dihoml4 - Mathbox for Norm Megill

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Theorem dihoml4 37451

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
dihoml4	⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))

**Proof of Theorem dihoml4**
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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