![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cssval | Structured version Visualization version GIF version |
Description: The set of closed subspaces 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 |
---|---|
cssval | ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3480 | . 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) | |
2 | cssval.c | . . 3 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | fveq2 6896 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊)) | |
4 | cssval.o | . . . . . . . 8 ⊢ ⊥ = (ocv‘𝑊) | |
5 | 3, 4 | eqtr4di 2783 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = ⊥ ) |
6 | 5 | fveq1d 6898 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑠) = ( ⊥ ‘𝑠)) |
7 | 5, 6 | fveq12d 6903 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑠))) |
8 | 7 | eqeq2d 2736 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) ↔ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)))) |
9 | 8 | abbidv 2794 | . . . 4 ⊢ (𝑤 = 𝑊 → {𝑠 ∣ 𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))} = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}) |
10 | df-css 21613 | . . . 4 ⊢ ClSubSp = (𝑤 ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))}) | |
11 | fvex 6909 | . . . . . 6 ⊢ (Base‘𝑊) ∈ V | |
12 | 11 | pwex 5380 | . . . . 5 ⊢ 𝒫 (Base‘𝑊) ∈ V |
13 | id 22 | . . . . . . 7 ⊢ (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) → 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))) | |
14 | eqid 2725 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
15 | 14, 4 | ocvss 21619 | . . . . . . . 8 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ⊆ (Base‘𝑊) |
16 | fvex 6909 | . . . . . . . . 9 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ∈ V | |
17 | 16 | elpw 4608 | . . . . . . . 8 ⊢ (( ⊥ ‘( ⊥ ‘𝑠)) ∈ 𝒫 (Base‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑠)) ⊆ (Base‘𝑊)) |
18 | 15, 17 | mpbir 230 | . . . . . . 7 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ∈ 𝒫 (Base‘𝑊) |
19 | 13, 18 | eqeltrdi 2833 | . . . . . 6 ⊢ (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) → 𝑠 ∈ 𝒫 (Base‘𝑊)) |
20 | 19 | abssi 4063 | . . . . 5 ⊢ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ⊆ 𝒫 (Base‘𝑊) |
21 | 12, 20 | ssexi 5323 | . . . 4 ⊢ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ∈ V |
22 | 9, 10, 21 | fvmpt 7004 | . . 3 ⊢ (𝑊 ∈ V → (ClSubSp‘𝑊) = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}) |
23 | 2, 22 | eqtrid 2777 | . 2 ⊢ (𝑊 ∈ V → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}) |
24 | 1, 23 | syl 17 | 1 ⊢ (𝑊 ∈ 𝑋 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {cab 2702 Vcvv 3461 ⊆ wss 3944 𝒫 cpw 4604 ‘cfv 6549 Basecbs 17183 ocvcocv 21609 ClSubSpccss 21610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-ocv 21612 df-css 21613 |
This theorem is referenced by: iscss 21632 |
Copyright terms: Public domain | W3C validator |