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Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrcl | Structured version Visualization version GIF version |
Description: A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.) |
Ref | Expression |
---|---|
lkrcl.v | ⊢ 𝑉 = (Base‘𝑊) |
lkrcl.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrcl.k | ⊢ 𝐾 = (LKer‘𝑊) |
Ref | Expression |
---|---|
lkrcl | ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ (𝐾‘𝐺)) → 𝑋 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkrcl.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | eqid 2818 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2818 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
4 | lkrcl.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | lkrcl.k | . . . 4 ⊢ 𝐾 = (LKer‘𝑊) | |
6 | 1, 2, 3, 4, 5 | ellkr 36105 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = (0g‘(Scalar‘𝑊))))) |
7 | 6 | simprbda 499 | . 2 ⊢ (((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) ∧ 𝑋 ∈ (𝐾‘𝐺)) → 𝑋 ∈ 𝑉) |
8 | 7 | 3impa 1102 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ (𝐾‘𝐺)) → 𝑋 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 Basecbs 16471 Scalarcsca 16556 0gc0g 16701 LFnlclfn 36073 LKerclk 36101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-lfl 36074 df-lkr 36102 |
This theorem is referenced by: lkrlss 36111 lkrin 36180 |
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