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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 2823 | . . . 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 36227 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ (Base‘𝑊) ∧ (𝐺‘𝑋) = 0 ))) |
7 | 6 | simplbda 502 | . 2 ⊢ (((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) ∧ 𝑋 ∈ (𝐾‘𝐺)) → (𝐺‘𝑋) = 0 ) |
8 | 7 | 3impa 1106 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ (𝐾‘𝐺)) → (𝐺‘𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 Basecbs 16485 Scalarcsca 16570 0gc0g 16715 LFnlclfn 36195 LKerclk 36223 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-lfl 36196 df-lkr 36224 |
This theorem is referenced by: lkrlss 36233 lkrshp 36243 lkrin 36302 lcfrlem12N 38692 |
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