| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrval2 | Structured version Visualization version GIF version | ||
| Description: Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.) |
| Ref | Expression |
|---|---|
| lkrfval2.v | ⊢ 𝑉 = (Base‘𝑊) |
| lkrfval2.d | ⊢ 𝐷 = (Scalar‘𝑊) |
| lkrfval2.o | ⊢ 0 = (0g‘𝐷) |
| lkrfval2.f | ⊢ 𝐹 = (LFnl‘𝑊) |
| lkrfval2.k | ⊢ 𝐾 = (LKer‘𝑊) |
| Ref | Expression |
|---|---|
| lkrval2 | ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 }) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3468 | . 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) | |
| 2 | lkrfval2.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
| 3 | lkrfval2.d | . . . . 5 ⊢ 𝐷 = (Scalar‘𝑊) | |
| 4 | lkrfval2.o | . . . . 5 ⊢ 0 = (0g‘𝐷) | |
| 5 | lkrfval2.f | . . . . 5 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 6 | lkrfval2.k | . . . . 5 ⊢ 𝐾 = (LKer‘𝑊) | |
| 7 | 2, 3, 4, 5, 6 | ellkr 39082 | . . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝑥 ∈ (𝐾‘𝐺) ↔ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 ))) |
| 8 | 7 | eqabdv 2861 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )}) |
| 9 | df-rab 3406 | . . 3 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 } = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )} | |
| 10 | 8, 9 | eqtr4di 2782 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 }) |
| 11 | 1, 10 | sylan 580 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 }) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {cab 2707 {crab 3405 Vcvv 3447 ‘cfv 6511 Basecbs 17179 Scalarcsca 17223 0gc0g 17402 LFnlclfn 39050 LKerclk 39078 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-map 8801 df-lfl 39051 df-lkr 39079 |
| This theorem is referenced by: lkrlss 39088 |
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