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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > erng1r | Structured version Visualization version GIF version |
Description: The division ring unity of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) |
Ref | Expression |
---|---|
erng1r.h | ⊢ 𝐻 = (LHyp‘𝐾) |
erng1r.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
erng1r.d | ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) |
erng1r.r | ⊢ 1 = (1r‘𝐷) |
Ref | Expression |
---|---|
erng1r | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 1 = ( I ↾ 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erng1r.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | erng1r.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | eqid 2726 | . . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | tendoidcl 40478 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) |
5 | erng1r.d | . . . . 5 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) | |
6 | eqid 2726 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
7 | 1, 2, 3, 5, 6 | erngbase 40510 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = ((TEndo‘𝐾)‘𝑊)) |
8 | 4, 7 | eleqtrrd 2829 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ (Base‘𝐷)) |
9 | eqid 2726 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | eqid 2726 | . . . . 5 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) | |
11 | 9, 1, 2, 3, 10 | tendo1ne0 40537 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ (𝑓 ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))) |
12 | eqid 2726 | . . . . 5 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
13 | 9, 1, 2, 5, 10, 12 | erng0g 40703 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0g‘𝐷) = (𝑓 ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))) |
14 | 11, 13 | neeqtrrd 3005 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ (0g‘𝐷)) |
15 | id 22 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
16 | eqid 2726 | . . . . . 6 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
17 | 1, 2, 3, 5, 16 | erngmul 40515 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊) ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → (( I ↾ 𝑇)(.r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ ( I ↾ 𝑇))) |
18 | 15, 4, 4, 17 | syl12anc 835 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝑇)(.r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ ( I ↾ 𝑇))) |
19 | f1oi 6870 | . . . . 5 ⊢ ( I ↾ 𝑇):𝑇–1-1-onto→𝑇 | |
20 | f1of 6832 | . . . . 5 ⊢ (( I ↾ 𝑇):𝑇–1-1-onto→𝑇 → ( I ↾ 𝑇):𝑇⟶𝑇) | |
21 | fcoi2 6766 | . . . . 5 ⊢ (( I ↾ 𝑇):𝑇⟶𝑇 → (( I ↾ 𝑇) ∘ ( I ↾ 𝑇)) = ( I ↾ 𝑇)) | |
22 | 19, 20, 21 | mp2b 10 | . . . 4 ⊢ (( I ↾ 𝑇) ∘ ( I ↾ 𝑇)) = ( I ↾ 𝑇) |
23 | 18, 22 | eqtrdi 2782 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝑇)(.r‘𝐷)( I ↾ 𝑇)) = ( I ↾ 𝑇)) |
24 | 8, 14, 23 | 3jca 1125 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝑇) ∈ (Base‘𝐷) ∧ ( I ↾ 𝑇) ≠ (0g‘𝐷) ∧ (( I ↾ 𝑇)(.r‘𝐷)( I ↾ 𝑇)) = ( I ↾ 𝑇))) |
25 | 1, 5 | erngdv 40702 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing) |
26 | erng1r.r | . . . 4 ⊢ 1 = (1r‘𝐷) | |
27 | 6, 16, 12, 26 | drngid2 20723 | . . 3 ⊢ (𝐷 ∈ DivRing → ((( I ↾ 𝑇) ∈ (Base‘𝐷) ∧ ( I ↾ 𝑇) ≠ (0g‘𝐷) ∧ (( I ↾ 𝑇)(.r‘𝐷)( I ↾ 𝑇)) = ( I ↾ 𝑇)) ↔ 1 = ( I ↾ 𝑇))) |
28 | 25, 27 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((( I ↾ 𝑇) ∈ (Base‘𝐷) ∧ ( I ↾ 𝑇) ≠ (0g‘𝐷) ∧ (( I ↾ 𝑇)(.r‘𝐷)( I ↾ 𝑇)) = ( I ↾ 𝑇)) ↔ 1 = ( I ↾ 𝑇))) |
29 | 24, 28 | mpbid 231 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 1 = ( I ↾ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ↦ cmpt 5226 I cid 5569 ↾ cres 5674 ∘ ccom 5676 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7413 Basecbs 17205 .rcmulr 17259 0gc0g 17446 1rcur 20157 DivRingcdr 20700 HLchlt 39058 LHypclh 39693 LTrncltrn 39810 TEndoctendo 40461 EDRingcedring 40462 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 38661 |
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 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-iin 4996 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-tpos 8230 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-3 12319 df-n0 12516 df-z 12602 df-uz 12866 df-fz 13530 df-struct 17141 df-sets 17158 df-slot 17176 df-ndx 17188 df-base 17206 df-ress 17235 df-plusg 17271 df-mulr 17272 df-0g 17448 df-proset 18312 df-poset 18330 df-plt 18347 df-lub 18363 df-glb 18364 df-join 18365 df-meet 18366 df-p0 18442 df-p1 18443 df-lat 18449 df-clat 18516 df-mgm 18625 df-sgrp 18704 df-mnd 18720 df-grp 18923 df-minusg 18924 df-cmn 19773 df-abl 19774 df-mgp 20111 df-rng 20129 df-ur 20158 df-ring 20211 df-oppr 20309 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-drng 20702 df-oposet 38884 df-ol 38886 df-oml 38887 df-covers 38974 df-ats 38975 df-atl 39006 df-cvlat 39030 df-hlat 39059 df-llines 39207 df-lplanes 39208 df-lvols 39209 df-lines 39210 df-psubsp 39212 df-pmap 39213 df-padd 39505 df-lhyp 39697 df-laut 39698 df-ldil 39813 df-ltrn 39814 df-trl 39868 df-tendo 40464 df-edring 40466 |
This theorem is referenced by: tendolinv 40814 tendorinv 40815 dvhlveclem 40817 |
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