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Mirrors > Home > MPE Home > Th. List > Mathboxes > djaclN | Structured version Visualization version GIF version |
Description: Closure of subspace join for DVecA partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
djacl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
djacl.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
djacl.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
djacl.j | ⊢ 𝐽 = ((vA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
djaclN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djacl.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | djacl.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | djacl.i | . . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
4 | eqid 2820 | . . 3 ⊢ ((ocA‘𝐾)‘𝑊) = ((ocA‘𝐾)‘𝑊) | |
5 | djacl.j | . . 3 ⊢ 𝐽 = ((vA‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | djavalN 38311 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) = (((ocA‘𝐾)‘𝑊)‘((((ocA‘𝐾)‘𝑊)‘𝑋) ∩ (((ocA‘𝐾)‘𝑊)‘𝑌)))) |
7 | inss1 4198 | . . . 4 ⊢ ((((ocA‘𝐾)‘𝑊)‘𝑋) ∩ (((ocA‘𝐾)‘𝑊)‘𝑌)) ⊆ (((ocA‘𝐾)‘𝑊)‘𝑋) | |
8 | 1, 2, 3, 4 | docaclN 38300 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (((ocA‘𝐾)‘𝑊)‘𝑋) ∈ ran 𝐼) |
9 | 8 | adantrr 715 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (((ocA‘𝐾)‘𝑊)‘𝑋) ∈ ran 𝐼) |
10 | 1, 2, 3 | diaelrnN 38221 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (((ocA‘𝐾)‘𝑊)‘𝑋) ∈ ran 𝐼) → (((ocA‘𝐾)‘𝑊)‘𝑋) ⊆ 𝑇) |
11 | 9, 10 | syldan 593 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (((ocA‘𝐾)‘𝑊)‘𝑋) ⊆ 𝑇) |
12 | 7, 11 | sstrid 3971 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → ((((ocA‘𝐾)‘𝑊)‘𝑋) ∩ (((ocA‘𝐾)‘𝑊)‘𝑌)) ⊆ 𝑇) |
13 | 1, 2, 3, 4 | docaclN 38300 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((((ocA‘𝐾)‘𝑊)‘𝑋) ∩ (((ocA‘𝐾)‘𝑊)‘𝑌)) ⊆ 𝑇) → (((ocA‘𝐾)‘𝑊)‘((((ocA‘𝐾)‘𝑊)‘𝑋) ∩ (((ocA‘𝐾)‘𝑊)‘𝑌))) ∈ ran 𝐼) |
14 | 12, 13 | syldan 593 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (((ocA‘𝐾)‘𝑊)‘((((ocA‘𝐾)‘𝑊)‘𝑋) ∩ (((ocA‘𝐾)‘𝑊)‘𝑌))) ∈ ran 𝐼) |
15 | 6, 14 | eqeltrd 2912 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∩ cin 3928 ⊆ wss 3929 ran crn 5549 ‘cfv 6348 (class class class)co 7149 HLchlt 36526 LHypclh 37160 LTrncltrn 37277 DIsoAcdia 38204 ocAcocaN 38295 vAcdjaN 38307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-riotaBAD 36129 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-undef 7932 df-map 8401 df-proset 17531 df-poset 17549 df-plt 17561 df-lub 17577 df-glb 17578 df-join 17579 df-meet 17580 df-p0 17642 df-p1 17643 df-lat 17649 df-clat 17711 df-oposet 36352 df-ol 36354 df-oml 36355 df-covers 36442 df-ats 36443 df-atl 36474 df-cvlat 36498 df-hlat 36527 df-llines 36674 df-lplanes 36675 df-lvols 36676 df-lines 36677 df-psubsp 36679 df-pmap 36680 df-padd 36972 df-lhyp 37164 df-laut 37165 df-ldil 37280 df-ltrn 37281 df-trl 37335 df-disoa 38205 df-docaN 38296 df-djaN 38308 |
This theorem is referenced by: (None) |
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