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Mirrors > Home > MPE Home > Th. List > iscss2 | Structured version Visualization version GIF version |
Description: It is sufficient to prove that the double orthocomplement is a subset of the target set to show that the set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
cssss.v | ⊢ 𝑉 = (Base‘𝑊) |
cssss.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
ocvcss.o | ⊢ ⊥ = (ocv‘𝑊) |
Ref | Expression |
---|---|
iscss2 | ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocvcss.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
2 | cssss.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | 1, 2 | iscss 20437 | . . 3 ⊢ (𝑊 ∈ PreHil → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
4 | 3 | adantr 474 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
5 | cssss.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
6 | 5, 1 | ocvocv 20425 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
7 | eqss 3836 | . . . 4 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) ↔ (𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)) | |
8 | 7 | baib 531 | . . 3 ⊢ (𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) → (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)) |
9 | 6, 8 | syl 17 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)) |
10 | 4, 9 | bitrd 271 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 ‘cfv 6137 Basecbs 16266 PreHilcphl 20378 ocvcocv 20414 ClSubSpccss 20415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-ip 16367 df-0g 16499 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-mhm 17732 df-grp 17823 df-ghm 18053 df-mgp 18888 df-ur 18900 df-ring 18947 df-oppr 19021 df-rnghom 19115 df-staf 19248 df-srng 19249 df-lmod 19268 df-lmhm 19428 df-lvec 19509 df-sra 19580 df-rgmod 19581 df-phl 20380 df-ocv 20417 df-css 20418 |
This theorem is referenced by: ocvcss 20441 lsmcss 20446 cssmre 20447 |
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