Theorem lkrfval 34374

Description: The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
lkrfval.d	⊢ 𝐷 = (Scalar‘𝑊)
lkrfval.o	⊢ 0 = (0g‘𝐷)
lkrfval.f	⊢ 𝐹 = (LFnl‘𝑊)
lkrfval.k	⊢ 𝐾 = (LKer‘𝑊)

Assertion

Ref	Expression
lkrfval	⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))

Distinct variable groups:   𝑓,𝐹   𝑓,𝑊

Allowed substitution hints:   𝐷(𝑓)   𝐾(𝑓)   𝑋(𝑓)   0 (𝑓)

Proof of Theorem lkrfval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		lkrfval.k	. . 3 ⊢ 𝐾 = (LKer‘𝑊)
3		fveq2 6191	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
4		lkrfval.f	. . . . . 6 ⊢ 𝐹 = (LFnl‘𝑊)
5	3, 4	syl6eqr 2674	. . . . 5 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
6		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
7		lkrfval.d	. . . . . . . . . 10 ⊢ 𝐷 = (Scalar‘𝑊)
8	6, 7	syl6eqr 2674	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
9	8	fveq2d 6195	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g‘𝐷))
10		lkrfval.o	. . . . . . . 8 ⊢ 0 = (0g‘𝐷)
11	9, 10	syl6eqr 2674	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
12	11	sneqd 4189	. . . . . 6 ⊢ (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
13	12	imaeq2d 5466	. . . . 5 ⊢ (𝑤 = 𝑊 → (◡𝑓 “ {(0g‘(Scalar‘𝑤))}) = (◡𝑓 “ { 0 }))
14	5, 13	mpteq12dv 4733	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))})) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
15		df-lkr 34373	. . . 4 ⊢ LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))})))
16		fvex 6201	. . . . . 6 ⊢ (LFnl‘𝑊) ∈ V
17	4, 16	eqeltri 2697	. . . . 5 ⊢ 𝐹 ∈ V
18	17	mptex 6486	. . . 4 ⊢ (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) ∈ V
19	14, 15, 18	fvmpt 6282	. . 3 ⊢ (𝑊 ∈ V → (LKer‘𝑊) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
20	2, 19	syl5eq 2668	. 2 ⊢ (𝑊 ∈ V → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
21	1, 20	syl 17	1 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177   ↦ cmpt 4729  ^◡ccnv 5113   “ cima 5117  ‘cfv 5888  Scalarcsca 15944  0_gc0g 16100  LFnlclfn 34344  LKerclk 34372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-lkr 34373

This theorem is referenced by:  lkrval  34375