Theorem lkrval2 39550

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 3451	. 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 39549	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝑥 ∈ (𝐾‘𝐺) ↔ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )))
8	7	eqabdv 2870	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )})
9		df-rab 3391	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 } = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )}
10	8, 9	eqtr4di 2790	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 })
11	1, 10	sylan 581	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 })

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  {crab 3390  Vcvv 3430  ‘cfv 6492  Basecbs 17170  Scalarcsca 17214  0_gc0g 17393  LFnlclfn 39517  LKerclk 39545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8768  df-lfl 39518  df-lkr 39546

This theorem is referenced by:  lkrlss  39555