Metamath Proof Explorer

Theorem cdleml7

Description: Part of proof of Lemma L of Crawley p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013)

Ref Expression
Hypotheses cdleml6.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdleml6.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleml6.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleml6.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleml6.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdleml6.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdleml6.p ⊢ 𝑄 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
cdleml6.z ⊢ 𝑍 = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( ℎ ‘ 𝑄 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ◡ ( 𝑠 ‘ ℎ ) ) ) ) )
cdleml6.y ⊢ 𝑌 = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) )
cdleml6.x ⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ ( 𝑠 ‘ ℎ ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑄 ) = 𝑌 ) )
cdleml6.u ⊢ 𝑈 = ( 𝑔 ∈ 𝑇 ↦ if ( ( 𝑠 ‘ ℎ ) = ℎ , 𝑔 , 𝑋 ) )
cdleml6.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
cdleml6.o ⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
Assertion cdleml7 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( ( 𝑈 ∘ 𝑠 ) ‘ ℎ ) = ( ( I ↾ 𝑇 ) ‘ ℎ ) )

Proof

Step Hyp Ref Expression
1 cdleml6.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdleml6.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleml6.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleml6.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 cdleml6.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 cdleml6.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7 cdleml6.p ⊢ 𝑄 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
8 cdleml6.z ⊢ 𝑍 = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( ℎ ‘ 𝑄 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ◡ ( 𝑠 ‘ ℎ ) ) ) ) )
9 cdleml6.y ⊢ 𝑌 = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) )
10 cdleml6.x ⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ ( 𝑠 ‘ ℎ ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑄 ) = 𝑌 ) )
11 cdleml6.u ⊢ 𝑈 = ( 𝑔 ∈ 𝑇 ↦ if ( ( 𝑠 ‘ ℎ ) = ℎ , 𝑔 , 𝑋 ) )
12 cdleml6.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
13 cdleml6.o ⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
14 1 2 3 4 5 6 7 8 9 10 11 12 13 cdleml6 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( 𝑈 ∈ 𝐸 ∧ ( 𝑈 ‘ ( 𝑠 ‘ ℎ ) ) = ℎ ) )
15 14 simprd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( 𝑈 ‘ ( 𝑠 ‘ ℎ ) ) = ℎ )
16 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
17 14 simpld ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → 𝑈 ∈ 𝐸 )
18 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → 𝑠 ∈ 𝐸 )
19 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ℎ ∈ 𝑇 )
20 4 5 12 tendocoval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑠 ∈ 𝐸 ) ∧ ℎ ∈ 𝑇 ) → ( ( 𝑈 ∘ 𝑠 ) ‘ ℎ ) = ( 𝑈 ‘ ( 𝑠 ‘ ℎ ) ) )
21 16 17 18 19 20 syl121anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( ( 𝑈 ∘ 𝑠 ) ‘ ℎ ) = ( 𝑈 ‘ ( 𝑠 ‘ ℎ ) ) )
22 fvresi ⊢ ( ℎ ∈ 𝑇 → ( ( I ↾ 𝑇 ) ‘ ℎ ) = ℎ )
23 22 3ad2ant2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( ( I ↾ 𝑇 ) ‘ ℎ ) = ℎ )
24 15 21 23 3eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( ( 𝑈 ∘ 𝑠 ) ‘ ℎ ) = ( ( I ↾ 𝑇 ) ‘ ℎ ) )