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Theorem ellkr2 36728
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 36726 . . 3 ((𝑊𝑌𝐺𝐹) → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
91, 2, 8syl2anc 587 . 2 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
10 ellkr2.x . . 3 (𝜑𝑋𝑉)
1110biantrurd 536 . 2 (𝜑 → ((𝐺𝑋) = 0 ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
129, 11bitr4d 285 1 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝐺𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  cfv 6339  Basecbs 16586  Scalarcsca 16671  0gc0g 16816  LFnlclfn 36694  LKerclk 36722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-lfl 36695  df-lkr 36723
This theorem is referenced by:  lclkrlem2f  39149  lclkrlem2n  39157  lcfrlem3  39181  lcfrlem25  39204  hdmapellkr  39551  hdmapip0  39552  hdmapinvlem1  39555
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