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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 29476 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 35617 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
6 | 5 | eleq2d 2845 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ 𝑋 ∈ (◡𝐺 “ { 0 }))) |
7 | eqid 2772 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
8 | lkrfval2.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | 1, 7, 8, 3 | lflf 35592 | . . . 4 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝐷)) |
10 | ffn 6338 | . . . 4 ⊢ (𝐺:𝑉⟶(Base‘𝐷) → 𝐺 Fn 𝑉) | |
11 | elpreima 6647 | . . . 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 6506 | . . . . 5 ⊢ (𝐺‘𝑋) ∈ V | |
14 | 13 | elsn 4450 | . . . 4 ⊢ ((𝐺‘𝑋) ∈ { 0 } ↔ (𝐺‘𝑋) = 0 ) |
15 | 14 | anbi2i 613 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) ∈ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )) |
16 | 12, 15 | syl6bb 279 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (◡𝐺 “ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
17 | 6, 16 | bitrd 271 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 {csn 4435 ◡ccnv 5399 “ cima 5403 Fn wfn 6177 ⟶wf 6178 ‘cfv 6182 Basecbs 16329 Scalarcsca 16414 0gc0g 16559 LFnlclfn 35586 LKerclk 35614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-map 8200 df-lfl 35587 df-lkr 35615 |
This theorem is referenced by: lkrval2 35619 ellkr2 35620 lkrcl 35621 lkrf0 35622 lkrlss 35624 lkrsc 35626 eqlkr 35628 lkrlsp 35631 lkrlsp2 35632 lshpkr 35646 lkrin 35693 dochfln0 38006 |
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