![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4388 | . . 3 ⊢ ∅ ⊆ 𝑉 | |
2 | ocvz.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | eqid 2724 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
4 | eqid 2724 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2724 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
6 | ocvz.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
7 | 2, 3, 4, 5, 6 | ocvval 21528 | . . 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 4504 | . . . 4 ⊢ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) | |
10 | 9 | rgenw 3057 | . . 3 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) |
11 | rabid2 3456 | . . 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 230 | . 2 ⊢ 𝑉 = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} |
13 | 8, 12 | eqtr4i 2755 | 1 ⊢ ( ⊥ ‘∅) = 𝑉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∀wral 3053 {crab 3424 ⊆ wss 3940 ∅c0 4314 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 Scalarcsca 17199 ·𝑖cip 17201 0gc0g 17384 ocvcocv 21521 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-ocv 21524 |
This theorem is referenced by: ocvz 21539 css1 21551 |
Copyright terms: Public domain | W3C validator |