dochdmm1 - Mathbox for Norm Megill

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Theorem dochdmm1 41057

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 40925	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ 𝑉)
8	1, 2, 7	syl2anc 582	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
9		dochdmm1.o	. . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
10	3, 4, 6, 9	dochssv 41002	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉)
11	1, 8, 10	syl2anc 582	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ 𝑉)
12		dochdmm1.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)
13	3, 4, 5, 6	dihrnss 40925	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → 𝑌 ⊆ 𝑉)
14	1, 12, 13	syl2anc 582	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑉)
15	3, 4, 6, 9	dochssv 41002	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘𝑌) ⊆ 𝑉)
16	1, 14, 15	syl2anc 582	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ 𝑉)
17	3, 4, 6, 9	dochdmj1 41037	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉 ∧ ( ⊥ ‘𝑌) ⊆ 𝑉) → ( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))) = (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘( ⊥ ‘𝑌))))
18	1, 11, 16, 17	syl3anc 1368	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))) = (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘( ⊥ ‘𝑌))))
19	3, 5, 9	dochoc 41014	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
20	1, 2, 19	syl2anc 582	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
21	3, 5, 9	dochoc 41014	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
22	1, 12, 21	syl2anc 582	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
23	20, 22	ineq12d 4213	. . . 4 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘( ⊥ ‘𝑌))) = (𝑋 ∩ 𝑌))
24	18, 23	eqtr2d 2766	. . 3 ⊢ (𝜑 → (𝑋 ∩ 𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌))))
25	24	fveq2d 6904	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌)))))
26		dochdmm1.j	. . . 4 ⊢ ∨ = ((joinH‘𝐾)‘𝑊)
27	3, 4, 6, 9, 26	djhval2 41046	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉 ∧ ( ⊥ ‘𝑌) ⊆ 𝑉) → (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)) = ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌)))))
28	1, 11, 16, 27	syl3anc 1368	. 2 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)) = ( ⊥ ‘( ⊥ ‘(( ⊥ ‘𝑋) ∪ ( ⊥ ‘𝑌)))))
29	25, 28	eqtr4d 2768	1 ⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3944 ∩ cin 3945 ⊆ wss 3946 ran crn 5682 ‘cfv 6553 (class class class)co 7423 Basecbs 17208 HLchlt 38996 LHypclh 39631 DVecHcdvh 40725 DIsoHcdih 40875 ocHcoch 40994 joinHcdjh 41041

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-riotaBAD 38599

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-tpos 8240 df-undef 8287 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-n0 12520 df-z 12606 df-uz 12870 df-fz 13534 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-0g 17451 df-proset 18315 df-poset 18333 df-plt 18350 df-lub 18366 df-glb 18367 df-join 18368 df-meet 18369 df-p0 18445 df-p1 18446 df-lat 18452 df-clat 18519 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19112 df-cntz 19306 df-lsm 19629 df-cmn 19775 df-abl 19776 df-mgp 20113 df-rng 20131 df-ur 20160 df-ring 20213 df-oppr 20311 df-dvdsr 20334 df-unit 20335 df-invr 20365 df-dvr 20378 df-drng 20666 df-lmod 20785 df-lss 20856 df-lsp 20896 df-lvec 21028 df-lsatoms 38622 df-oposet 38822 df-ol 38824 df-oml 38825 df-covers 38912 df-ats 38913 df-atl 38944 df-cvlat 38968 df-hlat 38997 df-llines 39145 df-lplanes 39146 df-lvols 39147 df-lines 39148 df-psubsp 39150 df-pmap 39151 df-padd 39443 df-lhyp 39635 df-laut 39636 df-ldil 39751 df-ltrn 39752 df-trl 39806 df-tendo 40402 df-edring 40404 df-disoa 40676 df-dvech 40726 df-dib 40786 df-dic 40820 df-dih 40876 df-doch 40995 df-djh 41042

This theorem is referenced by: lclkrlem2c 41156 lclkrslem2 41185 lcfrlem23 41212

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