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Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrfval | Structured version Visualization version GIF version |
Description: The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
lkrfval.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lkrfval.o | ⊢ 0 = (0g‘𝐷) |
lkrfval.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrfval.k | ⊢ 𝐾 = (LKer‘𝑊) |
Ref | Expression |
---|---|
lkrfval | ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) | |
2 | lkrfval.k | . . 3 ⊢ 𝐾 = (LKer‘𝑊) | |
3 | fveq2 6103 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊)) | |
4 | lkrfval.f | . . . . . 6 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | 3, 4 | syl6eqr 2662 | . . . . 5 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹) |
6 | fveq2 6103 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊)) | |
7 | lkrfval.d | . . . . . . . . . 10 ⊢ 𝐷 = (Scalar‘𝑊) | |
8 | 6, 7 | syl6eqr 2662 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷) |
9 | 8 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g‘𝐷)) |
10 | lkrfval.o | . . . . . . . 8 ⊢ 0 = (0g‘𝐷) | |
11 | 9, 10 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 ) |
12 | 11 | sneqd 4137 | . . . . . 6 ⊢ (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 }) |
13 | 12 | imaeq2d 5385 | . . . . 5 ⊢ (𝑤 = 𝑊 → (◡𝑓 “ {(0g‘(Scalar‘𝑤))}) = (◡𝑓 “ { 0 })) |
14 | 5, 13 | mpteq12dv 4663 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))})) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
15 | df-lkr 33391 | . . . 4 ⊢ LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))}))) | |
16 | fvex 6113 | . . . . . 6 ⊢ (LFnl‘𝑊) ∈ V | |
17 | 4, 16 | eqeltri 2684 | . . . . 5 ⊢ 𝐹 ∈ V |
18 | 17 | mptex 6390 | . . . 4 ⊢ (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) ∈ V |
19 | 14, 15, 18 | fvmpt 6191 | . . 3 ⊢ (𝑊 ∈ V → (LKer‘𝑊) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
20 | 2, 19 | syl5eq 2656 | . 2 ⊢ (𝑊 ∈ V → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
21 | 1, 20 | syl 17 | 1 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 ‘cfv 5804 Scalarcsca 15771 0gc0g 15923 LFnlclfn 33362 LKerclk 33390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-lkr 33391 |
This theorem is referenced by: lkrval 33393 |
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