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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 3924 | . . 3 ⊢ ∅ ⊆ 𝑉 | |
2 | ocvz.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | eqid 2610 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
4 | eqid 2610 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2610 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
6 | ocvz.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
7 | 2, 3, 4, 5, 6 | ocvval 19830 | . . 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 4028 | . . . 4 ⊢ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) | |
10 | 9 | rgenw 2908 | . . 3 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) |
11 | rabid2 3096 | . . 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 220 | . 2 ⊢ 𝑉 = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} |
13 | 8, 12 | eqtr4i 2635 | 1 ⊢ ( ⊥ ‘∅) = 𝑉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∀wral 2896 {crab 2900 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑖cip 15773 0gc0g 15923 ocvcocv 19823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-ocv 19826 |
This theorem is referenced by: ocvz 19841 css1 19853 |
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