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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 3426 | . 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) | |
2 | cssval.c | . . 3 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | fveq2 6717 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊)) | |
4 | cssval.o | . . . . . . . 8 ⊢ ⊥ = (ocv‘𝑊) | |
5 | 3, 4 | eqtr4di 2796 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (ocv‘𝑤) = ⊥ ) |
6 | 5 | fveq1d 6719 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑠) = ( ⊥ ‘𝑠)) |
7 | 5, 6 | fveq12d 6724 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) = ( ⊥ ‘( ⊥ ‘𝑠))) |
8 | 7 | eqeq2d 2748 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠)) ↔ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)))) |
9 | 8 | abbidv 2807 | . . . 4 ⊢ (𝑤 = 𝑊 → {𝑠 ∣ 𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))} = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}) |
10 | df-css 20626 | . . . 4 ⊢ ClSubSp = (𝑤 ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘𝑤)‘((ocv‘𝑤)‘𝑠))}) | |
11 | fvex 6730 | . . . . . 6 ⊢ (Base‘𝑊) ∈ V | |
12 | 11 | pwex 5273 | . . . . 5 ⊢ 𝒫 (Base‘𝑊) ∈ V |
13 | id 22 | . . . . . . 7 ⊢ (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) → 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))) | |
14 | eqid 2737 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
15 | 14, 4 | ocvss 20632 | . . . . . . . 8 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ⊆ (Base‘𝑊) |
16 | fvex 6730 | . . . . . . . . 9 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ∈ V | |
17 | 16 | elpw 4517 | . . . . . . . 8 ⊢ (( ⊥ ‘( ⊥ ‘𝑠)) ∈ 𝒫 (Base‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑠)) ⊆ (Base‘𝑊)) |
18 | 15, 17 | mpbir 234 | . . . . . . 7 ⊢ ( ⊥ ‘( ⊥ ‘𝑠)) ∈ 𝒫 (Base‘𝑊) |
19 | 13, 18 | eqeltrdi 2846 | . . . . . 6 ⊢ (𝑠 = ( ⊥ ‘( ⊥ ‘𝑠)) → 𝑠 ∈ 𝒫 (Base‘𝑊)) |
20 | 19 | abssi 3983 | . . . . 5 ⊢ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ⊆ 𝒫 (Base‘𝑊) |
21 | 12, 20 | ssexi 5215 | . . . 4 ⊢ {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))} ∈ V |
22 | 9, 10, 21 | fvmpt 6818 | . . 3 ⊢ (𝑊 ∈ V → (ClSubSp‘𝑊) = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}) |
23 | 2, 22 | syl5eq 2790 | . 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 1543 ∈ wcel 2110 {cab 2714 Vcvv 3408 ⊆ wss 3866 𝒫 cpw 4513 ‘cfv 6380 Basecbs 16760 ocvcocv 20622 ClSubSpccss 20623 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-ov 7216 df-ocv 20625 df-css 20626 |
This theorem is referenced by: iscss 20645 |
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