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Mirrors > Home > MPE Home > Th. List > dchrptlem3 | Structured version Visualization version GIF version |
Description: Lemma for dchrpt 25845. (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 11958 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
4 | dchrpt.z | . . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
5 | 4 | zncrng 20693 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
6 | 3, 5 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ CRing) |
7 | crngring 19310 | . . . . . . . . . 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 19419 | . . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp) |
12 | 8, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp) |
13 | grpmnd 18112 | . . . . . . . 8 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
15 | dchrpt.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈) | |
16 | 15 | dmexd 7617 | . . . . . . 7 ⊢ (𝜑 → dom 𝑊 ∈ V) |
17 | eqid 2823 | . . . . . . . 8 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
18 | 17 | gsumz 18002 | . . . . . . 7 ⊢ ((𝐻 ∈ Mnd ∧ dom 𝑊 ∈ V) → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
19 | 14, 16, 18 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
20 | dchrpt.1 | . . . . . . . . . 10 ⊢ 1 = (1r‘𝑍) | |
21 | 9, 10, 20 | unitgrpid 19421 | . . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 1 = (0g‘𝐻)) |
22 | 8, 21 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 = (0g‘𝐻)) |
23 | 22 | mpteq2dv 5164 | . . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻))) |
24 | 23 | oveq2d 7174 | . . . . . 6 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) = (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻)))) |
25 | 19, 24, 22 | 3eqtr4d 2868 | . . . . 5 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) = 1 ) |
26 | 1, 25 | neeqtrrd 3092 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 ))) |
27 | dchrpt.2 | . . . . . 6 ⊢ (𝜑 → 𝐻dom DProd 𝑆) | |
28 | zex 11993 | . . . . . . . . . 10 ⊢ ℤ ∈ V | |
29 | 28 | mptex 6988 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V |
30 | 29 | rnex 7619 | . . . . . . . 8 ⊢ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V |
31 | dchrpt.s | . . . . . . . 8 ⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘)))) | |
32 | 30, 31 | dmmpti 6494 | . . . . . . 7 ⊢ dom 𝑆 = dom 𝑊 |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom 𝑊) |
34 | eqid 2823 | . . . . . 6 ⊢ (𝐻dProj𝑆) = (𝐻dProj𝑆) | |
35 | dchrpt.au | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
36 | dchrpt.3 | . . . . . . 7 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈) | |
37 | 35, 36 | eleqtrrd 2918 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐻 DProd 𝑆)) |
38 | eqid 2823 | . . . . . 6 ⊢ {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐻)} = {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐻)} | |
39 | 22 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 = (0g‘𝐻)) |
40 | 27, 33 | dprdf2 19131 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆:dom 𝑊⟶(SubGrp‘𝐻)) |
41 | 40 | ffvelrnda 6853 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (𝑆‘𝑎) ∈ (SubGrp‘𝐻)) |
42 | 17 | subg0cl 18289 | . . . . . . . . 9 ⊢ ((𝑆‘𝑎) ∈ (SubGrp‘𝐻) → (0g‘𝐻) ∈ (𝑆‘𝑎)) |
43 | 41, 42 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (0g‘𝐻) ∈ (𝑆‘𝑎)) |
44 | 39, 43 | eqeltrd 2915 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 ∈ (𝑆‘𝑎)) |
45 | 20 | fvexi 6686 | . . . . . . . . . 10 ⊢ 1 ∈ V |
46 | 45 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ V) |
47 | 16, 46 | fczfsuppd 8853 | . . . . . . . 8 ⊢ (𝜑 → (dom 𝑊 × { 1 }) finSupp 1 ) |
48 | fconstmpt 5616 | . . . . . . . . . 10 ⊢ (dom 𝑊 × { 1 }) = (𝑎 ∈ dom 𝑊 ↦ 1 ) | |
49 | 48 | eqcomi 2832 | . . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 }) |
50 | 49 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 })) |
51 | 22 | eqcomd 2829 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐻) = 1 ) |
52 | 47, 50, 51 | 3brtr4d 5100 | . . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) finSupp (0g‘𝐻)) |
53 | 38, 27, 33, 44, 52 | dprdwd 19135 | . . . . . 6 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) ∈ {ℎ ∈ X𝑖 ∈ dom 𝑊(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐻)}) |
54 | 27, 33, 34, 37, 17, 38, 53 | dpjeq 19183 | . . . . 5 ⊢ (𝜑 → (𝐴 = (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )) |
55 | 54 | necon3abid 3054 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )) |
56 | 26, 55 | mpbid 234 | . . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ) |
57 | rexnal 3240 | . . 3 ⊢ (∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ↔ ¬ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ) | |
58 | 56, 57 | sylibr 236 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ) |
59 | df-ne 3019 | . . . 4 ⊢ ((((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 ↔ ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 ) | |
60 | dchrpt.g | . . . . . 6 ⊢ 𝐺 = (DChr‘𝑁) | |
61 | dchrpt.d | . . . . . 6 ⊢ 𝐷 = (Base‘𝐺) | |
62 | dchrpt.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑍) | |
63 | 2 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑁 ∈ ℕ) |
64 | 1 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ≠ 1 ) |
65 | dchrpt.m | . . . . . 6 ⊢ · = (.g‘𝐻) | |
66 | 35 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐴 ∈ 𝑈) |
67 | 15 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑊 ∈ Word 𝑈) |
68 | 27 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝐻dom DProd 𝑆) |
69 | 36 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (𝐻 DProd 𝑆) = 𝑈) |
70 | eqid 2823 | . . . . . 6 ⊢ (od‘𝐻) = (od‘𝐻) | |
71 | eqid 2823 | . . . . . 6 ⊢ (-1↑𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) = (-1↑𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) | |
72 | simprl 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑎 ∈ dom 𝑊) | |
73 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 ) | |
74 | eqid 2823 | . . . . . 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)))) | |
75 | 60, 4, 61, 62, 20, 63, 64, 9, 10, 65, 31, 66, 67, 68, 69, 34, 70, 71, 72, 73, 74 | dchrptlem2 25843 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1) |
76 | 75 | expr 459 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → ((((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)) |
77 | 59, 76 | syl5bir 245 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)) |
78 | 77 | rexlimdva 3286 | . 2 ⊢ (𝜑 → (∃𝑎 ∈ dom 𝑊 ¬ (((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)) |
79 | 58, 78 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 Vcvv 3496 {csn 4569 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 dom cdm 5557 ran crn 5558 ℩cio 6314 ‘cfv 6357 (class class class)co 7158 Xcixp 8463 finSupp cfsupp 8835 1c1 10540 -cneg 10873 / cdiv 11299 ℕcn 11640 2c2 11695 ℕ0cn0 11900 ℤcz 11984 ↑cexp 13432 Word cword 13864 Basecbs 16485 ↾s cress 16486 0gc0g 16715 Σg cgsu 16716 Mndcmnd 17913 Grpcgrp 18105 .gcmg 18226 SubGrpcsubg 18275 odcod 18654 DProd cdprd 19117 dProjcdpj 19118 mulGrpcmgp 19241 1rcur 19253 Ringcrg 19299 CRingccrg 19300 Unitcui 19391 ℤ/nℤczn 20652 ↑𝑐ccxp 25141 DChrcdchr 25810 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-word 13865 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-dvds 15610 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-qus 16784 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-nsg 18279 df-eqg 18280 df-ghm 18358 df-gim 18401 df-cntz 18449 df-oppg 18476 df-od 18658 df-lsm 18763 df-pj1 18764 df-cmn 18910 df-abl 18911 df-dprd 19119 df-dpj 19120 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-rsp 19949 df-2idl 20007 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-zn 20656 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-cxp 25143 df-dchr 25811 |
This theorem is referenced by: dchrpt 25845 |
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