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Mirrors > Home > MPE Home > Th. List > ocvin | Structured version Visualization version GIF version |
Description: An orthocomplement has trivial intersection with the original subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
ocv2ss.o | ⊢ ⊥ = (ocv‘𝑊) |
ocvin.l | ⊢ 𝐿 = (LSubSp‘𝑊) |
ocvin.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
ocvin | ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2622 | . . . . . . . . 9 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2622 | . . . . . . . . 9 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2622 | . . . . . . . . 9 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
5 | ocv2ss.o | . . . . . . . . 9 ⊢ ⊥ = (ocv‘𝑊) | |
6 | 1, 2, 3, 4, 5 | ocvi 20013 | . . . . . . . 8 ⊢ ((𝑥 ∈ ( ⊥ ‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
7 | 6 | ancoms 469 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆)) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
8 | 7 | adantl 482 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
9 | simpll 790 | . . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → 𝑊 ∈ PreHil) | |
10 | ocvin.l | . . . . . . . . 9 ⊢ 𝐿 = (LSubSp‘𝑊) | |
11 | 1, 10 | lssel 18938 | . . . . . . . 8 ⊢ ((𝑆 ∈ 𝐿 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝑊)) |
12 | 11 | ad2ant2lr 784 | . . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → 𝑥 ∈ (Base‘𝑊)) |
13 | ocvin.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑊) | |
14 | 3, 2, 1, 4, 13 | ipeq0 19983 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = 0 )) |
15 | 9, 12, 14 | syl2anc 693 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = 0 )) |
16 | 8, 15 | mpbid 222 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → 𝑥 = 0 ) |
17 | 16 | ex 450 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆)) → 𝑥 = 0 )) |
18 | elin 3796 | . . . 4 ⊢ (𝑥 ∈ (𝑆 ∩ ( ⊥ ‘𝑆)) ↔ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) | |
19 | velsn 4193 | . . . 4 ⊢ (𝑥 ∈ { 0 } ↔ 𝑥 = 0 ) | |
20 | 17, 18, 19 | 3imtr4g 285 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑥 ∈ (𝑆 ∩ ( ⊥ ‘𝑆)) → 𝑥 ∈ { 0 })) |
21 | 20 | ssrdv 3609 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ⊆ { 0 }) |
22 | phllmod 19975 | . . . 4 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
23 | 22 | adantr 481 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → 𝑊 ∈ LMod) |
24 | 1, 10 | lssss 18937 | . . . . 5 ⊢ (𝑆 ∈ 𝐿 → 𝑆 ⊆ (Base‘𝑊)) |
25 | 1, 5, 10 | ocvlss 20016 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ (Base‘𝑊)) → ( ⊥ ‘𝑆) ∈ 𝐿) |
26 | 24, 25 | sylan2 491 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → ( ⊥ ‘𝑆) ∈ 𝐿) |
27 | 10 | lssincl 18965 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝐿 ∧ ( ⊥ ‘𝑆) ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) |
28 | 22, 27 | syl3an1 1359 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿 ∧ ( ⊥ ‘𝑆) ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) |
29 | 26, 28 | mpd3an3 1425 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) |
30 | 13, 10 | lss0ss 18949 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) → { 0 } ⊆ (𝑆 ∩ ( ⊥ ‘𝑆))) |
31 | 23, 29, 30 | syl2anc 693 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → { 0 } ⊆ (𝑆 ∩ ( ⊥ ‘𝑆))) |
32 | 21, 31 | eqssd 3620 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 {csn 4177 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Scalarcsca 15944 ·𝑖cip 15946 0gc0g 16100 LModclmod 18863 LSubSpclss 18932 PreHilcphl 19969 ocvcocv 20004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-sca 15957 df-vsca 15958 df-ip 15959 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-ghm 17658 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 df-lmhm 19022 df-lvec 19103 df-sra 19172 df-rgmod 19173 df-phl 19971 df-ocv 20007 |
This theorem is referenced by: ocv1 20023 pjdm2 20055 pjff 20056 pjf2 20058 pjfo 20059 obselocv 20072 |
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