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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 unit 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 2771 | . . . . 5 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | tendoidcl 37387 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) |
5 | erng1r.d | . . . . 5 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) | |
6 | eqid 2771 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
7 | 1, 2, 3, 5, 6 | erngbase 37419 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = ((TEndo‘𝐾)‘𝑊)) |
8 | 4, 7 | eleqtrrd 2862 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ (Base‘𝐷)) |
9 | eqid 2771 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | eqid 2771 | . . . . 5 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾))) | |
11 | 9, 1, 2, 3, 10 | tendo1ne0 37446 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ (𝑓 ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))) |
12 | eqid 2771 | . . . . 5 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
13 | 9, 1, 2, 5, 10, 12 | erng0g 37612 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0g‘𝐷) = (𝑓 ∈ 𝑇 ↦ ( I ↾ (Base‘𝐾)))) |
14 | 11, 13 | neeqtrrd 3034 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ (0g‘𝐷)) |
15 | id 22 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
16 | eqid 2771 | . . . . . 6 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
17 | 1, 2, 3, 5, 16 | erngmul 37424 | . . . . 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 825 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝑇)(.r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ ( I ↾ 𝑇))) |
19 | f1oi 6478 | . . . . 5 ⊢ ( I ↾ 𝑇):𝑇–1-1-onto→𝑇 | |
20 | f1of 6441 | . . . . 5 ⊢ (( I ↾ 𝑇):𝑇–1-1-onto→𝑇 → ( I ↾ 𝑇):𝑇⟶𝑇) | |
21 | fcoi2 6379 | . . . . 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 | syl6eq 2823 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝑇)(.r‘𝐷)( I ↾ 𝑇)) = ( I ↾ 𝑇)) |
24 | 8, 14, 23 | 3jca 1109 | . 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 37611 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing) |
26 | erng1r.r | . . . 4 ⊢ 1 = (1r‘𝐷) | |
27 | 6, 16, 12, 26 | drngid2 19253 | . . 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 224 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 1 = ( I ↾ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 ↦ cmpt 5004 I cid 5307 ↾ cres 5405 ∘ ccom 5407 ⟶wf 6181 –1-1-onto→wf1o 6184 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 .rcmulr 16420 0gc0g 16567 1rcur 18986 DivRingcdr 19237 HLchlt 35968 LHypclh 36602 LTrncltrn 36719 TEndoctendo 37370 EDRingcedring 37371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-riotaBAD 35571 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-tpos 7693 df-undef 7740 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-0g 16569 df-proset 17408 df-poset 17426 df-plt 17438 df-lub 17454 df-glb 17455 df-join 17456 df-meet 17457 df-p0 17519 df-p1 17520 df-lat 17526 df-clat 17588 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-grp 17906 df-minusg 17907 df-mgp 18975 df-ur 18987 df-ring 19034 df-oppr 19108 df-dvdsr 19126 df-unit 19127 df-invr 19157 df-dvr 19168 df-drng 19239 df-oposet 35794 df-ol 35796 df-oml 35797 df-covers 35884 df-ats 35885 df-atl 35916 df-cvlat 35940 df-hlat 35969 df-llines 36116 df-lplanes 36117 df-lvols 36118 df-lines 36119 df-psubsp 36121 df-pmap 36122 df-padd 36414 df-lhyp 36606 df-laut 36607 df-ldil 36722 df-ltrn 36723 df-trl 36777 df-tendo 37373 df-edring 37375 |
This theorem is referenced by: tendolinv 37723 tendorinv 37724 dvhlveclem 37726 |
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