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 38779 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸) |
5 | erng1.d | . . . 4 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) | |
6 | eqid 2740 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
7 | 1, 2, 3, 5, 6 | erngbase 38811 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸) |
8 | 4, 7 | eleqtrrd 2844 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ (Base‘𝐷)) |
9 | 7 | eleq2d 2826 | . . . 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 2740 | . . . . . . . . 9 ⊢ (.r‘𝐷) = (.r‘𝐷) | |
14 | 1, 2, 3, 5, 13 | erngmul 38816 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (( I ↾ 𝑇)(.r‘𝐷)𝑢) = (( I ↾ 𝑇) ∘ 𝑢)) |
15 | 10, 11, 12, 14 | syl12anc 834 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇)(.r‘𝐷)𝑢) = (( I ↾ 𝑇) ∘ 𝑢)) |
16 | 1, 2, 3 | tendo1mul 38780 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇) ∘ 𝑢) = 𝑢) |
17 | 15, 16 | eqtrd 2780 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢) |
18 | 1, 2, 3, 5, 13 | erngmul 38816 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑢 ∈ 𝐸 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑢(.r‘𝐷)( I ↾ 𝑇)) = (𝑢 ∘ ( I ↾ 𝑇))) |
19 | 10, 12, 11, 18 | syl12anc 834 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢(.r‘𝐷)( I ↾ 𝑇)) = (𝑢 ∘ ( I ↾ 𝑇))) |
20 | 1, 2, 3 | tendo1mulr 38781 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢 ∘ ( I ↾ 𝑇)) = 𝑢) |
21 | 19, 20 | eqtrd 2780 | . . . . . 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 239 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑢 ∈ (Base‘𝐷) → ((( I ↾ 𝑇)(.r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(.r‘𝐷)( I ↾ 𝑇)) = 𝑢))) |
25 | 24 | ralrimiv 3109 | . 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 2740 | . . . 4 ⊢ (1r‘𝐷) = (1r‘𝐷) | |
28 | 6, 13, 27 | isringid 19810 | . . 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 709 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1r‘𝐷) = ( I ↾ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 I cid 5489 ↾ cres 5592 ∘ ccom 5594 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 .rcmulr 16961 1rcur 19735 Ringcrg 19781 HLchlt 37360 LHypclh 37994 LTrncltrn 38111 TEndoctendo 38762 EDRingcedring 38763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-riotaBAD 36963 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-undef 8080 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-0g 17150 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mgp 19719 df-ur 19736 df-ring 19783 df-oposet 37186 df-ol 37188 df-oml 37189 df-covers 37276 df-ats 37277 df-atl 37308 df-cvlat 37332 df-hlat 37361 df-llines 37508 df-lplanes 37509 df-lvols 37510 df-lines 37511 df-psubsp 37513 df-pmap 37514 df-padd 37806 df-lhyp 37998 df-laut 37999 df-ldil 38114 df-ltrn 38115 df-trl 38169 df-tendo 38765 df-edring 38767 |
This theorem is referenced by: erngdvlem4 39001 |
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