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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 36838 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
6 | 5 | fveq1d 6719 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝐾‘𝐺) = ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺)) |
7 | cnvexg 7702 | . . . 4 ⊢ (𝐺 ∈ 𝐹 → ◡𝐺 ∈ V) | |
8 | imaexg 7693 | . . . 4 ⊢ (◡𝐺 ∈ V → (◡𝐺 “ { 0 }) ∈ V) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐺 ∈ 𝐹 → (◡𝐺 “ { 0 }) ∈ V) |
10 | cnveq 5742 | . . . . 5 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺) | |
11 | 10 | imaeq1d 5928 | . . . 4 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ { 0 }) = (◡𝐺 “ { 0 })) |
12 | eqid 2737 | . . . 4 ⊢ (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) | |
13 | 11, 12 | fvmptg 6816 | . . 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 2798 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 {csn 4541 ↦ cmpt 5135 ◡ccnv 5550 “ cima 5554 ‘cfv 6380 Scalarcsca 16805 0gc0g 16944 LFnlclfn 36808 LKerclk 36836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-lkr 36837 |
This theorem is referenced by: ellkr 36840 lkr0f 36845 |
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