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Theorem ellkr2 36107
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 36105 . . 3 ((𝑊𝑌𝐺𝐹) → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
91, 2, 8syl2anc 584 . 2 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
10 ellkr2.x . . 3 (𝜑𝑋𝑉)
1110biantrurd 533 . 2 (𝜑 → ((𝐺𝑋) = 0 ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
129, 11bitr4d 283 1 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝐺𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  cfv 6348  Basecbs 16471  Scalarcsca 16556  0gc0g 16701  LFnlclfn 36073  LKerclk 36101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-lfl 36074  df-lkr 36102
This theorem is referenced by:  lclkrlem2f  38528  lclkrlem2n  38536  lcfrlem3  38560  lcfrlem25  38583  hdmapellkr  38930  hdmapip0  38931  hdmapinvlem1  38934
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