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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 21102 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}) |
4 | 3 | eleq2d 2820 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 ∈ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))})) |
5 | id 22 | . . . 4 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))) | |
6 | fvex 6856 | . . . 4 ⊢ ( ⊥ ‘( ⊥ ‘𝑆)) ∈ V | |
7 | 5, 6 | eqeltrdi 2842 | . . 3 ⊢ (𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)) → 𝑆 ∈ V) |
8 | id 22 | . . . 4 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆) | |
9 | 2fveq3 6848 | . . . 4 ⊢ (𝑠 = 𝑆 → ( ⊥ ‘( ⊥ ‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑆))) | |
10 | 8, 9 | eqeq12d 2749 | . . 3 ⊢ (𝑠 = 𝑆 → (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
11 | 7, 10 | elab3 3639 | . 2 ⊢ (𝑆 ∈ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))) |
12 | 4, 11 | bitrdi 287 | 1 ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 {cab 2710 Vcvv 3444 ‘cfv 6497 ocvcocv 21080 ClSubSpccss 21081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-ocv 21083 df-css 21084 |
This theorem is referenced by: cssi 21104 iscss2 21106 obslbs 21152 hlhillcs 40471 |
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