![]() |
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 31957 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 39070 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
6 | 5 | eleq2d 2825 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ 𝑋 ∈ (◡𝐺 “ { 0 }))) |
7 | eqid 2735 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
8 | lkrfval2.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | 1, 7, 8, 3 | lflf 39045 | . . . 4 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝐷)) |
10 | ffn 6737 | . . . 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 6920 | . . . . 5 ⊢ (𝐺‘𝑋) ∈ V | |
14 | 13 | elsn 4646 | . . . 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 1537 ∈ wcel 2106 {csn 4631 ◡ccnv 5688 “ cima 5692 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 Basecbs 17245 Scalarcsca 17301 0gc0g 17486 LFnlclfn 39039 LKerclk 39067 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-lfl 39040 df-lkr 39068 |
This theorem is referenced by: lkrval2 39072 ellkr2 39073 lkrcl 39074 lkrf0 39075 lkrlss 39077 lkrsc 39079 eqlkr 39081 lkrlsp 39084 lkrlsp2 39085 lshpkr 39099 lkrin 39146 dochfln0 41460 |
Copyright terms: Public domain | W3C validator |