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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochdmm1 | Structured version Visualization version GIF version |
Description: De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.) |
Ref | Expression |
---|---|
dochdmm1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochdmm1.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dochdmm1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochdmm1.v | ⊢ 𝑉 = (Base‘𝑈) |
dochdmm1.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochdmm1.j | ⊢ ∨ = ((joinH‘𝐾)‘𝑊) |
dochdmm1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochdmm1.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
dochdmm1.y | ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) |
Ref | Expression |
---|---|
dochdmm1 | ⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochdmm1.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dochdmm1.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
3 | dochdmm1.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dochdmm1.u | . . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | dochdmm1.i | . . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | dochdmm1.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈) | |
7 | 3, 4, 5, 6 | dihrnss 37441 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ 𝑉) |
8 | 1, 2, 7 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
9 | dochdmm1.o | . . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
10 | 3, 4, 6, 9 | dochssv 37518 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉) |
11 | 1, 8, 10 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ 𝑉) |
12 | dochdmm1.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) | |
13 | 3, 4, 5, 6 | dihrnss 37441 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → 𝑌 ⊆ 𝑉) |
14 | 1, 12, 13 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑉) |
15 | 3, 4, 6, 9 | dochssv 37518 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘𝑌) ⊆ 𝑉) |
16 | 1, 14, 15 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ 𝑉) |
17 | 3, 4, 6, 9 | dochdmj1 37553 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉 ∧ ( ⊥ ‘𝑌) ⊆ 𝑉) → ( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))) = (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘( ⊥ ‘𝑌)))) |
18 | 1, 11, 16, 17 | syl3anc 1439 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))) = (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘( ⊥ ‘𝑌)))) |
19 | 3, 5, 9 | dochoc 37530 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
20 | 1, 2, 19 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
21 | 3, 5, 9 | dochoc 37530 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
22 | 1, 12, 21 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
23 | 20, 22 | ineq12d 4038 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘( ⊥ ‘𝑌))) = (𝑋 ∩ 𝑌)) |
24 | 18, 23 | eqtr2d 2815 | . . 3 ⊢ (𝜑 → (𝑋 ∩ 𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌)))) |
25 | 24 | fveq2d 6452 | . 2 ⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))))) |
26 | dochdmm1.j | . . . 4 ⊢ ∨ = ((joinH‘𝐾)‘𝑊) | |
27 | 3, 4, 6, 9, 26 | djhval2 37562 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉 ∧ ( ⊥ ‘𝑌) ⊆ 𝑉) → (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)) = ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))))) |
28 | 1, 11, 16, 27 | syl3anc 1439 | . 2 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)) = ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))))) |
29 | 25, 28 | eqtr4d 2817 | 1 ⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∪ cun 3790 ∩ cin 3791 ⊆ wss 3792 ran crn 5358 ‘cfv 6137 (class class class)co 6924 Basecbs 16266 HLchlt 35513 LHypclh 36147 DVecHcdvh 37241 DIsoHcdih 37391 ocHcoch 37510 joinHcdjh 37557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-riotaBAD 35116 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-undef 7683 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-0g 16499 df-proset 17325 df-poset 17343 df-plt 17355 df-lub 17371 df-glb 17372 df-join 17373 df-meet 17374 df-p0 17436 df-p1 17437 df-lat 17443 df-clat 17505 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-grp 17823 df-minusg 17824 df-sbg 17825 df-subg 17986 df-cntz 18144 df-lsm 18446 df-cmn 18592 df-abl 18593 df-mgp 18888 df-ur 18900 df-ring 18947 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-drng 19152 df-lmod 19268 df-lss 19336 df-lsp 19378 df-lvec 19509 df-lsatoms 35139 df-oposet 35339 df-ol 35341 df-oml 35342 df-covers 35429 df-ats 35430 df-atl 35461 df-cvlat 35485 df-hlat 35514 df-llines 35661 df-lplanes 35662 df-lvols 35663 df-lines 35664 df-psubsp 35666 df-pmap 35667 df-padd 35959 df-lhyp 36151 df-laut 36152 df-ldil 36267 df-ltrn 36268 df-trl 36322 df-tendo 36918 df-edring 36920 df-disoa 37192 df-dvech 37242 df-dib 37302 df-dic 37336 df-dih 37392 df-doch 37511 df-djh 37558 |
This theorem is referenced by: lclkrlem2c 37672 lclkrslem2 37701 lcfrlem23 37728 |
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