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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochval2 | Structured version Visualization version GIF version |
Description: Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Apr-2014.) |
Ref | Expression |
---|---|
dochval2.o | ⊢ ⊥ = (oc‘𝐾) |
dochval2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochval2.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dochval2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochval2.v | ⊢ 𝑉 = (Base‘𝑈) |
dochval2.n | ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dochval2 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2734 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | dochval2.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
4 | dochval2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dochval2.i | . . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | dochval2.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | dochval2.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
8 | dochval2.n | . . 3 ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | dochval 41333 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)})))) |
10 | hlclat 39339 | . . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
11 | 10 | ad2antrr 726 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝐾 ∈ CLat) |
12 | ssrab2 4089 | . . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)} ⊆ (Base‘𝐾) | |
13 | 1, 2 | clatglbcl 18562 | . . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ {𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)} ⊆ (Base‘𝐾)) → ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}) ∈ (Base‘𝐾)) |
14 | 11, 12, 13 | sylancl 586 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}) ∈ (Base‘𝐾)) |
15 | 1, 4, 5 | dihcnvid1 41254 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}) ∈ (Base‘𝐾)) → (◡𝐼‘(𝐼‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))) = ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)})) |
16 | 14, 15 | syldan 591 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (◡𝐼‘(𝐼‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))) = ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)})) |
17 | 1, 2, 4, 5, 6, 7 | dihglb2 41324 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝐼‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)})) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
18 | 17 | fveq2d 6910 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (◡𝐼‘(𝐼‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) |
19 | 16, 18 | eqtr3d 2776 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) |
20 | 19 | fveq2d 6910 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)})) = ( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))) |
21 | 20 | fveq2d 6910 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝐼‘( ⊥ ‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))) = (𝐼‘( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) |
22 | 9, 21 | eqtrd 2774 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {crab 3432 ⊆ wss 3962 ∩ cint 4950 ◡ccnv 5687 ran crn 5689 ‘cfv 6562 Basecbs 17244 occoc 17305 glbcglb 18367 CLatccla 18555 HLchlt 39331 LHypclh 39966 DVecHcdvh 41060 DIsoHcdih 41210 ocHcoch 41329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-riotaBAD 38934 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-undef 8296 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17487 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cntz 19347 df-lsm 19668 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lvec 21119 df-lsatoms 38957 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 df-tendo 40737 df-edring 40739 df-disoa 41011 df-dvech 41061 df-dib 41121 df-dic 41155 df-dih 41211 df-doch 41330 |
This theorem is referenced by: doch2val2 41346 dochocss 41348 |
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