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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 6929 | . . . 4 ⊢ (𝑆 ∈ (ClSubSp‘𝑊) → 𝑊 ∈ dom ClSubSp) | |
2 | cssval.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | 1, 2 | eleq2s 2847 | . . 3 ⊢ (𝑆 ∈ 𝐶 → 𝑊 ∈ dom ClSubSp) |
4 | cssval.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
5 | 4, 2 | iscss 21609 | . . 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 205 = wceq 1534 ∈ wcel 2099 dom cdm 5673 ‘cfv 6543 ocvcocv 21586 ClSubSpccss 21587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7418 df-ocv 21589 df-css 21590 |
This theorem is referenced by: cssss 21611 cssincl 21614 csslss 21617 cssmre 21619 mrccss 21620 ocvpj 21645 csscld 25171 |
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