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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 = (CSubSp‘𝑊) |
Ref | Expression |
---|---|
cssi | ⊢ (𝑆 ∈ 𝐶 → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6130 | . . . 4 ⊢ (𝑆 ∈ (CSubSp‘𝑊) → 𝑊 ∈ dom CSubSp) | |
2 | cssval.c | . . . 4 ⊢ 𝐶 = (CSubSp‘𝑊) | |
3 | 1, 2 | eleq2s 2706 | . . 3 ⊢ (𝑆 ∈ 𝐶 → 𝑊 ∈ dom CSubSp) |
4 | cssval.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
5 | 4, 2 | iscss 19846 | . . 3 ⊢ (𝑊 ∈ dom CSubSp → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
6 | 3, 5 | syl 17 | . 2 ⊢ (𝑆 ∈ 𝐶 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
7 | 6 | ibi 255 | 1 ⊢ (𝑆 ∈ 𝐶 → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 dom cdm 5038 ‘cfv 5804 ocvcocv 19823 CSubSpccss 19824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-ocv 19826 df-css 19827 |
This theorem is referenced by: cssss 19848 cssincl 19851 csslss 19854 cssmre 19856 mrccss 19857 ocvpj 19880 csscld 22856 |
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