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Mirrors > Home > MPE Home > Th. List > ocvpj | Structured version Visualization version GIF version |
Description: The orthocomplement of a projection subspace is a projection subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
ocvpj.k | ⊢ 𝐾 = (proj‘𝑊) |
ocvpj.o | ⊢ ⊥ = (ocv‘𝑊) |
Ref | Expression |
---|---|
ocvpj | ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ( ⊥ ‘𝑇) ∈ dom 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocvpj.k | . . . . . 6 ⊢ 𝐾 = (proj‘𝑊) | |
2 | eqid 2738 | . . . . . 6 ⊢ (ClSubSp‘𝑊) = (ClSubSp‘𝑊) | |
3 | 1, 2 | pjcss 20921 | . . . . 5 ⊢ (𝑊 ∈ PreHil → dom 𝐾 ⊆ (ClSubSp‘𝑊)) |
4 | 3 | sselda 3922 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (ClSubSp‘𝑊)) |
5 | eqid 2738 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
6 | 5, 2 | cssss 20888 | . . . 4 ⊢ (𝑇 ∈ (ClSubSp‘𝑊) → 𝑇 ⊆ (Base‘𝑊)) |
7 | 4, 6 | syl 17 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ (Base‘𝑊)) |
8 | ocvpj.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
9 | eqid 2738 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
10 | 5, 8, 9 | ocvlss 20875 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ (Base‘𝑊)) → ( ⊥ ‘𝑇) ∈ (LSubSp‘𝑊)) |
11 | 7, 10 | syldan 591 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ( ⊥ ‘𝑇) ∈ (LSubSp‘𝑊)) |
12 | phllmod 20833 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
13 | 12 | adantr 481 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod) |
14 | lmodabl 20168 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel) |
16 | 9 | lsssssubg 20218 | . . . . . 6 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
17 | 13, 16 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
18 | 17, 11 | sseldd 3923 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ( ⊥ ‘𝑇) ∈ (SubGrp‘𝑊)) |
19 | 2, 9 | csslss 20894 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (ClSubSp‘𝑊)) → 𝑇 ∈ (LSubSp‘𝑊)) |
20 | 4, 19 | syldan 591 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (LSubSp‘𝑊)) |
21 | 17, 20 | sseldd 3923 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊)) |
22 | eqid 2738 | . . . . 5 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
23 | 22 | lsmcom 19457 | . . . 4 ⊢ ((𝑊 ∈ Abel ∧ ( ⊥ ‘𝑇) ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊)) → (( ⊥ ‘𝑇)(LSSum‘𝑊)𝑇) = (𝑇(LSSum‘𝑊)( ⊥ ‘𝑇))) |
24 | 15, 18, 21, 23 | syl3anc 1370 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (( ⊥ ‘𝑇)(LSSum‘𝑊)𝑇) = (𝑇(LSSum‘𝑊)( ⊥ ‘𝑇))) |
25 | 8, 2 | cssi 20887 | . . . . 5 ⊢ (𝑇 ∈ (ClSubSp‘𝑊) → 𝑇 = ( ⊥ ‘( ⊥ ‘𝑇))) |
26 | 4, 25 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 = ( ⊥ ‘( ⊥ ‘𝑇))) |
27 | 26 | oveq2d 7293 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (( ⊥ ‘𝑇)(LSSum‘𝑊)𝑇) = (( ⊥ ‘𝑇)(LSSum‘𝑊)( ⊥ ‘( ⊥ ‘𝑇)))) |
28 | 5, 9, 8, 22, 1 | pjdm2 20916 | . . . 4 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(LSSum‘𝑊)( ⊥ ‘𝑇)) = (Base‘𝑊)))) |
29 | 28 | simplbda 500 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(LSSum‘𝑊)( ⊥ ‘𝑇)) = (Base‘𝑊)) |
30 | 24, 27, 29 | 3eqtr3d 2786 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (( ⊥ ‘𝑇)(LSSum‘𝑊)( ⊥ ‘( ⊥ ‘𝑇))) = (Base‘𝑊)) |
31 | 5, 9, 8, 22, 1 | pjdm2 20916 | . . 3 ⊢ (𝑊 ∈ PreHil → (( ⊥ ‘𝑇) ∈ dom 𝐾 ↔ (( ⊥ ‘𝑇) ∈ (LSubSp‘𝑊) ∧ (( ⊥ ‘𝑇)(LSSum‘𝑊)( ⊥ ‘( ⊥ ‘𝑇))) = (Base‘𝑊)))) |
32 | 31 | adantr 481 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (( ⊥ ‘𝑇) ∈ dom 𝐾 ↔ (( ⊥ ‘𝑇) ∈ (LSubSp‘𝑊) ∧ (( ⊥ ‘𝑇)(LSSum‘𝑊)( ⊥ ‘( ⊥ ‘𝑇))) = (Base‘𝑊)))) |
33 | 11, 30, 32 | mpbir2and 710 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ( ⊥ ‘𝑇) ∈ dom 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⊆ wss 3888 dom cdm 5591 ‘cfv 6435 (class class class)co 7277 Basecbs 16910 SubGrpcsubg 18747 LSSumclsm 19237 Abelcabl 19385 LModclmod 20121 LSubSpclss 20191 PreHilcphl 20827 ocvcocv 20863 ClSubSpccss 20864 projcpj 20905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8040 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-map 8615 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-ghm 18830 df-cntz 18921 df-lsm 19239 df-pj1 19240 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-rnghom 19957 df-staf 20103 df-srng 20104 df-lmod 20123 df-lss 20192 df-lmhm 20282 df-lvec 20363 df-sra 20432 df-rgmod 20433 df-phl 20829 df-ocv 20866 df-css 20867 df-pj 20908 |
This theorem is referenced by: (None) |
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