Theorem lkrfval 39902

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 3479	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		lkrfval.k	. . 3 ⊢ 𝐾 = (LKer‘𝑊)
3		fveq2 6888	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
4		lkrfval.f	. . . . . 6 ⊢ 𝐹 = (LFnl‘𝑊)
5	3, 4	eqtr4di 2819	. . . . 5 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
6		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
7		lkrfval.d	. . . . . . . . . 10 ⊢ 𝐷 = (Scalar‘𝑊)
8	6, 7	eqtr4di 2819	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
9	8	fveq2d 6892	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g‘𝐷))
10		lkrfval.o	. . . . . . . 8 ⊢ 0 = (0g‘𝐷)
11	9, 10	eqtr4di 2819	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
12	11	sneqd 4606	. . . . . 6 ⊢ (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
13	12	imaeq2d 6067	. . . . 5 ⊢ (𝑤 = 𝑊 → (◡𝑓 “ {(0g‘(Scalar‘𝑤))}) = (◡𝑓 “ { 0 }))
14	5, 13	mpteq12dv 5203	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))})) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
15		df-lkr 39901	. . . 4 ⊢ LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))})))
16	14, 15, 4	mptfvmpt 7233	. . 3 ⊢ (𝑊 ∈ V → (LKer‘𝑊) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
17	2, 16	eqtrid 2813	. 2 ⊢ (𝑊 ∈ V → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
18	1, 17	syl 18	1 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3458  {csn 4594   ↦ cmpt 5197  ^◡ccnv 5665   “ cima 5669  ‘cfv 6543  Scalarcsca 17338  0_gc0g 17517  LFnlclfn 39872  LKerclk 39900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-lkr 39901

This theorem is used by:  lkrval  39903