![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrval | Structured version Visualization version GIF version |
Description: Value of the kernel of a functional. (Contributed by NM, 15-Apr-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 |
---|---|
lkrval | ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkrfval.d | . . . 4 ⊢ 𝐷 = (Scalar‘𝑊) | |
2 | lkrfval.o | . . . 4 ⊢ 0 = (0g‘𝐷) | |
3 | lkrfval.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
4 | lkrfval.k | . . . 4 ⊢ 𝐾 = (LKer‘𝑊) | |
5 | 1, 2, 3, 4 | lkrfval 39068 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
6 | 5 | fveq1d 6908 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝐾‘𝐺) = ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺)) |
7 | cnvexg 7946 | . . . 4 ⊢ (𝐺 ∈ 𝐹 → ◡𝐺 ∈ V) | |
8 | imaexg 7935 | . . . 4 ⊢ (◡𝐺 ∈ V → (◡𝐺 “ { 0 }) ∈ V) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐺 ∈ 𝐹 → (◡𝐺 “ { 0 }) ∈ V) |
10 | cnveq 5886 | . . . . 5 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺) | |
11 | 10 | imaeq1d 6078 | . . . 4 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ { 0 }) = (◡𝐺 “ { 0 })) |
12 | eqid 2734 | . . . 4 ⊢ (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) | |
13 | 11, 12 | fvmptg 7013 | . . 3 ⊢ ((𝐺 ∈ 𝐹 ∧ (◡𝐺 “ { 0 }) ∈ V) → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 })) |
14 | 9, 13 | mpdan 687 | . 2 ⊢ (𝐺 ∈ 𝐹 → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 })) |
15 | 6, 14 | sylan9eq 2794 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 {csn 4630 ↦ cmpt 5230 ◡ccnv 5687 “ cima 5691 ‘cfv 6562 Scalarcsca 17300 0gc0g 17485 LFnlclfn 39038 LKerclk 39066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-lkr 39067 |
This theorem is referenced by: ellkr 39070 lkr0f 39075 |
Copyright terms: Public domain | W3C validator |