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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 29705 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 36239 | . . 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 36214 | . . . 4 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝐷)) |
10 | ffn 6514 | . . . 4 ⊢ (𝐺:𝑉⟶(Base‘𝐷) → 𝐺 Fn 𝑉) | |
11 | elpreima 6828 | . . . 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 6683 | . . . . 5 ⊢ (𝐺‘𝑋) ∈ V | |
14 | 13 | elsn 4582 | . . . 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 1537 ∈ wcel 2114 {csn 4567 ◡ccnv 5554 “ cima 5558 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 Basecbs 16483 Scalarcsca 16568 0gc0g 16713 LFnlclfn 36208 LKerclk 36236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-lfl 36209 df-lkr 36237 |
This theorem is referenced by: lkrval2 36241 ellkr2 36242 lkrcl 36243 lkrf0 36244 lkrlss 36246 lkrsc 36248 eqlkr 36250 lkrlsp 36253 lkrlsp2 36254 lshpkr 36268 lkrin 36315 dochfln0 38628 |
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