Theorem ellkr2 39089

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 39087	. . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )))
9	1, 2, 8	syl2anc 584	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6486  Basecbs 17139  Scalarcsca 17183  0_gc0g 17362  LFnlclfn 39055  LKerclk 39083

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-ov 7356  df-oprab 7357  df-mpo 7358  df-map 8762  df-lfl 39056  df-lkr 39084

This theorem is referenced by:  lclkrlem2f  41511  lclkrlem2n  41519  lcfrlem3  41543  lcfrlem25  41566  hdmapellkr  41913  hdmapip0  41914  hdmapinvlem1  41917