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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ellkr | Structured version Visualization version GIF version | ||
| Description: Membership in the kernel of a functional. (elnlfn 31947 analog.) (Contributed by NM, 16-Apr-2014.) | 
| 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) | 
| Ref | Expression | 
|---|---|
| ellkr | ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lkrfval2.d | . . . 4 ⊢ 𝐷 = (Scalar‘𝑊) | |
| 2 | lkrfval2.o | . . . 4 ⊢ 0 = (0g‘𝐷) | |
| 3 | lkrfval2.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 4 | lkrfval2.k | . . . 4 ⊢ 𝐾 = (LKer‘𝑊) | |
| 5 | 1, 2, 3, 4 | lkrval 39089 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) | 
| 6 | 5 | eleq2d 2827 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ 𝑋 ∈ (◡𝐺 “ { 0 }))) | 
| 7 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
| 8 | lkrfval2.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
| 9 | 1, 7, 8, 3 | lflf 39064 | . . . 4 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝐷)) | 
| 10 | ffn 6736 | . . . 4 ⊢ (𝐺:𝑉⟶(Base‘𝐷) → 𝐺 Fn 𝑉) | |
| 11 | elpreima 7078 | . . . 4 ⊢ (𝐺 Fn 𝑉 → (𝑋 ∈ (◡𝐺 “ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) ∈ { 0 }))) | |
| 12 | 9, 10, 11 | 3syl 18 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (◡𝐺 “ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) ∈ { 0 }))) | 
| 13 | fvex 6919 | . . . . 5 ⊢ (𝐺‘𝑋) ∈ V | |
| 14 | 13 | elsn 4641 | . . . 4 ⊢ ((𝐺‘𝑋) ∈ { 0 } ↔ (𝐺‘𝑋) = 0 ) | 
| 15 | 14 | anbi2i 623 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) ∈ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )) | 
| 16 | 12, 15 | bitrdi 287 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (◡𝐺 “ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) | 
| 17 | 6, 16 | bitrd 279 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 ◡ccnv 5684 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 Basecbs 17247 Scalarcsca 17300 0gc0g 17484 LFnlclfn 39058 LKerclk 39086 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-lfl 39059 df-lkr 39087 | 
| This theorem is referenced by: lkrval2 39091 ellkr2 39092 lkrcl 39093 lkrf0 39094 lkrlss 39096 lkrsc 39098 eqlkr 39100 lkrlsp 39103 lkrlsp2 39104 lshpkr 39118 lkrin 39165 dochfln0 41479 | 
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