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Mirrors > Home > MPE Home > Th. List > Mathboxes > doch2val2 | Structured version Visualization version GIF version |
Description: Double orthocomplement for DVecH vector space. (Contributed by NM, 26-Jul-2014.) |
Ref | Expression |
---|---|
doch2val2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
doch2val2.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
doch2val2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
doch2val2.v | ⊢ 𝑉 = (Base‘𝑈) |
doch2val2.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
doch2val2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
doch2val2.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
Ref | Expression |
---|---|
doch2val2 | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | doch2val2.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | doch2val2.x | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) | |
3 | eqid 2798 | . . . . 5 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
4 | doch2val2.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | doch2val2.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | doch2val2.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | doch2val2.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
8 | doch2val2.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
9 | 3, 4, 5, 6, 7, 8 | dochval2 37366 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) = (𝐼‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) |
10 | 1, 2, 9 | syl2anc 580 | . . 3 ⊢ (𝜑 → ( ⊥ ‘𝑋) = (𝐼‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) |
11 | 10 | fveq2d 6414 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘(𝐼‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))))) |
12 | 1 | simpld 489 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
13 | hlop 35376 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ OP) |
15 | ssrab2 3882 | . . . . . . 7 ⊢ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ⊆ ran 𝐼 | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ⊆ ran 𝐼) |
17 | 4, 5, 6, 7 | dih1rn 37301 | . . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑉 ∈ ran 𝐼) |
18 | 1, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ran 𝐼) |
19 | sseq2 3822 | . . . . . . . . 9 ⊢ (𝑧 = 𝑉 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑉)) | |
20 | 19 | elrab 3555 | . . . . . . . 8 ⊢ (𝑉 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ↔ (𝑉 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑉)) |
21 | 18, 2, 20 | sylanbrc 579 | . . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
22 | 21 | ne0d 4121 | . . . . . 6 ⊢ (𝜑 → {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ≠ ∅) |
23 | 4, 5 | dihintcl 37358 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ({𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ⊆ ran 𝐼 ∧ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ≠ ∅)) → ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼) |
24 | 1, 16, 22, 23 | syl12anc 866 | . . . . 5 ⊢ (𝜑 → ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼) |
25 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
26 | 25, 4, 5 | dihcnvcl 37285 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼) → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾)) |
27 | 1, 24, 26 | syl2anc 580 | . . . 4 ⊢ (𝜑 → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾)) |
28 | 25, 3 | opoccl 35208 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∈ (Base‘𝐾)) |
29 | 14, 27, 28 | syl2anc 580 | . . 3 ⊢ (𝜑 → ((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∈ (Base‘𝐾)) |
30 | 25, 3, 4, 5, 8 | dochvalr2 37376 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∈ (Base‘𝐾)) → ( ⊥ ‘(𝐼‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) = (𝐼‘((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))))) |
31 | 1, 29, 30 | syl2anc 580 | . 2 ⊢ (𝜑 → ( ⊥ ‘(𝐼‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) = (𝐼‘((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))))) |
32 | 25, 3 | opococ 35209 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) |
33 | 14, 27, 32 | syl2anc 580 | . . . 4 ⊢ (𝜑 → ((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) |
34 | 33 | fveq2d 6414 | . . 3 ⊢ (𝜑 → (𝐼‘((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) = (𝐼‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))) |
35 | 4, 5 | dihcnvid2 37287 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼) → (𝐼‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
36 | 1, 24, 35 | syl2anc 580 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
37 | 34, 36 | eqtrd 2832 | . 2 ⊢ (𝜑 → (𝐼‘((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
38 | 11, 31, 37 | 3eqtrd 2836 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2970 {crab 3092 ⊆ wss 3768 ∅c0 4114 ∩ cint 4666 ◡ccnv 5310 ran crn 5312 ‘cfv 6100 Basecbs 16181 occoc 16272 OPcops 35186 HLchlt 35364 LHypclh 35998 DVecHcdvh 37092 DIsoHcdih 37242 ocHcoch 37361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 ax-riotaBAD 34967 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-iin 4712 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-tpos 7589 df-undef 7636 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-oadd 7802 df-er 7981 df-map 8096 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-nn 11312 df-2 11373 df-3 11374 df-4 11375 df-5 11376 df-6 11377 df-n0 11578 df-z 11664 df-uz 11928 df-fz 12578 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-sca 16280 df-vsca 16281 df-0g 16414 df-proset 17240 df-poset 17258 df-plt 17270 df-lub 17286 df-glb 17287 df-join 17288 df-meet 17289 df-p0 17351 df-p1 17352 df-lat 17358 df-clat 17420 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-submnd 17648 df-grp 17738 df-minusg 17739 df-sbg 17740 df-subg 17901 df-cntz 18059 df-lsm 18361 df-cmn 18507 df-abl 18508 df-mgp 18803 df-ur 18815 df-ring 18862 df-oppr 18936 df-dvdsr 18954 df-unit 18955 df-invr 18985 df-dvr 18996 df-drng 19064 df-lmod 19180 df-lss 19248 df-lsp 19290 df-lvec 19421 df-lsatoms 34990 df-oposet 35190 df-ol 35192 df-oml 35193 df-covers 35280 df-ats 35281 df-atl 35312 df-cvlat 35336 df-hlat 35365 df-llines 35512 df-lplanes 35513 df-lvols 35514 df-lines 35515 df-psubsp 35517 df-pmap 35518 df-padd 35810 df-lhyp 36002 df-laut 36003 df-ldil 36118 df-ltrn 36119 df-trl 36173 df-tendo 36769 df-edring 36771 df-disoa 37043 df-dvech 37093 df-dib 37153 df-dic 37187 df-dih 37243 df-doch 37362 |
This theorem is referenced by: dochspss 37392 |
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