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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ellkr2 | Structured version Visualization version GIF version | ||
| Description: Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| lkrfval2.v | ⊢ 𝑉 = (Base‘𝑊) | 
| lkrfval2.d | ⊢ 𝐷 = (Scalar‘𝑊) | 
| lkrfval2.o | ⊢ 0 = (0g‘𝐷) | 
| lkrfval2.f | ⊢ 𝐹 = (LFnl‘𝑊) | 
| lkrfval2.k | ⊢ 𝐾 = (LKer‘𝑊) | 
| ellkr2.w | ⊢ (𝜑 → 𝑊 ∈ 𝑌) | 
| ellkr2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) | 
| ellkr2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) | 
| Ref | Expression | 
|---|---|
| ellkr2 | ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 )) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ellkr2.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑌) | |
| 2 | ellkr2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
| 3 | lkrfval2.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
| 4 | lkrfval2.d | . . . 4 ⊢ 𝐷 = (Scalar‘𝑊) | |
| 5 | lkrfval2.o | . . . 4 ⊢ 0 = (0g‘𝐷) | |
| 6 | lkrfval2.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 7 | lkrfval2.k | . . . 4 ⊢ 𝐾 = (LKer‘𝑊) | |
| 8 | 3, 4, 5, 6, 7 | ellkr 39091 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) | 
| 9 | 1, 2, 8 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) | 
| 10 | ellkr2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 11 | 10 | biantrurd 532 | . 2 ⊢ (𝜑 → ((𝐺‘𝑋) = 0 ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) | 
| 12 | 9, 11 | bitr4d 282 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 Basecbs 17248 Scalarcsca 17301 0gc0g 17485 LFnlclfn 39059 LKerclk 39087 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-lfl 39060 df-lkr 39088 | 
| This theorem is referenced by: lclkrlem2f 41515 lclkrlem2n 41523 lcfrlem3 41547 lcfrlem25 41570 hdmapellkr 41917 hdmapip0 41918 hdmapinvlem1 41921 | 
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