Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ellkr | Structured version Visualization version GIF version |
Description: Membership in the kernel of a functional. (elnlfn 29699 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 36218 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
6 | 5 | eleq2d 2898 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ 𝑋 ∈ (◡𝐺 “ { 0 }))) |
7 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
8 | lkrfval2.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | 1, 7, 8, 3 | lflf 36193 | . . . 4 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝐷)) |
10 | ffn 6509 | . . . 4 ⊢ (𝐺:𝑉⟶(Base‘𝐷) → 𝐺 Fn 𝑉) | |
11 | elpreima 6823 | . . . 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 6678 | . . . . 5 ⊢ (𝐺‘𝑋) ∈ V | |
14 | 13 | elsn 4576 | . . . 4 ⊢ ((𝐺‘𝑋) ∈ { 0 } ↔ (𝐺‘𝑋) = 0 ) |
15 | 14 | anbi2i 624 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) ∈ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )) |
16 | 12, 15 | syl6bb 289 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (◡𝐺 “ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
17 | 6, 16 | bitrd 281 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {csn 4561 ◡ccnv 5549 “ cima 5553 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 Basecbs 16477 Scalarcsca 16562 0gc0g 16707 LFnlclfn 36187 LKerclk 36215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-lfl 36188 df-lkr 36216 |
This theorem is referenced by: lkrval2 36220 ellkr2 36221 lkrcl 36222 lkrf0 36223 lkrlss 36225 lkrsc 36227 eqlkr 36229 lkrlsp 36232 lkrlsp2 36233 lshpkr 36247 lkrin 36294 dochfln0 38607 |
Copyright terms: Public domain | W3C validator |