Theorem lkrval2 39129

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 3457	. 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 39128	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝑥 ∈ (𝐾‘𝐺) ↔ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )))
8	7	eqabdv 2864	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )})
9		df-rab 3396	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 } = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝐺‘𝑥) = 0 )}
10	8, 9	eqtr4di 2784	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 })
11	1, 10	sylan 580	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 })

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  {crab 3395  Vcvv 3436  ‘cfv 6476  Basecbs 17115  Scalarcsca 17159  0_gc0g 17338  LFnlclfn 39096  LKerclk 39124

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-map 8747  df-lfl 39097  df-lkr 39125

This theorem is referenced by:  lkrlss  39134