dochdmm1 - Mathbox for Norm Megill

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Theorem dochdmm1 37573

Description: De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)

**Hypotheses**
Ref	Expression
dochdmm1.h	⊢ 𝐻 = (LHyp‘𝐾)
dochdmm1.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dochdmm1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochdmm1.v	⊢ 𝑉 = (Base‘𝑈)
dochdmm1.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochdmm1.j	⊢ ∨ = ((joinH‘𝐾)‘𝑊)
dochdmm1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochdmm1.x	⊢ (𝜑 → 𝑋 ∈ ran 𝐼)
dochdmm1.y	⊢ (𝜑 → 𝑌 ∈ ran 𝐼)

**Assertion**
Ref	Expression
dochdmm1	⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)))

**Proof of Theorem dochdmm1**
Step	Hyp	Ref	Expression
1		dochdmm1.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dochdmm1.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)
3		dochdmm1.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
4		dochdmm1.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		dochdmm1.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
6		dochdmm1.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
7	3, 4, 5, 6	dihrnss 37441	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ 𝑉)
8	1, 2, 7	syl2anc 579	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
9		dochdmm1.o	. . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
10	3, 4, 6, 9	dochssv 37518	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉)
11	1, 8, 10	syl2anc 579	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ 𝑉)
12		dochdmm1.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)
13	3, 4, 5, 6	dihrnss 37441	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → 𝑌 ⊆ 𝑉)
14	1, 12, 13	syl2anc 579	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑉)
15	3, 4, 6, 9	dochssv 37518	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘𝑌) ⊆ 𝑉)
16	1, 14, 15	syl2anc 579	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ 𝑉)
17	3, 4, 6, 9	dochdmj1 37553	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉 ∧ ( ⊥ ‘𝑌) ⊆ 𝑉) → ( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))) = (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘( ⊥ ‘𝑌))))
18	1, 11, 16, 17	syl3anc 1439	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))) = (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘( ⊥ ‘𝑌))))
19	3, 5, 9	dochoc 37530	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
20	1, 2, 19	syl2anc 579	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
21	3, 5, 9	dochoc 37530	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
22	1, 12, 21	syl2anc 579	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
23	20, 22	ineq12d 4038	. . . 4 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘( ⊥ ‘𝑌))) = (𝑋 ∩ 𝑌))
24	18, 23	eqtr2d 2815	. . 3 ⊢ (𝜑 → (𝑋 ∩ 𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))))
25	24	fveq2d 6452	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌)))))
26		dochdmm1.j	. . . 4 ⊢ ∨ = ((joinH‘𝐾)‘𝑊)
27	3, 4, 6, 9, 26	djhval2 37562	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉 ∧ ( ⊥ ‘𝑌) ⊆ 𝑉) → (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)) = ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌)))))
28	1, 11, 16, 27	syl3anc 1439	. 2 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)) = ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌)))))
29	25, 28	eqtr4d 2817	1 ⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∪ cun 3790 ∩ cin 3791 ⊆ wss 3792 ran crn 5358 ‘cfv 6137 (class class class)co 6924 Basecbs 16266 HLchlt 35513 LHypclh 36147 DVecHcdvh 37241 DIsoHcdih 37391 ocHcoch 37510 joinHcdjh 37557

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-riotaBAD 35116

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-undef 7683 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-0g 16499 df-proset 17325 df-poset 17343 df-plt 17355 df-lub 17371 df-glb 17372 df-join 17373 df-meet 17374 df-p0 17436 df-p1 17437 df-lat 17443 df-clat 17505 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-grp 17823 df-minusg 17824 df-sbg 17825 df-subg 17986 df-cntz 18144 df-lsm 18446 df-cmn 18592 df-abl 18593 df-mgp 18888 df-ur 18900 df-ring 18947 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-drng 19152 df-lmod 19268 df-lss 19336 df-lsp 19378 df-lvec 19509 df-lsatoms 35139 df-oposet 35339 df-ol 35341 df-oml 35342 df-covers 35429 df-ats 35430 df-atl 35461 df-cvlat 35485 df-hlat 35514 df-llines 35661 df-lplanes 35662 df-lvols 35663 df-lines 35664 df-psubsp 35666 df-pmap 35667 df-padd 35959 df-lhyp 36151 df-laut 36152 df-ldil 36267 df-ltrn 36268 df-trl 36322 df-tendo 36918 df-edring 36920 df-disoa 37192 df-dvech 37242 df-dib 37302 df-dic 37336 df-dih 37392 df-doch 37511 df-djh 37558

This theorem is referenced by: lclkrlem2c 37672 lclkrslem2 37701 lcfrlem23 37728

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