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| Description: Lemma for dchrpt 27312. (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 12589 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | 
| 4 | dchrpt.z | . . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
| 5 | 4 | zncrng 21564 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) | 
| 6 | 3, 5 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ CRing) | 
| 7 | crngring 20243 | . . . . . . . . . 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 20384 | . . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp) | 
| 12 | 8, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Grp) | 
| 13 | 12 | grpmndd 18965 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Mnd) | 
| 14 | dchrpt.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈) | |
| 15 | 14 | dmexd 7926 | . . . . . . 7 ⊢ (𝜑 → dom 𝑊 ∈ V) | 
| 16 | eqid 2736 | . . . . . . . 8 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
| 17 | 16 | gsumz 18850 | . . . . . . 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 20386 | . . . . . . . . 9 ⊢ (𝑍 ∈ Ring → 1 = (0g‘𝐻)) | 
| 21 | 8, 20 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 1 = (0g‘𝐻)) | 
| 22 | 21 | mpteq2dv 5243 | . . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻))) | 
| 23 | 22 | oveq2d 7448 | . . . . . 6 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) = (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ (0g‘𝐻)))) | 
| 24 | 18, 23, 21 | 3eqtr4d 2786 | . . . . 5 ⊢ (𝜑 → (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) = 1 ) | 
| 25 | 1, 24 | neeqtrrd 3014 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 ))) | 
| 26 | dchrpt.2 | . . . . . 6 ⊢ (𝜑 → 𝐻dom DProd 𝑆) | |
| 27 | zex 12624 | . . . . . . . . . 10 ⊢ ℤ ∈ V | |
| 28 | 27 | mptex 7244 | . . . . . . . . 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 6711 | . . . . . . 7 ⊢ dom 𝑆 = dom 𝑊 | 
| 32 | 31 | a1i 11 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 = dom 𝑊) | 
| 33 | eqid 2736 | . . . . . 6 ⊢ (𝐻dProj𝑆) = (𝐻dProj𝑆) | |
| 34 | dchrpt.au | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 35 | dchrpt.3 | . . . . . . 7 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈) | |
| 36 | 34, 35 | eleqtrrd 2843 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐻 DProd 𝑆)) | 
| 37 | eqid 2736 | . . . . . 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 20028 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆:dom 𝑊⟶(SubGrp‘𝐻)) | 
| 40 | 39 | ffvelcdmda 7103 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → (𝑆‘𝑎) ∈ (SubGrp‘𝐻)) | 
| 41 | 16 | subg0cl 19153 | . . . . . . . . 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 2840 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ dom 𝑊) → 1 ∈ (𝑆‘𝑎)) | 
| 44 | 19 | fvexi 6919 | . . . . . . . . . 10 ⊢ 1 ∈ V | 
| 45 | 44 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ V) | 
| 46 | 15, 45 | fczfsuppd 9427 | . . . . . . . 8 ⊢ (𝜑 → (dom 𝑊 × { 1 }) finSupp 1 ) | 
| 47 | fconstmpt 5746 | . . . . . . . . . 10 ⊢ (dom 𝑊 × { 1 }) = (𝑎 ∈ dom 𝑊 ↦ 1 ) | |
| 48 | 47 | eqcomi 2745 | . . . . . . . . 9 ⊢ (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 }) | 
| 49 | 48 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) = (dom 𝑊 × { 1 })) | 
| 50 | 21 | eqcomd 2742 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝐻) = 1 ) | 
| 51 | 46, 49, 50 | 3brtr4d 5174 | . . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ dom 𝑊 ↦ 1 ) finSupp (0g‘𝐻)) | 
| 52 | 37, 26, 32, 43, 51 | dprdwd 20032 | . . . . . 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 20080 | . . . . 5 ⊢ (𝜑 → (𝐴 = (𝐻 Σg (𝑎 ∈ dom 𝑊 ↦ 1 )) ↔ ∀𝑎 ∈ dom 𝑊(((𝐻dProj𝑆)‘𝑎)‘𝐴) = 1 )) | 
| 54 | 53 | necon3abid 2976 | . . . 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 3099 | . . 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 2940 | . . . 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 2736 | . . . . . 6 ⊢ (od‘𝐻) = (od‘𝐻) | |
| 70 | eqid 2736 | . . . . . 6 ⊢ (-1↑𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) = (-1↑𝑐(2 / ((od‘𝐻)‘(𝑊‘𝑎)))) | |
| 71 | simprl 770 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → 𝑎 ∈ dom 𝑊) | |
| 72 | simprr 772 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ dom 𝑊 ∧ (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 )) → (((𝐻dProj𝑆)‘𝑎)‘𝐴) ≠ 1 ) | |
| 73 | eqid 2736 | . . . . . 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 27310 | . . . . 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 3154 | . 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 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3435 Vcvv 3479 {csn 4625 class class class wbr 5142 ↦ cmpt 5224 × cxp 5682 dom cdm 5684 ran crn 5685 ℩cio 6511 ‘cfv 6560 (class class class)co 7432 Xcixp 8938 finSupp cfsupp 9402 1c1 11157 -cneg 11494 / cdiv 11921 ℕcn 12267 2c2 12322 ℕ0cn0 12528 ℤcz 12615 ↑cexp 14103 Word cword 14553 Basecbs 17248 ↾s cress 17275 0gc0g 17485 Σg cgsu 17486 Mndcmnd 18748 Grpcgrp 18952 .gcmg 19086 SubGrpcsubg 19139 odcod 19543 DProd cdprd 20014 dProjcdpj 20015 mulGrpcmgp 20138 1rcur 20179 Ringcrg 20231 CRingccrg 20232 Unitcui 20356 ℤ/nℤczn 21514 ↑𝑐ccxp 26598 DChrcdchr 27277 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-word 14554 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-dvds 16292 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 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 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-qus 17555 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-nsg 19143 df-eqg 19144 df-ghm 19232 df-gim 19278 df-cntz 19336 df-oppg 19365 df-od 19547 df-lsm 19655 df-pj1 19656 df-cmn 19801 df-abl 19802 df-dprd 20016 df-dpj 20017 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-lmod 20861 df-lss 20931 df-lsp 20971 df-sra 21173 df-rgmod 21174 df-lidl 21219 df-rsp 21220 df-2idl 21261 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-zring 21459 df-zrh 21515 df-zn 21518 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-log 26599 df-cxp 26600 df-dchr 27278 | 
| This theorem is referenced by: dchrpt 27312 | 
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