Theorem lkrval 39903

Description: Value of the kernel of a functional. (Contributed by NM, 15-Apr-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
lkrval	⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 }))

Proof of Theorem lkrval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lkrfval.d	. . . 4 ⊢ 𝐷 = (Scalar‘𝑊)
2		lkrfval.o	. . . 4 ⊢ 0 = (0g‘𝐷)
3		lkrfval.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
4		lkrfval.k	. . . 4 ⊢ 𝐾 = (LKer‘𝑊)
5	1, 2, 3, 4	lkrfval 39902	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
6	5	fveq1d 6890	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝐾‘𝐺) = ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺))
7		cnvexg 7930	. . . 4 ⊢ (𝐺 ∈ 𝐹 → ◡𝐺 ∈ V)
8		imaexg 7919	. . . 4 ⊢ (◡𝐺 ∈ V → (◡𝐺 “ { 0 }) ∈ V)
9	7, 8	syl 18	. . 3 ⊢ (𝐺 ∈ 𝐹 → (◡𝐺 “ { 0 }) ∈ V)
10		cnveq 5864	. . . . 5 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺)
11	10	imaeq1d 6066	. . . 4 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ { 0 }) = (◡𝐺 “ { 0 }))
12		eqid 2766	. . . 4 ⊢ (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))
13	11, 12	fvmptg 6994	. . 3 ⊢ ((𝐺 ∈ 𝐹 ∧ (◡𝐺 “ { 0 }) ∈ V) → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 }))
14	9, 13	mpdan 700	. 2 ⊢ (𝐺 ∈ 𝐹 → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 }))
15	6, 14	sylan9eq 2821	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 }))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = 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-pow 5341  ax-pr 5409  ax-un 7745

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-pw 4569  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:  ellkr  39904  lkr0f  39909