Theorem lkrfval 39043

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 3509	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		lkrfval.k	. . 3 ⊢ 𝐾 = (LKer‘𝑊)
3		fveq2 6920	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
4		lkrfval.f	. . . . . 6 ⊢ 𝐹 = (LFnl‘𝑊)
5	3, 4	eqtr4di 2798	. . . . 5 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
6		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
7		lkrfval.d	. . . . . . . . . 10 ⊢ 𝐷 = (Scalar‘𝑊)
8	6, 7	eqtr4di 2798	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
9	8	fveq2d 6924	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g‘𝐷))
10		lkrfval.o	. . . . . . . 8 ⊢ 0 = (0g‘𝐷)
11	9, 10	eqtr4di 2798	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
12	11	sneqd 4660	. . . . . 6 ⊢ (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
13	12	imaeq2d 6089	. . . . 5 ⊢ (𝑤 = 𝑊 → (◡𝑓 “ {(0g‘(Scalar‘𝑤))}) = (◡𝑓 “ { 0 }))
14	5, 13	mpteq12dv 5257	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))})) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
15		df-lkr 39042	. . . 4 ⊢ LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))})))
16	14, 15, 4	mptfvmpt 7265	. . 3 ⊢ (𝑊 ∈ V → (LKer‘𝑊) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
17	2, 16	eqtrid 2792	. 2 ⊢ (𝑊 ∈ V → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
18	1, 17	syl 17	1 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  Vcvv 3488  {csn 4648   ↦ cmpt 5249  ^◡ccnv 5699   “ cima 5703  ‘cfv 6573  Scalarcsca 17314  0_gc0g 17499  LFnlclfn 39013  LKerclk 39041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-lkr 39042

This theorem is referenced by:  lkrval  39044