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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > erng1lem | Structured version Visualization version GIF version |
Description: Value of the endomorphism division ring unit. (Contributed by NM, 12-Oct-2013.) |
Ref | Expression |
---|---|
erng1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
erng1.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
erng1.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
erng1.d | ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) |
erng1.r | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Ring) |
Ref | Expression |
---|---|
erng1lem | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1r‘𝐷) = ( I ↾ 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erng1.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | erng1.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | erng1.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | tendoidcl 37457 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸) |
5 | erng1.d | . . . 4 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) | |
6 | eqid 2797 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
7 | 1, 2, 3, 5, 6 | erngbase 37489 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸) |
8 | 4, 7 | eleqtrrd 2888 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ (Base‘𝐷)) |
9 | 7 | eleq2d 2870 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑢 ∈ (Base‘𝐷) ↔ 𝑢 ∈ 𝐸)) |
10 | simpl 483 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | 4 | adantr 481 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → ( I ↾ 𝑇) ∈ 𝐸) |
12 | simpr 485 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → 𝑢 ∈ 𝐸) | |
13 | eqid 2797 | . . . . . . . . 9 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
14 | 1, 2, 3, 5, 13 | erngmul 37494 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (( I ↾ 𝑇)(.r‘𝐷)𝑢) = (( I ↾ 𝑇) ∘ 𝑢)) |
15 | 10, 11, 12, 14 | syl12anc 833 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇)(.r‘𝐷)𝑢) = (( I ↾ 𝑇) ∘ 𝑢)) |
16 | 1, 2, 3 | tendo1mul 37458 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇) ∘ 𝑢) = 𝑢) |
17 | 15, 16 | eqtrd 2833 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢) |
18 | 1, 2, 3, 5, 13 | erngmul 37494 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑢 ∈ 𝐸 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑢(.r‘𝐷)( I ↾ 𝑇)) = (𝑢 ∘ ( I ↾ 𝑇))) |
19 | 10, 12, 11, 18 | syl12anc 833 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢(.r‘𝐷)( I ↾ 𝑇)) = (𝑢 ∘ ( I ↾ 𝑇))) |
20 | 1, 2, 3 | tendo1mulr 37459 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢 ∘ ( I ↾ 𝑇)) = 𝑢) |
21 | 19, 20 | eqtrd 2833 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢) |
22 | 17, 21 | jca 512 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → ((( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢)) |
23 | 22 | ex 413 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑢 ∈ 𝐸 → ((( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢))) |
24 | 9, 23 | sylbid 241 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑢 ∈ (Base‘𝐷) → ((( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢))) |
25 | 24 | ralrimiv 3150 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∀𝑢 ∈ (Base‘𝐷)((( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢)) |
26 | erng1.r | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Ring) | |
27 | eqid 2797 | . . . 4 ⊢ (1r‘𝐷) = (1r‘𝐷) | |
28 | 6, 13, 27 | isringid 19017 | . . 3 ⊢ (𝐷 ∈ Ring → ((( I ↾ 𝑇) ∈ (Base‘𝐷) ∧ ∀𝑢 ∈ (Base‘𝐷)((( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢)) ↔ (1r‘𝐷) = ( I ↾ 𝑇))) |
29 | 26, 28 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((( I ↾ 𝑇) ∈ (Base‘𝐷) ∧ ∀𝑢 ∈ (Base‘𝐷)((( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢)) ↔ (1r‘𝐷) = ( I ↾ 𝑇))) |
30 | 8, 25, 29 | mpbi2and 708 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1r‘𝐷) = ( I ↾ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∀wral 3107 I cid 5354 ↾ cres 5452 ∘ ccom 5454 ‘cfv 6232 (class class class)co 7023 Basecbs 16316 .rcmulr 16399 1rcur 18945 Ringcrg 18991 HLchlt 36038 LHypclh 36672 LTrncltrn 36789 TEndoctendo 37440 EDRingcedring 37441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-riotaBAD 35641 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-undef 7797 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-plusg 16411 df-mulr 16412 df-0g 16548 df-proset 17371 df-poset 17389 df-plt 17401 df-lub 17417 df-glb 17418 df-join 17419 df-meet 17420 df-p0 17482 df-p1 17483 df-lat 17489 df-clat 17551 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-mgp 18934 df-ur 18946 df-ring 18993 df-oposet 35864 df-ol 35866 df-oml 35867 df-covers 35954 df-ats 35955 df-atl 35986 df-cvlat 36010 df-hlat 36039 df-llines 36186 df-lplanes 36187 df-lvols 36188 df-lines 36189 df-psubsp 36191 df-pmap 36192 df-padd 36484 df-lhyp 36676 df-laut 36677 df-ldil 36792 df-ltrn 36793 df-trl 36847 df-tendo 37443 df-edring 37445 |
This theorem is referenced by: erngdvlem4 37679 |
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