![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrf0 | Structured version Visualization version GIF version |
Description: The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.) |
Ref | Expression |
---|---|
lkrf0.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lkrf0.o | ⊢ 0 = (0g‘𝐷) |
lkrf0.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrf0.k | ⊢ 𝐾 = (LKer‘𝑊) |
Ref | Expression |
---|---|
lkrf0 | ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ (𝐾‘𝐺)) → (𝐺‘𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lkrf0.d | . . . 4 ⊢ 𝐷 = (Scalar‘𝑊) | |
3 | lkrf0.o | . . . 4 ⊢ 0 = (0g‘𝐷) | |
4 | lkrf0.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | lkrf0.k | . . . 4 ⊢ 𝐾 = (LKer‘𝑊) | |
6 | 1, 2, 3, 4, 5 | ellkr 37482 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ (Base‘𝑊) ∧ (𝐺‘𝑋) = 0 ))) |
7 | 6 | simplbda 500 | . 2 ⊢ (((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) ∧ 𝑋 ∈ (𝐾‘𝐺)) → (𝐺‘𝑋) = 0 ) |
8 | 7 | 3impa 1110 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ (𝐾‘𝐺)) → (𝐺‘𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 Basecbs 17042 Scalarcsca 17095 0gc0g 17280 LFnlclfn 37450 LKerclk 37478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-map 8725 df-lfl 37451 df-lkr 37479 |
This theorem is referenced by: lkrlss 37488 lkrshp 37498 lkrin 37557 lcfrlem12N 39948 |
Copyright terms: Public domain | W3C validator |