Theorem lkrval 39969

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 39968	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
6	5	fveq1d 6884	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝐾‘𝐺) = ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺))
7		cnvexg 7925	. . . 4 ⊢ (𝐺 ∈ 𝐹 → ◡𝐺 ∈ V)
8		imaexg 7914	. . . 4 ⊢ (◡𝐺 ∈ V → (◡𝐺 “ { 0 }) ∈ V)
9	7, 8	syl 18	. . 3 ⊢ (𝐺 ∈ 𝐹 → (◡𝐺 “ { 0 }) ∈ V)
10		cnveq 5857	. . . . 5 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺)
11	10	imaeq1d 6059	. . . 4 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ { 0 }) = (◡𝐺 “ { 0 }))
12		eqid 2762	. . . 4 ⊢ (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))
13	11, 12	fvmptg 6988	. . 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 2817	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 }))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  {csn 4587   ↦ cmpt 5190  ^◡ccnv 5658   “ cima 5662  ‘cfv 6537  Scalarcsca 17351  0_gc0g 17530  LFnlclfn 39938  LKerclk 39966

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-lkr 39967

This theorem is used by:  ellkr  39970  lkr0f  39975