Theorem ellkr2 38472

Description: Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)

Hypotheses

Ref	Expression
lkrfval2.v	⊢ 𝑉 = (Base‘𝑊)
lkrfval2.d	⊢ 𝐷 = (Scalar‘𝑊)
lkrfval2.o	⊢ 0 = (0g‘𝐷)
lkrfval2.f	⊢ 𝐹 = (LFnl‘𝑊)
lkrfval2.k	⊢ 𝐾 = (LKer‘𝑊)
ellkr2.w	⊢ (𝜑 → 𝑊 ∈ 𝑌)
ellkr2.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
ellkr2.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)

Assertion

Ref	Expression
ellkr2	⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 ))

Proof of Theorem ellkr2

Step	Hyp	Ref	Expression
1		ellkr2.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑌)
2		ellkr2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
3		lkrfval2.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
4		lkrfval2.d	. . . 4 ⊢ 𝐷 = (Scalar‘𝑊)
5		lkrfval2.o	. . . 4 ⊢ 0 = (0g‘𝐷)
6		lkrfval2.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
7		lkrfval2.k	. . . 4 ⊢ 𝐾 = (LKer‘𝑊)
8	3, 4, 5, 6, 7	ellkr 38470	. . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )))
9	1, 2, 8	syl2anc 583	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )))
10		ellkr2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
11	10	biantrurd 532	. 2 ⊢ (𝜑 → ((𝐺‘𝑋) = 0 ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )))
12	9, 11	bitr4d 282	1 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ‘cfv 6536  Basecbs 17151  Scalarcsca 17207  0_gc0g 17392  LFnlclfn 38438  LKerclk 38466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-lfl 38439  df-lkr 38467

This theorem is referenced by:  lclkrlem2f  40894  lclkrlem2n  40902  lcfrlem3  40926  lcfrlem25  40949  hdmapellkr  41296  hdmapip0  41297  hdmapinvlem1  41300