Theorem lkrval 36239

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 36238	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
6	5	fveq1d 6672	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝐾‘𝐺) = ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺))
7		cnvexg 7629	. . . 4 ⊢ (𝐺 ∈ 𝐹 → ◡𝐺 ∈ V)
8		imaexg 7620	. . . 4 ⊢ (◡𝐺 ∈ V → (◡𝐺 “ { 0 }) ∈ V)
9	7, 8	syl 17	. . 3 ⊢ (𝐺 ∈ 𝐹 → (◡𝐺 “ { 0 }) ∈ V)
10		cnveq 5744	. . . . 5 ⊢ (𝑓 = 𝐺 → ◡𝑓 = ◡𝐺)
11	10	imaeq1d 5928	. . . 4 ⊢ (𝑓 = 𝐺 → (◡𝑓 “ { 0 }) = (◡𝐺 “ { 0 }))
12		eqid 2821	. . . 4 ⊢ (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })) = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))
13	11, 12	fvmptg 6766	. . 3 ⊢ ((𝐺 ∈ 𝐹 ∧ (◡𝐺 “ { 0 }) ∈ V) → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 }))
14	9, 13	mpdan 685	. 2 ⊢ (𝐺 ∈ 𝐹 → ((𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 }))‘𝐺) = (◡𝐺 “ { 0 }))
15	6, 14	sylan9eq 2876	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 }))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  {csn 4567   ↦ cmpt 5146  ^◡ccnv 5554   “ cima 5558  ‘cfv 6355  Scalarcsca 16568  0_gc0g 16713  LFnlclfn 36208  LKerclk 36236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-lkr 36237

This theorem is referenced by:  ellkr  36240  lkr0f  36245