Metamath Proof Explorer

Theorem lclkrlem2i

Description: Lemma for lclkr . Eliminate the ( LE ) =/= ( LG ) hypothesis. (Contributed by NM, 17-Jan-2015)

Ref Expression
Hypotheses lclkrlem2f.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
lclkrlem2f.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
lclkrlem2f.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
lclkrlem2f.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
lclkrlem2f.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
lclkrlem2f.q ⊢ 𝑄 = ( 0g ‘ 𝑆 )
lclkrlem2f.z ⊢ 0 = ( 0g ‘ 𝑈 )
lclkrlem2f.a ⊢ ⊕ = ( LSSum ‘ 𝑈 )
lclkrlem2f.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
lclkrlem2f.f ⊢ 𝐹 = ( LFnl ‘ 𝑈 )
lclkrlem2f.j ⊢ 𝐽 = ( LSHyp ‘ 𝑈 )
lclkrlem2f.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
lclkrlem2f.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
lclkrlem2f.p ⊢ + = ( +g ‘ 𝐷 )
lclkrlem2f.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
lclkrlem2f.b ⊢ ( 𝜑 → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) )
lclkrlem2f.e ⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
lclkrlem2f.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
lclkrlem2f.le ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
lclkrlem2f.lg ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
lclkrlem2f.kb ⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 )
lclkrlem2f.nx ⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) )
lclkrlem2i.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
lclkrlem2i.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion lclkrlem2i ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 lclkrlem2f.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 lclkrlem2f.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 lclkrlem2f.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 lclkrlem2f.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
5 lclkrlem2f.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
6 lclkrlem2f.q ⊢ 𝑄 = ( 0g ‘ 𝑆 )
7 lclkrlem2f.z ⊢ 0 = ( 0g ‘ 𝑈 )
8 lclkrlem2f.a ⊢ ⊕ = ( LSSum ‘ 𝑈 )
9 lclkrlem2f.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
10 lclkrlem2f.f ⊢ 𝐹 = ( LFnl ‘ 𝑈 )
11 lclkrlem2f.j ⊢ 𝐽 = ( LSHyp ‘ 𝑈 )
12 lclkrlem2f.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
13 lclkrlem2f.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
14 lclkrlem2f.p ⊢ + = ( +g ‘ 𝐷 )
15 lclkrlem2f.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
16 lclkrlem2f.b ⊢ ( 𝜑 → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) )
17 lclkrlem2f.e ⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
18 lclkrlem2f.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
19 lclkrlem2f.le ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
20 lclkrlem2f.lg ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
21 lclkrlem2f.kb ⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 )
22 lclkrlem2f.nx ⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) )
23 lclkrlem2i.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
24 lclkrlem2i.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
25 15 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) = ( 𝐿 ‘ 𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
26 23 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) = ( 𝐿 ‘ 𝐺 ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
27 17 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) = ( 𝐿 ‘ 𝐺 ) ) → 𝐸 ∈ 𝐹 )
28 18 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) = ( 𝐿 ‘ 𝐺 ) ) → 𝐺 ∈ 𝐹 )
29 19 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) = ( 𝐿 ‘ 𝐺 ) ) → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
30 simpr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) = ( 𝐿 ‘ 𝐺 ) ) → ( 𝐿 ‘ 𝐸 ) = ( 𝐿 ‘ 𝐺 ) )
31 1 2 3 4 7 10 12 13 14 25 26 27 28 29 30 lclkrlem2e ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) = ( 𝐿 ‘ 𝐺 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )
32 15 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
33 16 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) )
34 17 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → 𝐸 ∈ 𝐹 )
35 18 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → 𝐺 ∈ 𝐹 )
36 19 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
37 20 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
38 21 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 )
39 22 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) )
40 23 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
41 24 adantr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
42 simpr ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) )
43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 32 33 34 35 36 37 38 39 40 41 42 lclkrlem2h ⊢ ( ( 𝜑 ∧ ( 𝐿 ‘ 𝐸 ) ≠ ( 𝐿 ‘ 𝐺 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )
44 31 43 pm2.61dane ⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )