dihoml4 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dihoml4		Structured version Visualization version GIF version

Theorem dihoml4 41378

Description: Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 39954 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 2730	. . . . . . . . 9 ⊢ (Base‘𝑈) = (Base‘𝑈)
4		dihoml4.s	. . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑈)
5	3, 4	lssss 20849	. . . . . . . 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 2730	. . . . . . . 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 41354	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
12	1, 6, 11	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
13	7, 8, 10	dochoc 41368	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋))
14	1, 12, 13	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋))
15	14	ineq1d 4185	. . . 4 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌) = (( ⊥ ‘𝑋) ∩ 𝑌))
16	15	fveq2d 6865	. . 3 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)))
17	16	ineq1d 4185	. 2 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) ∩ 𝑌) = (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌))
18	7, 9, 3, 10	dochssv 41356	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ⊆ (Base‘𝑈))
19	1, 6, 18	syl2anc 584	. . . 4 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ (Base‘𝑈))
20	7, 8, 9, 3, 10	dochcl 41354	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
21	1, 19, 20	syl2anc 584	. . 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 20849	. . . . . 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 41370	. . . 4 ⊢ (𝜑 → (𝑌 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌))
27	22, 26	mpbird 257	. . 3 ⊢ (𝜑 → 𝑌 ∈ ran ((DIsoH‘𝐾)‘𝑊))
28		dihoml4.l	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
29	7, 9, 3, 10	dochss 41366	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ (Base‘𝑈) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
30	1, 25, 28, 29	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
31	7, 9, 3, 10	dochss 41366	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘𝑈) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌)))
32	1, 19, 30, 31	syl3anc 1373	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌)))
33	32, 22	sseqtrd 3986	. . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝑌)
34	7, 8, 10, 1, 21, 27, 33	dihoml4c 41377	. 2 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))
35	17, 34	eqtr3d 2767	1 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∩ cin 3916 ⊆ wss 3917 ran crn 5642 ‘cfv 6514 Basecbs 17186 LSubSpclss 20844 HLchlt 39350 LHypclh 39985 DVecHcdvh 41079 DIsoHcdih 41229 ocHcoch 41348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 38953

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17411 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38976 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160 df-tendo 40756 df-edring 40758 df-disoa 41030 df-dvech 41080 df-dib 41140 df-dic 41174 df-dih 41230 df-doch 41349

This theorem is referenced by: dochexmidlem6 41466

W3C validator