Metamath Proof Explorer

Theorem cdlemn6

Description: Part of proof of Lemma N of Crawley p. 121 line 35. (Contributed by NM, 26-Feb-2014)

Ref Expression
Hypotheses cdlemn8.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemn8.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemn8.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemn8.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemn8.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
cdlemn8.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
cdlemn8.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemn8.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
cdlemn8.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
cdlemn8.s ⊢ + = ( +g ‘ 𝑈 )
cdlemn8.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 )
Assertion cdlemn6 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) = ⟨ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) , 𝑠 ⟩ )

Proof

Step Hyp Ref Expression
1 cdlemn8.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemn8.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemn8.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 cdlemn8.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 cdlemn8.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
6 cdlemn8.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
7 cdlemn8.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemn8.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemn8.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
10 cdlemn8.s ⊢ + = ( +g ‘ 𝑈 )
11 cdlemn8.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑄 )
12 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
13 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → 𝑠 ∈ 𝐸 )
14 2 3 4 5 lhpocnel2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
15 12 14 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
16 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
17 2 3 4 7 11 ltrniotacl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
18 12 15 16 17 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → 𝐹 ∈ 𝑇 )
19 4 7 8 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑠 ‘ 𝐹 ) ∈ 𝑇 )
20 12 13 18 19 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( 𝑠 ‘ 𝐹 ) ∈ 𝑇 )
21 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → 𝑔 ∈ 𝑇 )
22 1 4 7 8 6 tendo0cl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑂 ∈ 𝐸 )
23 12 22 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → 𝑂 ∈ 𝐸 )
24 eqid ⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
25 eqid ⊢ ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( +g ‘ ( Scalar ‘ 𝑈 ) )
26 4 7 8 9 24 10 25 dvhopvadd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑠 ‘ 𝐹 ) ∈ 𝑇 ∧ 𝑠 ∈ 𝐸 ) ∧ ( 𝑔 ∈ 𝑇 ∧ 𝑂 ∈ 𝐸 ) ) → ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) = ⟨ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) , ( 𝑠 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) ⟩ )
27 12 20 13 21 23 26 syl122anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) = ⟨ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) , ( 𝑠 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) ⟩ )
28 eqid ⊢ ( 𝑡 ∈ 𝐸 , 𝑢 ∈ 𝐸 ↦ ( ℎ ∈ 𝑇 ↦ ( ( 𝑡 ‘ ℎ ) ∘ ( 𝑢 ‘ ℎ ) ) ) ) = ( 𝑡 ∈ 𝐸 , 𝑢 ∈ 𝐸 ↦ ( ℎ ∈ 𝑇 ↦ ( ( 𝑡 ‘ ℎ ) ∘ ( 𝑢 ‘ ℎ ) ) ) )
29 4 7 8 9 24 28 25 dvhfplusr ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( 𝑡 ∈ 𝐸 , 𝑢 ∈ 𝐸 ↦ ( ℎ ∈ 𝑇 ↦ ( ( 𝑡 ‘ ℎ ) ∘ ( 𝑢 ‘ ℎ ) ) ) ) )
30 12 29 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( 𝑡 ∈ 𝐸 , 𝑢 ∈ 𝐸 ↦ ( ℎ ∈ 𝑇 ↦ ( ( 𝑡 ‘ ℎ ) ∘ ( 𝑢 ‘ ℎ ) ) ) ) )
31 30 oveqd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( 𝑠 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) = ( 𝑠 ( 𝑡 ∈ 𝐸 , 𝑢 ∈ 𝐸 ↦ ( ℎ ∈ 𝑇 ↦ ( ( 𝑡 ‘ ℎ ) ∘ ( 𝑢 ‘ ℎ ) ) ) ) 𝑂 ) )
32 1 4 7 8 6 28 tendo0plr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ) → ( 𝑠 ( 𝑡 ∈ 𝐸 , 𝑢 ∈ 𝐸 ↦ ( ℎ ∈ 𝑇 ↦ ( ( 𝑡 ‘ ℎ ) ∘ ( 𝑢 ‘ ℎ ) ) ) ) 𝑂 ) = 𝑠 )
33 12 13 32 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( 𝑠 ( 𝑡 ∈ 𝐸 , 𝑢 ∈ 𝐸 ↦ ( ℎ ∈ 𝑇 ↦ ( ( 𝑡 ‘ ℎ ) ∘ ( 𝑢 ‘ ℎ ) ) ) ) 𝑂 ) = 𝑠 )
34 31 33 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( 𝑠 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) = 𝑠 )
35 34 opeq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ⟨ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) , ( 𝑠 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑂 ) ⟩ = ⟨ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) , 𝑠 ⟩ )
36 27 35 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ) → ( ⟨ ( 𝑠 ‘ 𝐹 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) = ⟨ ( ( 𝑠 ‘ 𝐹 ) ∘ 𝑔 ) , 𝑠 ⟩ )