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Metamath Proof Explorer

Theorem cdlemn11a

Description: Part of proof of Lemma N of Crawley p. 121 line 37. (Contributed by NM, 27-Feb-2014)

Ref Expression
Hypotheses cdlemn11a.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemn11a.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemn11a.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemn11a.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemn11a.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemn11a.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
cdlemn11a.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
cdlemn11a.d ⊢ + = ( +g ‘ 𝑈 )
cdlemn11a.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
cdlemn11a.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 )
cdlemn11a.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 )
Assertion cdlemn11a ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ ∈ ( 𝐽 ‘ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 cdlemn11a.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemn11a.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemn11a.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemn11a.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemn11a.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemn11a.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemn11a.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
8 cdlemn11a.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemn11a.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
10 cdlemn11a.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
11 cdlemn11a.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
12 cdlemn11a.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
13 cdlemn11a.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
14 cdlemn11a.d ⊢ + = ( +g ‘ 𝑈 )
15 cdlemn11a.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
16 cdlemn11a.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 )
17 cdlemn11a.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 )
18 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
19 2 4 5 6 lhpocnel2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
20 19 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
21 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) )
22 2 4 5 8 17 ltrniotacl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) → 𝐺 ∈ 𝑇 )
23 18 20 21 22 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → 𝐺 ∈ 𝑇 )
24 fvresi ⊢ ( 𝐺 ∈ 𝑇 → ( ( I ↾ 𝑇 ) ‘ 𝐺 ) = 𝐺 )
25 23 24 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( ( I ↾ 𝑇 ) ‘ 𝐺 ) = 𝐺 )
26 25 eqcomd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → 𝐺 = ( ( I ↾ 𝑇 ) ‘ 𝐺 ) )
27 5 8 10 tendoidcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝑇 ) ∈ 𝐸 )
28 27 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( I ↾ 𝑇 ) ∈ 𝐸 )
29 riotaex ⊢ ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 ) ∈ V
30 17 29 eqeltri ⊢ 𝐺 ∈ V
31 8 fvexi ⊢ 𝑇 ∈ V
32 resiexg ⊢ ( 𝑇 ∈ V → ( I ↾ 𝑇 ) ∈ V )
33 31 32 ax-mp ⊢ ( I ↾ 𝑇 ) ∈ V
34 2 4 5 6 8 10 12 17 30 33 dicopelval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ ∈ ( 𝐽 ‘ 𝑁 ) ↔ ( 𝐺 = ( ( I ↾ 𝑇 ) ‘ 𝐺 ) ∧ ( I ↾ 𝑇 ) ∈ 𝐸 ) ) )
35 18 21 34 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ( ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ ∈ ( 𝐽 ‘ 𝑁 ) ↔ ( 𝐺 = ( ( I ↾ 𝑇 ) ‘ 𝐺 ) ∧ ( I ↾ 𝑇 ) ∈ 𝐸 ) ) )
36 26 28 35 mpbir2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑁 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → ⟨ 𝐺 , ( I ↾ 𝑇 ) ⟩ ∈ ( 𝐽 ‘ 𝑁 ) )