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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 37249 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 2798 | . . . . . . . . 9 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
4 | dihoml4.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑈) | |
5 | 3, 4 | lssss 19701 | . . . . . . . 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 2798 | . . . . . . . 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 38649 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
12 | 1, 6, 11 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
13 | 7, 8, 10 | dochoc 38663 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) |
14 | 1, 12, 13 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) |
15 | 14 | ineq1d 4138 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌) = (( ⊥ ‘𝑋) ∩ 𝑌)) |
16 | 15 | fveq2d 6649 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌))) |
17 | 16 | ineq1d 4138 | . 2 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) ∩ 𝑌) = (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌)) |
18 | 7, 9, 3, 10 | dochssv 38651 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) |
19 | 1, 6, 18 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) |
20 | 7, 8, 9, 3, 10 | dochcl 38649 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
21 | 1, 19, 20 | syl2anc 587 | . . 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 19701 | . . . . . 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 38665 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)) |
27 | 22, 26 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
28 | dihoml4.l | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌) | |
29 | 7, 9, 3, 10 | dochss 38661 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ (Base‘𝑈) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) |
30 | 1, 25, 28, 29 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) |
31 | 7, 9, 3, 10 | dochss 38661 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘𝑈) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌))) |
32 | 1, 19, 30, 31 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌))) |
33 | 32, 22 | sseqtrd 3955 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝑌) |
34 | 7, 8, 10, 1, 21, 27, 33 | dihoml4c 38672 | . 2 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) |
35 | 17, 34 | eqtr3d 2835 | 1 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 ran crn 5520 ‘cfv 6324 Basecbs 16475 LSubSpclss 19696 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 DIsoHcdih 38524 ocHcoch 38643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tendo 38051 df-edring 38053 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 |
This theorem is referenced by: dochexmidlem6 38761 |
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