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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 35780 | . . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))) |
6 | 5 | fveq1d 6545 | . 2 ⊢ (𝑊 ∈ 𝑋 → (𝐾‘𝐺) = ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺)) |
7 | cnvexg 7490 | . . . 4 ⊢ (𝐺 ∈ 𝐹 → ◡𝐺 ∈ V) | |
8 | imaexg 7481 | . . . 4 ⊢ (◡𝐺 ∈ V → (◡𝐺 “ { 0 }) ∈ V) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐺 ∈ 𝐹 → (◡𝐺 “ { 0 }) ∈ V) |
10 | cnveq 5635 | . . . . 5 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺) | |
11 | 10 | imaeq1d 5810 | . . . 4 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ { 0 }) = (◡𝐺 “ { 0 })) |
12 | eqid 2795 | . . . 4 ⊢ (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) | |
13 | 11, 12 | fvmptg 6638 | . . 3 ⊢ ((𝐺 ∈ 𝐹 ∧ (◡𝐺 “ { 0 }) ∈ V) → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 })) |
14 | 9, 13 | mpdan 683 | . 2 ⊢ (𝐺 ∈ 𝐹 → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 })) |
15 | 6, 14 | sylan9eq 2851 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 {csn 4476 ↦ cmpt 5045 ◡ccnv 5447 “ cima 5451 ‘cfv 6230 Scalarcsca 16402 0gc0g 16547 LFnlclfn 35750 LKerclk 35778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-id 5353 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-lkr 35779 |
This theorem is referenced by: ellkr 35782 lkr0f 35787 |
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