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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 21620 | . . 3 ⊢ (𝑊 ∈ PreHil → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
4 | 3 | adantr 479 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
5 | cssss.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
6 | 5, 1 | ocvocv 21608 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) |
7 | eqss 3995 | . . . 4 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) ↔ (𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆)) ∧ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)) | |
8 | 7 | baib 534 | . . 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 278 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3947 ‘cfv 6551 Basecbs 17185 PreHilcphl 21561 ocvcocv 21597 ClSubSpccss 21598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-tpos 8236 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-plusg 17251 df-mulr 17252 df-sca 17254 df-vsca 17255 df-ip 17256 df-0g 17428 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18745 df-grp 18898 df-ghm 19173 df-mgp 20080 df-ur 20127 df-ring 20180 df-oppr 20278 df-rhm 20416 df-staf 20730 df-srng 20731 df-lmod 20750 df-lmhm 20912 df-lvec 20993 df-sra 21063 df-rgmod 21064 df-phl 21563 df-ocv 21600 df-css 21601 |
This theorem is referenced by: ocvcss 21624 lsmcss 21629 cssmre 21630 |
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