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Theorem ellkr2 33897
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 = (0g𝐷)
lkrfval2.f 𝐹 = (LFnl‘𝑊)
lkrfval2.k 𝐾 = (LKer‘𝑊)
ellkr2.w (𝜑𝑊𝑌)
ellkr2.g (𝜑𝐺𝐹)
ellkr2.x (𝜑𝑋𝑉)
Assertion
Ref Expression
ellkr2 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝐺𝑋) = 0 ))

Proof of Theorem ellkr2
StepHypRef Expression
1 ellkr2.w . . 3 (𝜑𝑊𝑌)
2 ellkr2.g . . 3 (𝜑𝐺𝐹)
3 lkrfval2.v . . . 4 𝑉 = (Base‘𝑊)
4 lkrfval2.d . . . 4 𝐷 = (Scalar‘𝑊)
5 lkrfval2.o . . . 4 0 = (0g𝐷)
6 lkrfval2.f . . . 4 𝐹 = (LFnl‘𝑊)
7 lkrfval2.k . . . 4 𝐾 = (LKer‘𝑊)
83, 4, 5, 6, 7ellkr 33895 . . 3 ((𝑊𝑌𝐺𝐹) → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
91, 2, 8syl2anc 692 . 2 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
10 ellkr2.x . . 3 (𝜑𝑋𝑉)
1110biantrurd 529 . 2 (𝜑 → ((𝐺𝑋) = 0 ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
129, 11bitr4d 271 1 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝐺𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cfv 5857  Basecbs 15800  Scalarcsca 15884  0gc0g 16040  LFnlclfn 33863  LKerclk 33891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-lfl 33864  df-lkr 33892
This theorem is referenced by:  lclkrlem2f  36320  lclkrlem2n  36328  lcfrlem3  36352  lcfrlem25  36375  hdmapellkr  36725  hdmapip0  36726  hdmapinvlem1  36729
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