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Mirrors > Home > MPE Home > Th. List > ocv2ss | Structured version Visualization version GIF version |
Description: Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
ocv2ss.o | ⊢ ⊥ = (ocv‘𝑊) |
Ref | Expression |
---|---|
ocv2ss | ⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3974 | . . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ (Base‘𝑊) → 𝑇 ⊆ (Base‘𝑊))) | |
2 | idd 24 | . . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝑥 ∈ (Base‘𝑊) → 𝑥 ∈ (Base‘𝑊))) | |
3 | ssralv 4033 | . . . 4 ⊢ (𝑇 ⊆ 𝑆 → (∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) → ∀𝑦 ∈ 𝑇 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) | |
4 | 1, 2, 3 | 3anim123d 1439 | . . 3 ⊢ (𝑇 ⊆ 𝑆 → ((𝑆 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) → (𝑇 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝑇 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))) |
5 | eqid 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
6 | eqid 2821 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
7 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
8 | eqid 2821 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
9 | ocv2ss.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
10 | 5, 6, 7, 8, 9 | elocv 20812 | . . 3 ⊢ (𝑥 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) |
11 | 5, 6, 7, 8, 9 | elocv 20812 | . . 3 ⊢ (𝑥 ∈ ( ⊥ ‘𝑇) ↔ (𝑇 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝑇 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) |
12 | 4, 10, 11 | 3imtr4g 298 | . 2 ⊢ (𝑇 ⊆ 𝑆 → (𝑥 ∈ ( ⊥ ‘𝑆) → 𝑥 ∈ ( ⊥ ‘𝑇))) |
13 | 12 | ssrdv 3973 | 1 ⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Scalarcsca 16568 ·𝑖cip 16570 0gc0g 16713 ocvcocv 20804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-ocv 20807 |
This theorem is referenced by: ocvsscon 20819 ocvlsp 20820 ocvcss 20831 cssmre 20837 mrccss 20838 clsocv 23853 |
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