| Mathbox for Norm Megill |
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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 3501 | . 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) | |
| 2 | lkrfval.k | . . 3 ⊢ 𝐾 = (LKer‘𝑊) | |
| 3 | fveq2 6906 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊)) | |
| 4 | lkrfval.f | . . . . . 6 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . 5 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹) |
| 6 | fveq2 6906 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊)) | |
| 7 | lkrfval.d | . . . . . . . . . 10 ⊢ 𝐷 = (Scalar‘𝑊) | |
| 8 | 6, 7 | eqtr4di 2795 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷) |
| 9 | 8 | fveq2d 6910 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g‘𝐷)) |
| 10 | lkrfval.o | . . . . . . . 8 ⊢ 0 = (0g‘𝐷) | |
| 11 | 9, 10 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 ) |
| 12 | 11 | sneqd 4638 | . . . . . 6 ⊢ (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 }) |
| 13 | 12 | imaeq2d 6078 | . . . . 5 ⊢ (𝑤 = 𝑊 → (◡𝑓 “ {(0g‘(Scalar‘𝑤))}) = (◡𝑓 “ { 0 })) |
| 14 | 5, 13 | mpteq12dv 5233 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))})) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
| 15 | df-lkr 39087 | . . . 4 ⊢ LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))}))) | |
| 16 | 14, 15, 4 | mptfvmpt 7248 | . . 3 ⊢ (𝑊 ∈ V → (LKer‘𝑊) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
| 17 | 2, 16 | eqtrid 2789 | . 2 ⊢ (𝑊 ∈ V → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
| 18 | 1, 17 | syl 17 | 1 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 Scalarcsca 17300 0gc0g 17484 LFnlclfn 39058 LKerclk 39086 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-lkr 39087 |
| This theorem is referenced by: lkrval 39089 |
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