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Mirrors > Home > MPE Home > Th. List > cssi | Structured version Visualization version GIF version |
Description: Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
cssval.o | ⊢ ⊥ = (ocv‘𝑊) |
cssval.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
Ref | Expression |
---|---|
cssi | ⊢ (𝑆 ∈ 𝐶 → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6959 | . . . 4 ⊢ (𝑆 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ dom ClSubSp) | |
2 | cssval.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | 1, 2 | eleq2s 2862 | . . 3 ⊢ (𝑆 ∈ 𝐶 → 𝑊 ∈ dom ClSubSp) |
4 | cssval.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
5 | 4, 2 | iscss 21726 | . . 3 ⊢ (𝑊 ∈ dom ClSubSp → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
6 | 3, 5 | syl 17 | . 2 ⊢ (𝑆 ∈ 𝐶 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
7 | 6 | ibi 267 | 1 ⊢ (𝑆 ∈ 𝐶 → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2108 dom cdm 5700 ‘cfv 6575 ocvcocv 21703 ClSubSpccss 21704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-fv 6583 df-ov 7453 df-ocv 21706 df-css 21707 |
This theorem is referenced by: cssss 21728 cssincl 21731 csslss 21734 cssmre 21736 mrccss 21737 ocvpj 21762 csscld 25304 |
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