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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochdmm1 | Structured version Visualization version GIF version |
Description: De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.) |
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) |
Ref | Expression |
---|---|
dochdmm1 | ⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌))) |
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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