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Mirrors > Home > MPE Home > Th. List > ocv0 | Structured version Visualization version GIF version |
Description: The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
ocvz.v | ⊢ 𝑉 = (Base‘𝑊) |
ocvz.o | ⊢ ⊥ = (ocv‘𝑊) |
Ref | Expression |
---|---|
ocv0 | ⊢ ( ⊥ ‘∅) = 𝑉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4406 | . . 3 ⊢ ∅ ⊆ 𝑉 | |
2 | ocvz.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | eqid 2735 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
4 | eqid 2735 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2735 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
6 | ocvz.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
7 | 2, 3, 4, 5, 6 | ocvval 21703 | . . 3 ⊢ (∅ ⊆ 𝑉 → ( ⊥ ‘∅) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}) |
8 | 1, 7 | ax-mp 5 | . 2 ⊢ ( ⊥ ‘∅) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} |
9 | ral0 4519 | . . . 4 ⊢ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) | |
10 | 9 | rgenw 3063 | . . 3 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) |
11 | rabid2 3468 | . . 3 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) | |
12 | 10, 11 | mpbir 231 | . 2 ⊢ 𝑉 = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} |
13 | 8, 12 | eqtr4i 2766 | 1 ⊢ ( ⊥ ‘∅) = 𝑉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∀wral 3059 {crab 3433 ⊆ wss 3963 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Scalarcsca 17301 ·𝑖cip 17303 0gc0g 17486 ocvcocv 21696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-ocv 21699 |
This theorem is referenced by: ocvz 21714 css1 21726 |
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