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Mirrors > Home > MPE Home > Th. List > Mathboxes > djhexmid | Structured version Visualization version GIF version |
Description: Excluded middle property of DVecH vector space closed subspace join. (Contributed by NM, 22-Jul-2014.) |
Ref | Expression |
---|---|
djhexmid.h | ⊢ 𝐻 = (LHyp‘𝐾) |
djhexmid.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
djhexmid.v | ⊢ 𝑉 = (Base‘𝑈) |
djhexmid.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
djhexmid.j | ⊢ ∨ = ((joinH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
djhexmid | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simpr 484 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑋 ⊆ 𝑉) | |
3 | djhexmid.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | djhexmid.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | djhexmid.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
6 | djhexmid.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
7 | 3, 4, 5, 6 | dochssv 40542 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉) |
8 | djhexmid.j | . . . 4 ⊢ ∨ = ((joinH‘𝐾)‘𝑊) | |
9 | 3, 4, 5, 6, 8 | djhval 40585 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ ( ⊥ ‘𝑋) ⊆ 𝑉)) → (𝑋 ∨ ( ⊥ ‘𝑋)) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))))) |
10 | 1, 2, 7, 9 | syl12anc 834 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑋 ∨ ( ⊥ ‘𝑋)) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))))) |
11 | eqid 2731 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
12 | 3, 4, 5, 11, 6 | dochlss 40541 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈)) |
13 | eqid 2731 | . . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
14 | 3, 4, 11, 13, 6 | dochnoncon 40578 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈)) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))) = {(0g‘𝑈)}) |
15 | 12, 14 | syldan 590 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))) = {(0g‘𝑈)}) |
16 | 3, 4, 6, 5, 13 | doch1 40546 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑉) = {(0g‘𝑈)}) |
17 | 16 | adantr 480 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑉) = {(0g‘𝑈)}) |
18 | 15, 17 | eqtr4d 2774 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑉)) |
19 | 18 | fveq2d 6895 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋)))) = ( ⊥ ‘( ⊥ ‘𝑉))) |
20 | eqid 2731 | . . . . 5 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
21 | 3, 20, 4, 5 | dih1rn 40474 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑉 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
22 | 21 | adantr 480 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑉 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
23 | 3, 20, 6 | dochoc 40554 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉) |
24 | 22, 23 | syldan 590 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉) |
25 | 10, 19, 24 | 3eqtrd 2775 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∩ cin 3947 ⊆ wss 3948 {csn 4628 ran crn 5677 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 0gc0g 17392 LSubSpclss 20690 HLchlt 38536 LHypclh 39171 DVecHcdvh 40265 DIsoHcdih 40415 ocHcoch 40534 joinHcdjh 40581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38139 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-proset 18255 df-poset 18273 df-plt 18290 df-lub 18306 df-glb 18307 df-join 18308 df-meet 18309 df-p0 18385 df-p1 18386 df-lat 18392 df-clat 18459 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-submnd 18709 df-grp 18861 df-minusg 18862 df-sbg 18863 df-subg 19043 df-cntz 19226 df-lsm 19549 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-oppr 20229 df-dvdsr 20252 df-unit 20253 df-invr 20283 df-dvr 20296 df-drng 20506 df-lmod 20620 df-lss 20691 df-lsp 20731 df-lvec 20862 df-lsatoms 38162 df-oposet 38362 df-ol 38364 df-oml 38365 df-covers 38452 df-ats 38453 df-atl 38484 df-cvlat 38508 df-hlat 38537 df-llines 38685 df-lplanes 38686 df-lvols 38687 df-lines 38688 df-psubsp 38690 df-pmap 38691 df-padd 38983 df-lhyp 39175 df-laut 39176 df-ldil 39291 df-ltrn 39292 df-trl 39346 df-tendo 39942 df-edring 39944 df-disoa 40216 df-dvech 40266 df-dib 40326 df-dic 40360 df-dih 40416 df-doch 40535 df-djh 40582 |
This theorem is referenced by: dochsatshp 40638 |
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