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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 36840 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ (Base‘𝑊) ∧ (𝐺‘𝑋) = 0 ))) |
7 | 6 | simplbda 503 | . 2 ⊢ (((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) ∧ 𝑋 ∈ (𝐾‘𝐺)) → (𝐺‘𝑋) = 0 ) |
8 | 7 | 3impa 1112 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ (𝐾‘𝐺)) → (𝐺‘𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 Basecbs 16760 Scalarcsca 16805 0gc0g 16944 LFnlclfn 36808 LKerclk 36836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-map 8510 df-lfl 36809 df-lkr 36837 |
This theorem is referenced by: lkrlss 36846 lkrshp 36856 lkrin 36915 lcfrlem12N 39305 |
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