsm 19669 df-pj1 19670 df-cmn 19815 df-abl 19816 df-dprd 20030 df-dpj 20031 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-zn 21535 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614 df-dchr 27292 |
This theorem is referenced by: dchrpt 27326 |
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