Theorem lkrval 39066

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 39065	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
6	5	fveq1d 6828	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝐾‘𝐺) = ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺))
7		cnvexg 7864	. . . 4 ⊢ (𝐺 ∈ 𝐹 → ◡𝐺 ∈ V)
8		imaexg 7853	. . . 4 ⊢ (◡𝐺 ∈ V → (◡𝐺 “ { 0 }) ∈ V)
9	7, 8	syl 17	. . 3 ⊢ (𝐺 ∈ 𝐹 → (◡𝐺 “ { 0 }) ∈ V)
10		cnveq 5820	. . . . 5 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺)
11	10	imaeq1d 6014	. . . 4 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ { 0 }) = (◡𝐺 “ { 0 }))
12		eqid 2729	. . . 4 ⊢ (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))
13	11, 12	fvmptg 6932	. . 3 ⊢ ((𝐺 ∈ 𝐹 ∧ (◡𝐺 “ { 0 }) ∈ V) → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 }))
14	9, 13	mpdan 687	. 2 ⊢ (𝐺 ∈ 𝐹 → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 }))
15	6, 14	sylan9eq 2784	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 }))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  {csn 4579   ↦ cmpt 5176  ^◡ccnv 5622   “ cima 5626  ‘cfv 6486  Scalarcsca 17182  0_gc0g 17361  LFnlclfn 39035  LKerclk 39063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-lkr 39064

This theorem is referenced by:  ellkr  39067  lkr0f  39072