Metamath Proof Explorer

Theorem lcfrlem3

Description: Lemma for lcfr . (Contributed by NM, 27-Feb-2015)

Ref Expression
Hypotheses lcfrlem1.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
lcfrlem1.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
lcfrlem1.q ⊢ × = ( .r ‘ 𝑆 )
lcfrlem1.z ⊢ 0 = ( 0g ‘ 𝑆 )
lcfrlem1.i ⊢ 𝐼 = ( invr ‘ 𝑆 )
lcfrlem1.f ⊢ 𝐹 = ( LFnl ‘ 𝑈 )
lcfrlem1.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
lcfrlem1.t ⊢ · = ( ·𝑠 ‘ 𝐷 )
lcfrlem1.m ⊢ − = ( -g ‘ 𝐷 )
lcfrlem1.u ⊢ ( 𝜑 → 𝑈 ∈ LVec )
lcfrlem1.e ⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
lcfrlem1.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
lcfrlem1.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
lcfrlem1.n ⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) ≠ 0 )
lcfrlem1.h ⊢ 𝐻 = ( 𝐸 − ( ( ( 𝐼 ‘ ( 𝐺 ‘ 𝑋 ) ) × ( 𝐸 ‘ 𝑋 ) ) · 𝐺 ) )
lcfrlem2.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
Assertion lcfrlem3 ( 𝜑 → 𝑋 ∈ ( 𝐿 ‘ 𝐻 ) )

Proof

Step Hyp Ref Expression
1 lcfrlem1.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
2 lcfrlem1.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
3 lcfrlem1.q ⊢ × = ( .r ‘ 𝑆 )
4 lcfrlem1.z ⊢ 0 = ( 0g ‘ 𝑆 )
5 lcfrlem1.i ⊢ 𝐼 = ( invr ‘ 𝑆 )
6 lcfrlem1.f ⊢ 𝐹 = ( LFnl ‘ 𝑈 )
7 lcfrlem1.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
8 lcfrlem1.t ⊢ · = ( ·𝑠 ‘ 𝐷 )
9 lcfrlem1.m ⊢ − = ( -g ‘ 𝐷 )
10 lcfrlem1.u ⊢ ( 𝜑 → 𝑈 ∈ LVec )
11 lcfrlem1.e ⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
12 lcfrlem1.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
13 lcfrlem1.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
14 lcfrlem1.n ⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) ≠ 0 )
15 lcfrlem1.h ⊢ 𝐻 = ( 𝐸 − ( ( ( 𝐼 ‘ ( 𝐺 ‘ 𝑋 ) ) × ( 𝐸 ‘ 𝑋 ) ) · 𝐺 ) )
16 lcfrlem2.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 lcfrlem1 ⊢ ( 𝜑 → ( 𝐻 ‘ 𝑋 ) = 0 )
18 lveclmod ⊢ ( 𝑈 ∈ LVec → 𝑈 ∈ LMod )
19 10 18 syl ⊢ ( 𝜑 → 𝑈 ∈ LMod )
20 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
21 2 lmodring ⊢ ( 𝑈 ∈ LMod → 𝑆 ∈ Ring )
22 19 21 syl ⊢ ( 𝜑 → 𝑆 ∈ Ring )
23 2 lvecdrng ⊢ ( 𝑈 ∈ LVec → 𝑆 ∈ DivRing )
24 10 23 syl ⊢ ( 𝜑 → 𝑆 ∈ DivRing )
25 2 20 1 6 lflcl ⊢ ( ( 𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) )
26 10 12 13 25 syl3anc ⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) )
27 20 4 5 drnginvrcl ⊢ ( ( 𝑆 ∈ DivRing ∧ ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑋 ) ) ∈ ( Base ‘ 𝑆 ) )
28 24 26 14 27 syl3anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝐺 ‘ 𝑋 ) ) ∈ ( Base ‘ 𝑆 ) )
29 2 20 1 6 lflcl ⊢ ( ( 𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐸 ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) )
30 10 11 13 29 syl3anc ⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) )
31 20 3 ringcl ⊢ ( ( 𝑆 ∈ Ring ∧ ( 𝐼 ‘ ( 𝐺 ‘ 𝑋 ) ) ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝐼 ‘ ( 𝐺 ‘ 𝑋 ) ) × ( 𝐸 ‘ 𝑋 ) ) ∈ ( Base ‘ 𝑆 ) )
32 22 28 30 31 syl3anc ⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝐺 ‘ 𝑋 ) ) × ( 𝐸 ‘ 𝑋 ) ) ∈ ( Base ‘ 𝑆 ) )
33 6 2 20 7 8 19 32 12 ldualvscl ⊢ ( 𝜑 → ( ( ( 𝐼 ‘ ( 𝐺 ‘ 𝑋 ) ) × ( 𝐸 ‘ 𝑋 ) ) · 𝐺 ) ∈ 𝐹 )
34 6 7 9 19 11 33 ldualvsubcl ⊢ ( 𝜑 → ( 𝐸 − ( ( ( 𝐼 ‘ ( 𝐺 ‘ 𝑋 ) ) × ( 𝐸 ‘ 𝑋 ) ) · 𝐺 ) ) ∈ 𝐹 )
35 15 34 eqeltrid ⊢ ( 𝜑 → 𝐻 ∈ 𝐹 )
36 1 2 4 6 16 10 35 13 ellkr2 ⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐿 ‘ 𝐻 ) ↔ ( 𝐻 ‘ 𝑋 ) = 0 ) )
37 17 36 mpbird ⊢ ( 𝜑 → 𝑋 ∈ ( 𝐿 ‘ 𝐻 ) )