Theorem ellkr2 36387

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 36385	. . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )))
9	1, 2, 8	syl2anc 587	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )))
10		ellkr2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
11	10	biantrurd 536	. 2 ⊢ (𝜑 → ((𝐺‘𝑋) = 0 ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )))
12	9, 11	bitr4d 285	1 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6324  Basecbs 16475  Scalarcsca 16560  0_gc0g 16705  LFnlclfn 36353  LKerclk 36381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-lfl 36354  df-lkr 36382

This theorem is referenced by:  lclkrlem2f  38808  lclkrlem2n  38816  lcfrlem3  38840  lcfrlem25  38863  hdmapellkr  39210  hdmapip0  39211  hdmapinvlem1  39214