Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ellkr2 Structured version   Visualization version   GIF version

Theorem ellkr2 38563
Description: Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
Hypotheses
Ref Expression
lkrfval2.v 𝑉 = (Baseβ€˜π‘Š)
lkrfval2.d 𝐷 = (Scalarβ€˜π‘Š)
lkrfval2.o 0 = (0gβ€˜π·)
lkrfval2.f 𝐹 = (LFnlβ€˜π‘Š)
lkrfval2.k 𝐾 = (LKerβ€˜π‘Š)
ellkr2.w (πœ‘ β†’ π‘Š ∈ π‘Œ)
ellkr2.g (πœ‘ β†’ 𝐺 ∈ 𝐹)
ellkr2.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
Assertion
Ref Expression
ellkr2 (πœ‘ β†’ (𝑋 ∈ (πΎβ€˜πΊ) ↔ (πΊβ€˜π‘‹) = 0 ))

Proof of Theorem ellkr2
StepHypRef Expression
1 ellkr2.w . . 3 (πœ‘ β†’ π‘Š ∈ π‘Œ)
2 ellkr2.g . . 3 (πœ‘ β†’ 𝐺 ∈ 𝐹)
3 lkrfval2.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
4 lkrfval2.d . . . 4 𝐷 = (Scalarβ€˜π‘Š)
5 lkrfval2.o . . . 4 0 = (0gβ€˜π·)
6 lkrfval2.f . . . 4 𝐹 = (LFnlβ€˜π‘Š)
7 lkrfval2.k . . . 4 𝐾 = (LKerβ€˜π‘Š)
83, 4, 5, 6, 7ellkr 38561 . . 3 ((π‘Š ∈ π‘Œ ∧ 𝐺 ∈ 𝐹) β†’ (𝑋 ∈ (πΎβ€˜πΊ) ↔ (𝑋 ∈ 𝑉 ∧ (πΊβ€˜π‘‹) = 0 )))
91, 2, 8syl2anc 583 . 2 (πœ‘ β†’ (𝑋 ∈ (πΎβ€˜πΊ) ↔ (𝑋 ∈ 𝑉 ∧ (πΊβ€˜π‘‹) = 0 )))
10 ellkr2.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝑉)
1110biantrurd 532 . 2 (πœ‘ β†’ ((πΊβ€˜π‘‹) = 0 ↔ (𝑋 ∈ 𝑉 ∧ (πΊβ€˜π‘‹) = 0 )))
129, 11bitr4d 282 1 (πœ‘ β†’ (𝑋 ∈ (πΎβ€˜πΊ) ↔ (πΊβ€˜π‘‹) = 0 ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  β€˜cfv 6548  Basecbs 17180  Scalarcsca 17236  0gc0g 17421  LFnlclfn 38529  LKerclk 38557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847  df-lfl 38530  df-lkr 38558
This theorem is referenced by:  lclkrlem2f  40985  lclkrlem2n  40993  lcfrlem3  41017  lcfrlem25  41040  hdmapellkr  41387  hdmapip0  41388  hdmapinvlem1  41391
  Copyright terms: Public domain W3C validator