![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ocvlsp | Structured version Visualization version GIF version |
Description: The orthocomplement of a linear span. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
ocvlsp.v | ⊢ 𝑉 = (Base‘𝑊) |
ocvlsp.o | ⊢ ⊥ = (ocv‘𝑊) |
ocvlsp.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
ocvlsp | ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘(𝑁‘𝑆)) = ( ⊥ ‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phllmod 20299 | . . . 4 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
2 | ocvlsp.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
3 | ocvlsp.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | 2, 3 | lspssid 19306 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ (𝑁‘𝑆)) |
5 | 1, 4 | sylan 576 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ (𝑁‘𝑆)) |
6 | ocvlsp.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
7 | 6 | ocv2ss 20342 | . . 3 ⊢ (𝑆 ⊆ (𝑁‘𝑆) → ( ⊥ ‘(𝑁‘𝑆)) ⊆ ( ⊥ ‘𝑆)) |
8 | 5, 7 | syl 17 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘(𝑁‘𝑆)) ⊆ ( ⊥ ‘𝑆)) |
9 | 2, 6 | ocvss 20339 | . . . . 5 ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉 |
10 | 9 | a1i 11 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ⊆ 𝑉) |
11 | 2, 6 | ocvocv 20340 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ ( ⊥ ‘𝑆) ⊆ 𝑉) → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑆)))) |
12 | 10, 11 | syldan 586 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑆)))) |
13 | 1 | adantr 473 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑊 ∈ LMod) |
14 | eqid 2799 | . . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
15 | 2, 6, 14 | ocvlss 20341 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ ( ⊥ ‘𝑆) ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑆)) ∈ (LSubSp‘𝑊)) |
16 | 10, 15 | syldan 586 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑆)) ∈ (LSubSp‘𝑊)) |
17 | 2, 6 | ocvocv 20340 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
18 | 14, 3 | lspssp 19309 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ ( ⊥ ‘( ⊥ ‘𝑆)) ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) → (𝑁‘𝑆) ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
19 | 13, 16, 17, 18 | syl3anc 1491 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑁‘𝑆) ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
20 | 6 | ocv2ss 20342 | . . . 4 ⊢ ((𝑁‘𝑆) ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑆))) ⊆ ( ⊥ ‘(𝑁‘𝑆))) |
21 | 19, 20 | syl 17 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑆))) ⊆ ( ⊥ ‘(𝑁‘𝑆))) |
22 | 12, 21 | sstrd 3808 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘(𝑁‘𝑆))) |
23 | 8, 22 | eqssd 3815 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘(𝑁‘𝑆)) = ( ⊥ ‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ⊆ wss 3769 ‘cfv 6101 Basecbs 16184 LModclmod 19181 LSubSpclss 19250 LSpanclspn 19292 PreHilcphl 20293 ocvcocv 20329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-tpos 7590 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-ip 16285 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-mhm 17650 df-grp 17741 df-ghm 17971 df-mgp 18806 df-ur 18818 df-ring 18865 df-oppr 18939 df-rnghom 19033 df-staf 19163 df-srng 19164 df-lmod 19183 df-lss 19251 df-lsp 19293 df-lmhm 19343 df-lvec 19424 df-sra 19495 df-rgmod 19496 df-phl 20295 df-ocv 20332 |
This theorem is referenced by: ocvz 20347 obselocv 20397 obslbs 20399 |
Copyright terms: Public domain | W3C validator |