Theorem lkrval2 37031

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 3440	. 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 37030	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝑥 ∈ (𝐾‘𝐺) ↔ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )))
8	7	abbi2dv 2876	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )})
9		df-rab 3072	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 } = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )}
10	8, 9	eqtr4di 2797	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 })
11	1, 10	sylan 579	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 })

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  {crab 3067  Vcvv 3422  ‘cfv 6418  Basecbs 16840  Scalarcsca 16891  0_gc0g 17067  LFnlclfn 36998  LKerclk 37026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-lfl 36999  df-lkr 37027

This theorem is referenced by:  lkrlss  37036