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Mirrors > Home > MPE Home > Th. List > iscss | Structured version Visualization version GIF version |
Description: The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
cssval.o | ⊢ ⊥ = (ocv‘𝑊) |
cssval.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
Ref | Expression |
---|---|
iscss | ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cssval.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
2 | cssval.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | 1, 2 | cssval 20598 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}) |
4 | 3 | eleq2d 2816 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 ∈ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})) |
5 | id 22 | . . . 4 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))) | |
6 | fvex 6708 | . . . 4 ⊢ ( ⊥ ‘( ⊥ ‘𝑆)) ∈ V | |
7 | 5, 6 | eqeltrdi 2839 | . . 3 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑆 ∈ V) |
8 | id 22 | . . . 4 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆) | |
9 | 2fveq3 6700 | . . . 4 ⊢ (𝑠 = 𝑆 → ( ⊥ ‘( ⊥ ‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑆))) | |
10 | 8, 9 | eqeq12d 2752 | . . 3 ⊢ (𝑠 = 𝑆 → (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
11 | 7, 10 | elab3 3584 | . 2 ⊢ (𝑆 ∈ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))) |
12 | 4, 11 | bitrdi 290 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 {cab 2714 Vcvv 3398 ‘cfv 6358 ocvcocv 20576 ClSubSpccss 20577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-ocv 20579 df-css 20580 |
This theorem is referenced by: cssi 20600 iscss2 20602 obslbs 20646 hlhillcs 39658 |
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