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Metamath Proof Explorer

Theorem cdlemi2

Description: Part of proof of Lemma I of Crawley p. 118. (Contributed by NM, 18-Jun-2013)

Ref Expression
Hypotheses cdlemi.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemi.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemi.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemi.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemi.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemi.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemi.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemi.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemi.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemi2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑈 ‘ 𝐺 ) ‘ 𝑃 ) ≤ ( ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemi.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemi.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemi.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemi.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemi.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemi.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemi.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemi.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemi.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
10 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ HL )
11 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 )
12 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑈 ∈ 𝐸 )
13 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐺 ∈ 𝑇 )
15 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
16 6 7 ltrncnv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ◡ 𝐹 ∈ 𝑇 )
17 13 15 16 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ◡ 𝐹 ∈ 𝑇 )
18 6 7 ltrnco ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ◡ 𝐹 ∈ 𝑇 ) → ( 𝐺 ∘ ◡ 𝐹 ) ∈ 𝑇 )
19 13 14 17 18 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐺 ∘ ◡ 𝐹 ) ∈ 𝑇 )
20 6 7 9 tendovalco ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑈 ∈ 𝐸 ) ∧ ( ( 𝐺 ∘ ◡ 𝐹 ) ∈ 𝑇 ∧ 𝐹 ∈ 𝑇 ) ) → ( 𝑈 ‘ ( ( 𝐺 ∘ ◡ 𝐹 ) ∘ 𝐹 ) ) = ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∘ ( 𝑈 ‘ 𝐹 ) ) )
21 10 11 12 19 15 20 syl32anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑈 ‘ ( ( 𝐺 ∘ ◡ 𝐹 ) ∘ 𝐹 ) ) = ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∘ ( 𝑈 ‘ 𝐹 ) ) )
22 coass ⊢ ( ( 𝐺 ∘ ◡ 𝐹 ) ∘ 𝐹 ) = ( 𝐺 ∘ ( ◡ 𝐹 ∘ 𝐹 ) )
23 1 6 7 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )
24 13 15 23 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )
25 f1ococnv1 ⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ 𝐵 ) )
26 24 25 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ 𝐵 ) )
27 26 coeq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐺 ∘ ( ◡ 𝐹 ∘ 𝐹 ) ) = ( 𝐺 ∘ ( I ↾ 𝐵 ) ) )
28 1 6 7 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → 𝐺 : 𝐵 –1-1-onto→ 𝐵 )
29 13 14 28 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐺 : 𝐵 –1-1-onto→ 𝐵 )
30 f1of ⊢ ( 𝐺 : 𝐵 –1-1-onto→ 𝐵 → 𝐺 : 𝐵 ⟶ 𝐵 )
31 fcoi1 ⊢ ( 𝐺 : 𝐵 ⟶ 𝐵 → ( 𝐺 ∘ ( I ↾ 𝐵 ) ) = 𝐺 )
32 29 30 31 3syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐺 ∘ ( I ↾ 𝐵 ) ) = 𝐺 )
33 27 32 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐺 ∘ ( ◡ 𝐹 ∘ 𝐹 ) ) = 𝐺 )
34 22 33 eqtrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ∘ ◡ 𝐹 ) ∘ 𝐹 ) = 𝐺 )
35 34 fveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑈 ‘ ( ( 𝐺 ∘ ◡ 𝐹 ) ∘ 𝐹 ) ) = ( 𝑈 ‘ 𝐺 ) )
36 21 35 eqtr3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∘ ( 𝑈 ‘ 𝐹 ) ) = ( 𝑈 ‘ 𝐺 ) )
37 36 fveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∘ ( 𝑈 ‘ 𝐹 ) ) ‘ 𝑃 ) = ( ( 𝑈 ‘ 𝐺 ) ‘ 𝑃 ) )
38 6 7 9 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ ( 𝐺 ∘ ◡ 𝐹 ) ∈ 𝑇 ) → ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝑇 )
39 13 12 19 38 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝑇 )
40 6 7 9 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 )
41 13 12 15 40 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 )
42 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ 𝐴 )
43 2 5 6 7 ltrncoval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝑇 ∧ ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∘ ( 𝑈 ‘ 𝐹 ) ) ‘ 𝑃 ) = ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ‘ ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ) )
44 13 39 41 42 43 syl121anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∘ ( 𝑈 ‘ 𝐹 ) ) ‘ 𝑃 ) = ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ‘ ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ) )
45 37 44 eqtr3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑈 ‘ 𝐺 ) ‘ 𝑃 ) = ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ‘ ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ) )
46 2 5 6 7 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ≤ 𝑊 ) )
47 41 46 syld3an2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ≤ 𝑊 ) )
48 1 2 3 4 5 6 7 8 9 cdlemi1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ ( 𝐺 ∘ ◡ 𝐹 ) ∈ 𝑇 ) ∧ ( ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ≤ 𝑊 ) ) → ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ‘ ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ) ≤ ( ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) )
49 13 12 19 47 48 syl121anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑈 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ‘ ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ) ≤ ( ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) )
50 45 49 eqbrtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑈 ‘ 𝐺 ) ‘ 𝑃 ) ≤ ( ( ( 𝑈 ‘ 𝐹 ) ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) )