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Mirrors > Home > HSE Home > Th. List > ocss | Structured version Visualization version GIF version |
Description: An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ocss | ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocsh 28698 | . 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Sℋ ) | |
2 | shss 28623 | . 2 ⊢ ((⊥‘𝐴) ∈ Sℋ → (⊥‘𝐴) ⊆ ℋ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ⊆ wss 3799 ‘cfv 6124 ℋchba 28332 Sℋ csh 28341 ⊥cort 28343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-hilex 28412 ax-hfvadd 28413 ax-hv0cl 28416 ax-hfvmul 28418 ax-hvmul0 28423 ax-hfi 28492 ax-his2 28496 ax-his3 28497 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-ltxr 10397 df-sh 28620 df-oc 28665 |
This theorem is referenced by: shocss 28701 occon2 28703 ocorth 28706 ococss 28708 occon3 28712 occllem 28718 hsupcl 28754 spanssoc 28764 sshjcl 28770 ococin 28823 ssjo 28862 |
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