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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngdv | Structured version Visualization version GIF version |
Description: An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013.) |
Ref | Expression |
---|---|
ernggrp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ernggrp.d | ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
erngdv | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | ernggrp.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2821 | . . 3 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | cdlemftr0 37719 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑓 ∈ ((LTrn‘𝐾)‘𝑊)𝑓 ≠ ( I ↾ (Base‘𝐾))) |
5 | ernggrp.d | . . 3 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) | |
6 | eqid 2821 | . . 3 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
7 | eqid 2821 | . . 3 ⊢ (𝑎 ∈ ((TEndo‘𝐾)‘𝑊), 𝑏 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))) = (𝑎 ∈ ((TEndo‘𝐾)‘𝑊), 𝑏 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))) | |
8 | eqid 2821 | . . 3 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
9 | eqid 2821 | . . 3 ⊢ (𝑎 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ◡(𝑎‘𝑓))) = (𝑎 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ◡(𝑎‘𝑓))) | |
10 | eqid 2821 | . . 3 ⊢ (𝑎 ∈ ((TEndo‘𝐾)‘𝑊), 𝑏 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑎 ∘ 𝑏)) = (𝑎 ∈ ((TEndo‘𝐾)‘𝑊), 𝑏 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑎 ∘ 𝑏)) | |
11 | eqid 2821 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
12 | eqid 2821 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
13 | eqid 2821 | . . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
14 | eqid 2821 | . . 3 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊) | |
15 | eqid 2821 | . . 3 ⊢ ((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑏))(meet‘𝐾)((𝑓‘((oc‘𝐾)‘𝑊))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑏 ∘ ◡(𝑠‘𝑓))))) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑏))(meet‘𝐾)((𝑓‘((oc‘𝐾)‘𝑊))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑏 ∘ ◡(𝑠‘𝑓))))) | |
16 | eqid 2821 | . . 3 ⊢ ((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑔))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑏))(meet‘𝐾)((𝑓‘((oc‘𝐾)‘𝑊))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑏 ∘ ◡(𝑠‘𝑓)))))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑔 ∘ ◡𝑏)))) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑔))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑏))(meet‘𝐾)((𝑓‘((oc‘𝐾)‘𝑊))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑏 ∘ ◡(𝑠‘𝑓)))))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑔 ∘ ◡𝑏)))) | |
17 | eqid 2821 | . . 3 ⊢ (℩𝑧 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑏 ∈ ((LTrn‘𝐾)‘𝑊)((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (((trL‘𝐾)‘𝑊)‘𝑏) ≠ (((trL‘𝐾)‘𝑊)‘(𝑠‘𝑓)) ∧ (((trL‘𝐾)‘𝑊)‘𝑏) ≠ (((trL‘𝐾)‘𝑊)‘𝑔)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑔))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑏))(meet‘𝐾)((𝑓‘((oc‘𝐾)‘𝑊))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑏 ∘ ◡(𝑠‘𝑓)))))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑔 ∘ ◡𝑏)))))) = (℩𝑧 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑏 ∈ ((LTrn‘𝐾)‘𝑊)((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (((trL‘𝐾)‘𝑊)‘𝑏) ≠ (((trL‘𝐾)‘𝑊)‘(𝑠‘𝑓)) ∧ (((trL‘𝐾)‘𝑊)‘𝑏) ≠ (((trL‘𝐾)‘𝑊)‘𝑔)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑔))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑏))(meet‘𝐾)((𝑓‘((oc‘𝐾)‘𝑊))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑏 ∘ ◡(𝑠‘𝑓)))))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑔 ∘ ◡𝑏)))))) | |
18 | eqid 2821 | . . 3 ⊢ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ if((𝑠‘𝑓) = 𝑓, 𝑔, (℩𝑧 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑏 ∈ ((LTrn‘𝐾)‘𝑊)((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (((trL‘𝐾)‘𝑊)‘𝑏) ≠ (((trL‘𝐾)‘𝑊)‘(𝑠‘𝑓)) ∧ (((trL‘𝐾)‘𝑊)‘𝑏) ≠ (((trL‘𝐾)‘𝑊)‘𝑔)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑔))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑏))(meet‘𝐾)((𝑓‘((oc‘𝐾)‘𝑊))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑏 ∘ ◡(𝑠‘𝑓)))))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑔 ∘ ◡𝑏)))))))) = (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ if((𝑠‘𝑓) = 𝑓, 𝑔, (℩𝑧 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑏 ∈ ((LTrn‘𝐾)‘𝑊)((𝑏 ≠ ( I ↾ (Base‘𝐾)) ∧ (((trL‘𝐾)‘𝑊)‘𝑏) ≠ (((trL‘𝐾)‘𝑊)‘(𝑠‘𝑓)) ∧ (((trL‘𝐾)‘𝑊)‘𝑏) ≠ (((trL‘𝐾)‘𝑊)‘𝑔)) → (𝑧‘((oc‘𝐾)‘𝑊)) = ((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑔))(meet‘𝐾)(((((oc‘𝐾)‘𝑊)(join‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑏))(meet‘𝐾)((𝑓‘((oc‘𝐾)‘𝑊))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑏 ∘ ◡(𝑠‘𝑓)))))(join‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑔 ∘ ◡𝑏)))))))) | |
19 | 2, 5, 1, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | erngdvlem4 38142 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑓 ≠ ( I ↾ (Base‘𝐾)))) → 𝐷 ∈ DivRing) |
20 | 4, 19 | rexlimddv 3291 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ifcif 4467 ↦ cmpt 5146 I cid 5459 ◡ccnv 5554 ↾ cres 5557 ∘ ccom 5559 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 ∈ cmpo 7158 Basecbs 16483 occoc 16573 joincjn 17554 meetcmee 17555 DivRingcdr 19502 HLchlt 36501 LHypclh 37135 LTrncltrn 37252 trLctrl 37309 TEndoctendo 37903 EDRingcedring 37904 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tendo 37906 df-edring 37908 |
This theorem is referenced by: erng1r 38146 dvalveclem 38176 dvhvaddass 38248 tendoinvcl 38255 tendolinv 38256 tendorinv 38257 dvhgrp 38258 dvhlveclem 38259 cdlemn4 38349 hlhildrng 39103 |
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