![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6956 | . . . 4 ⊢ (𝑆 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ dom ClSubSp) | |
2 | cssval.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | 1, 2 | eleq2s 2856 | . . 3 ⊢ (𝑆 ∈ 𝐶 → 𝑊 ∈ dom ClSubSp) |
4 | cssval.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
5 | 4, 2 | iscss 21719 | . . 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 2103 dom cdm 5699 ‘cfv 6572 ocvcocv 21696 ClSubSpccss 21697 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-fv 6580 df-ov 7448 df-ocv 21699 df-css 21700 |
This theorem is referenced by: cssss 21721 cssincl 21724 csslss 21727 cssmre 21729 mrccss 21730 ocvpj 21755 csscld 25295 |
Copyright terms: Public domain | W3C validator |