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Mathbox for Norm Megill |
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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 34692 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
6 | 5 | fveq1d 6231 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝐾‘𝐺) = ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺)) |
7 | cnvexg 7154 | . . . 4 ⊢ (𝐺 ∈ 𝐹 → ◡𝐺 ∈ V) | |
8 | imaexg 7145 | . . . 4 ⊢ (◡𝐺 ∈ V → (◡𝐺 “ { 0 }) ∈ V) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐺 ∈ 𝐹 → (◡𝐺 “ { 0 }) ∈ V) |
10 | cnveq 5328 | . . . . 5 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺) | |
11 | 10 | imaeq1d 5500 | . . . 4 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ { 0 }) = (◡𝐺 “ { 0 })) |
12 | eqid 2651 | . . . 4 ⊢ (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) | |
13 | 11, 12 | fvmptg 6319 | . . 3 ⊢ ((𝐺 ∈ 𝐹 ∧ (◡𝐺 “ { 0 }) ∈ V) → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 })) |
14 | 9, 13 | mpdan 703 | . 2 ⊢ (𝐺 ∈ 𝐹 → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 })) |
15 | 6, 14 | sylan9eq 2705 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 {csn 4210 ↦ cmpt 4762 ◡ccnv 5142 “ cima 5146 ‘cfv 5926 Scalarcsca 15991 0gc0g 16147 LFnlclfn 34662 LKerclk 34690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-lkr 34691 |
This theorem is referenced by: ellkr 34694 lkr0f 34699 |
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