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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 37083 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 2821 | . . . . . . . . 9 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
4 | dihoml4.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑈) | |
5 | 3, 4 | lssss 19702 | . . . . . . . 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 2821 | . . . . . . . 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 38483 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
12 | 1, 6, 11 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
13 | 7, 8, 10 | dochoc 38497 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) |
14 | 1, 12, 13 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋)) |
15 | 14 | ineq1d 4187 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌) = (( ⊥ ‘𝑋) ∩ 𝑌)) |
16 | 15 | fveq2d 6668 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌))) |
17 | 16 | ineq1d 4187 | . 2 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) ∩ 𝑌) = (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌)) |
18 | 7, 9, 3, 10 | dochssv 38485 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) |
19 | 1, 6, 18 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) |
20 | 7, 8, 9, 3, 10 | dochcl 38483 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
21 | 1, 19, 20 | syl2anc 586 | . . 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 19702 | . . . . . 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 38499 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)) |
27 | 22, 26 | mpbird 259 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
28 | dihoml4.l | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌) | |
29 | 7, 9, 3, 10 | dochss 38495 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ (Base‘𝑈) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) |
30 | 1, 25, 28, 29 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) |
31 | 7, 9, 3, 10 | dochss 38495 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘𝑈) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌))) |
32 | 1, 19, 30, 31 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌))) |
33 | 32, 22 | sseqtrd 4006 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝑌) |
34 | 7, 8, 10, 1, 21, 27, 33 | dihoml4c 38506 | . 2 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) |
35 | 17, 34 | eqtr3d 2858 | 1 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 ran crn 5550 ‘cfv 6349 Basecbs 16477 LSubSpclss 19697 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 DIsoHcdih 38358 ocHcoch 38477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-0g 16709 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tendo 37885 df-edring 37887 df-disoa 38159 df-dvech 38209 df-dib 38269 df-dic 38303 df-dih 38359 df-doch 38478 |
This theorem is referenced by: dochexmidlem6 38595 |
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