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 38469 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸) |
5 | erng1.d | . . . 4 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) | |
6 | eqid 2736 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
7 | 1, 2, 3, 5, 6 | erngbase 38501 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸) |
8 | 4, 7 | eleqtrrd 2834 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ (Base‘𝐷)) |
9 | 7 | eleq2d 2816 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑢 ∈ (Base‘𝐷) ↔ 𝑢 ∈ 𝐸)) |
10 | simpl 486 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | 4 | adantr 484 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → ( I ↾ 𝑇) ∈ 𝐸) |
12 | simpr 488 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → 𝑢 ∈ 𝐸) | |
13 | eqid 2736 | . . . . . . . . 9 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
14 | 1, 2, 3, 5, 13 | erngmul 38506 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (( I ↾ 𝑇)(.r‘𝐷)𝑢) = (( I ↾ 𝑇) ∘ 𝑢)) |
15 | 10, 11, 12, 14 | syl12anc 837 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇)(.r‘𝐷)𝑢) = (( I ↾ 𝑇) ∘ 𝑢)) |
16 | 1, 2, 3 | tendo1mul 38470 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇) ∘ 𝑢) = 𝑢) |
17 | 15, 16 | eqtrd 2771 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢) |
18 | 1, 2, 3, 5, 13 | erngmul 38506 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑢 ∈ 𝐸 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑢(.r‘𝐷)( I ↾ 𝑇)) = (𝑢 ∘ ( I ↾ 𝑇))) |
19 | 10, 12, 11, 18 | syl12anc 837 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢(.r‘𝐷)( I ↾ 𝑇)) = (𝑢 ∘ ( I ↾ 𝑇))) |
20 | 1, 2, 3 | tendo1mulr 38471 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢 ∘ ( I ↾ 𝑇)) = 𝑢) |
21 | 19, 20 | eqtrd 2771 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢) |
22 | 17, 21 | jca 515 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → ((( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢)) |
23 | 22 | ex 416 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑢 ∈ 𝐸 → ((( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢))) |
24 | 9, 23 | sylbid 243 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑢 ∈ (Base‘𝐷) → ((( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢))) |
25 | 24 | ralrimiv 3094 | . 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 2736 | . . . 4 ⊢ (1r‘𝐷) = (1r‘𝐷) | |
28 | 6, 13, 27 | isringid 19545 | . . 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 712 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1r‘𝐷) = ( I ↾ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 I cid 5439 ↾ cres 5538 ∘ ccom 5540 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 .rcmulr 16750 1rcur 19470 Ringcrg 19516 HLchlt 37050 LHypclh 37684 LTrncltrn 37801 TEndoctendo 38452 EDRingcedring 38453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-riotaBAD 36653 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-undef 7993 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-plusg 16762 df-mulr 16763 df-0g 16900 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-p1 17886 df-lat 17892 df-clat 17959 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mgp 19459 df-ur 19471 df-ring 19518 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-llines 37198 df-lplanes 37199 df-lvols 37200 df-lines 37201 df-psubsp 37203 df-pmap 37204 df-padd 37496 df-lhyp 37688 df-laut 37689 df-ldil 37804 df-ltrn 37805 df-trl 37859 df-tendo 38455 df-edring 38457 |
This theorem is referenced by: erngdvlem4 38691 |
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