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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochnel2 | Structured version Visualization version GIF version |
Description: A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.) |
Ref | Expression |
---|---|
dochnoncon.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochnoncon.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochnoncon.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
dochnoncon.z | ⊢ 0 = (0g‘𝑈) |
dochnoncon.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochnel2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochnel2.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
dochnel2.x | ⊢ (𝜑 → 𝑋 ∈ (𝑇 ∖ { 0 })) |
Ref | Expression |
---|---|
dochnel2 | ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochnel2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑇 ∖ { 0 })) | |
2 | 1 | eldifbd 3837 | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ { 0 }) |
3 | 1 | eldifad 3836 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑇) |
4 | elin 4052 | . . . . 5 ⊢ (𝑋 ∈ (𝑇 ∩ ( ⊥ ‘𝑇)) ↔ (𝑋 ∈ 𝑇 ∧ 𝑋 ∈ ( ⊥ ‘𝑇))) | |
5 | dochnel2.k | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | dochnel2.t | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
7 | dochnoncon.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | dochnoncon.u | . . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
9 | dochnoncon.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑈) | |
10 | dochnoncon.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑈) | |
11 | dochnoncon.o | . . . . . . . 8 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
12 | 7, 8, 9, 10, 11 | dochnoncon 38005 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑇 ∈ 𝑆) → (𝑇 ∩ ( ⊥ ‘𝑇)) = { 0 }) |
13 | 5, 6, 12 | syl2anc 576 | . . . . . 6 ⊢ (𝜑 → (𝑇 ∩ ( ⊥ ‘𝑇)) = { 0 }) |
14 | 13 | eleq2d 2846 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑇 ∩ ( ⊥ ‘𝑇)) ↔ 𝑋 ∈ { 0 })) |
15 | 4, 14 | syl5bbr 277 | . . . 4 ⊢ (𝜑 → ((𝑋 ∈ 𝑇 ∧ 𝑋 ∈ ( ⊥ ‘𝑇)) ↔ 𝑋 ∈ { 0 })) |
16 | 15 | biimpd 221 | . . 3 ⊢ (𝜑 → ((𝑋 ∈ 𝑇 ∧ 𝑋 ∈ ( ⊥ ‘𝑇)) → 𝑋 ∈ { 0 })) |
17 | 3, 16 | mpand 683 | . 2 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘𝑇) → 𝑋 ∈ { 0 })) |
18 | 2, 17 | mtod 190 | 1 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∖ cdif 3821 ∩ cin 3823 {csn 4436 ‘cfv 6186 0gc0g 16568 LSubSpclss 19438 HLchlt 35964 LHypclh 36598 DVecHcdvh 37692 ocHcoch 37961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-riotaBAD 35567 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-iin 4792 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-tpos 7694 df-undef 7741 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-oadd 7908 df-er 8088 df-map 8207 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-n0 11707 df-z 11793 df-uz 12058 df-fz 12708 df-struct 16340 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-ress 16346 df-plusg 16433 df-mulr 16434 df-sca 16436 df-vsca 16437 df-0g 16570 df-proset 17409 df-poset 17427 df-plt 17439 df-lub 17455 df-glb 17456 df-join 17457 df-meet 17458 df-p0 17520 df-p1 17521 df-lat 17527 df-clat 17589 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-submnd 17817 df-grp 17907 df-minusg 17908 df-sbg 17909 df-subg 18073 df-cntz 18231 df-lsm 18535 df-cmn 18681 df-abl 18682 df-mgp 18976 df-ur 18988 df-ring 19035 df-oppr 19109 df-dvdsr 19127 df-unit 19128 df-invr 19158 df-dvr 19169 df-drng 19240 df-lmod 19371 df-lss 19439 df-lsp 19479 df-lvec 19610 df-lsatoms 35590 df-oposet 35790 df-ol 35792 df-oml 35793 df-covers 35880 df-ats 35881 df-atl 35912 df-cvlat 35936 df-hlat 35965 df-llines 36112 df-lplanes 36113 df-lvols 36114 df-lines 36115 df-psubsp 36117 df-pmap 36118 df-padd 36410 df-lhyp 36602 df-laut 36603 df-ldil 36718 df-ltrn 36719 df-trl 36773 df-tendo 37369 df-edring 37371 df-disoa 37643 df-dvech 37693 df-dib 37753 df-dic 37787 df-dih 37843 df-doch 37962 |
This theorem is referenced by: dochnel 38007 |
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