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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihoml4 | Structured version Visualization version GIF version |
Description: Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 36027 analog.) (Contributed by NM, 15-Jan-2015.) |
Ref | Expression |
---|---|
dihoml4.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihoml4.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihoml4.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
dihoml4.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dihoml4.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihoml4.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
dihoml4.y | ⊢ (𝜑 → 𝑌 ∈ 𝑆) |
dihoml4.c | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
dihoml4.l | ⊢ (𝜑 → 𝑋 ⊆ 𝑌) |
Ref | Expression |
---|---|
dihoml4 | ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihoml4.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dihoml4.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
3 | eqid 2825 | . . . . . . . . 9 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
4 | dihoml4.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑈) | |
5 | 3, 4 | lssss 19300 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ (Base‘𝑈)) |
6 | 2, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑈)) |
7 | dihoml4.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | eqid 2825 | . . . . . . . 8 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
9 | dihoml4.u | . . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | dihoml4.o | . . . . . . . 8 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
11 | 7, 8, 9, 3, 10 | dochcl 37427 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
12 | 1, 6, 11 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
13 | 7, 8, 10 | dochoc 37441 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) |
14 | 1, 12, 13 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) |
15 | 14 | ineq1d 4042 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌) = (( ⊥ ‘𝑋) ∩ 𝑌)) |
16 | 15 | fveq2d 6441 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌))) |
17 | 16 | ineq1d 4042 | . 2 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) ∩ 𝑌) = (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌)) |
18 | 7, 9, 3, 10 | dochssv 37429 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) |
19 | 1, 6, 18 | syl2anc 579 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) |
20 | 7, 8, 9, 3, 10 | dochcl 37427 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
21 | 1, 19, 20 | syl2anc 579 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
22 | dihoml4.c | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) | |
23 | dihoml4.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
24 | 3, 4 | lssss 19300 | . . . . . 6 ⊢ (𝑌 ∈ 𝑆 → 𝑌 ⊆ (Base‘𝑈)) |
25 | 23, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ (Base‘𝑈)) |
26 | 7, 8, 9, 3, 10, 1, 25 | dochoccl 37443 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)) |
27 | 22, 26 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
28 | dihoml4.l | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌) | |
29 | 7, 9, 3, 10 | dochss 37439 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ (Base‘𝑈) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) |
30 | 1, 25, 28, 29 | syl3anc 1494 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) |
31 | 7, 9, 3, 10 | dochss 37439 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘𝑈) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌))) |
32 | 1, 19, 30, 31 | syl3anc 1494 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌))) |
33 | 32, 22 | sseqtrd 3866 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝑌) |
34 | 7, 8, 10, 1, 21, 27, 33 | dihoml4c 37450 | . 2 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) |
35 | 17, 34 | eqtr3d 2863 | 1 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∩ cin 3797 ⊆ wss 3798 ran crn 5347 ‘cfv 6127 Basecbs 16229 LSubSpclss 19295 HLchlt 35424 LHypclh 36058 DVecHcdvh 37152 DIsoHcdih 37302 ocHcoch 37421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-riotaBAD 35027 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-undef 7669 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-0g 16462 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-cntz 18107 df-lsm 18409 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lvec 19469 df-lsatoms 35050 df-oposet 35250 df-ol 35252 df-oml 35253 df-covers 35340 df-ats 35341 df-atl 35372 df-cvlat 35396 df-hlat 35425 df-llines 35572 df-lplanes 35573 df-lvols 35574 df-lines 35575 df-psubsp 35577 df-pmap 35578 df-padd 35870 df-lhyp 36062 df-laut 36063 df-ldil 36178 df-ltrn 36179 df-trl 36233 df-tendo 36829 df-edring 36831 df-disoa 37103 df-dvech 37153 df-dib 37213 df-dic 37247 df-dih 37303 df-doch 37422 |
This theorem is referenced by: dochexmidlem6 37539 |
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