|   | Mathbox for Norm Megill | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > diarnN | Structured version Visualization version GIF version | ||
| Description: Partial isomorphism A maps onto the set of all closed subspaces of partial vector space A. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| dvadia.h | ⊢ 𝐻 = (LHyp‘𝐾) | 
| dvadia.u | ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊) | 
| dvadia.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | 
| dvadia.n | ⊢ ⊥ = ((ocA‘𝐾)‘𝑊) | 
| dvadia.s | ⊢ 𝑆 = (LSubSp‘𝑈) | 
| Ref | Expression | 
|---|---|
| diarnN | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran 𝐼 = {𝑥 ∈ 𝑆 ∣ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dvadia.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 2 | dvadia.u | . . . 4 ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊) | |
| 3 | dvadia.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
| 4 | dvadia.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑈) | |
| 5 | 1, 2, 3, 4 | diasslssN 41062 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran 𝐼 ⊆ 𝑆) | 
| 6 | sseqin2 4222 | . . 3 ⊢ (ran 𝐼 ⊆ 𝑆 ↔ (𝑆 ∩ ran 𝐼) = ran 𝐼) | |
| 7 | 5, 6 | sylib 218 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑆 ∩ ran 𝐼) = ran 𝐼) | 
| 8 | dvadia.n | . . . . . . 7 ⊢ ⊥ = ((ocA‘𝐾)‘𝑊) | |
| 9 | 1, 3, 8 | doca3N 41130 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) | 
| 10 | 9 | ex 412 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑥 ∈ ran 𝐼 → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)) | 
| 11 | 10 | adantr 480 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ ran 𝐼 → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)) | 
| 12 | 1, 2, 3, 8, 4 | dvadiaN 41131 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑆 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)) → 𝑥 ∈ ran 𝐼) | 
| 13 | 12 | expr 456 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑆) → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 → 𝑥 ∈ ran 𝐼)) | 
| 14 | 11, 13 | impbid 212 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ ran 𝐼 ↔ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)) | 
| 15 | 14 | rabbi2dva 4225 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑆 ∩ ran 𝐼) = {𝑥 ∈ 𝑆 ∣ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥}) | 
| 16 | 7, 15 | eqtr3d 2778 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran 𝐼 = {𝑥 ∈ 𝑆 ∣ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 ∩ cin 3949 ⊆ wss 3950 ran crn 5685 ‘cfv 6560 LSubSpclss 20930 HLchlt 39352 LHypclh 39987 DVecAcdveca 41005 DIsoAcdia 41031 ocAcocaN 41122 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-riotaBAD 38955 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-undef 8299 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-lss 20931 df-oposet 39178 df-cmtN 39179 df-ol 39180 df-oml 39181 df-covers 39268 df-ats 39269 df-atl 39300 df-cvlat 39324 df-hlat 39353 df-llines 39501 df-lplanes 39502 df-lvols 39503 df-lines 39504 df-psubsp 39506 df-pmap 39507 df-padd 39799 df-lhyp 39991 df-laut 39992 df-ldil 40107 df-ltrn 40108 df-trl 40162 df-tendo 40758 df-edring 40760 df-dveca 41006 df-disoa 41032 df-docaN 41123 | 
| This theorem is referenced by: diaf1oN 41133 | 
| Copyright terms: Public domain | W3C validator |