Theorem lkrval2 36228

Description: Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)

Hypotheses

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)

Assertion

Ref	Expression
lkrval2	⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 })

Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝐾   𝑥,𝑊

Allowed substitution hints:   𝐷(𝑥)   𝑉(𝑥)   𝑋(𝑥)   0 (𝑥)

Proof of Theorem lkrval2

Step	Hyp	Ref	Expression
1		elex 3514	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		lkrfval2.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
3		lkrfval2.d	. . . . 5 ⊢ 𝐷 = (Scalar‘𝑊)
4		lkrfval2.o	. . . . 5 ⊢ 0 = (0g‘𝐷)
5		lkrfval2.f	. . . . 5 ⊢ 𝐹 = (LFnl‘𝑊)
6		lkrfval2.k	. . . . 5 ⊢ 𝐾 = (LKer‘𝑊)
7	2, 3, 4, 5, 6	ellkr 36227	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝑥 ∈ (𝐾‘𝐺) ↔ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )))
8	7	abbi2dv 2952	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )})
9		df-rab 3149	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 } = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )}
10	8, 9	syl6eqr 2876	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 })
11	1, 10	sylan 582	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 })

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2801  {crab 3144  Vcvv 3496  ‘cfv 6357  Basecbs 16485  Scalarcsca 16570  0_gc0g 16715  LFnlclfn 36195  LKerclk 36223

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 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-lfl 36196  df-lkr 36224

This theorem is referenced by:  lkrlss  36233