Metamath Proof Explorer


Theorem cdleml6

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 ( 𝑅𝑏 ) ) ( ( 𝑄 ) ( 𝑅 ‘ ( 𝑏 ( 𝑠 ) ) ) ) )
cdleml6.y 𝑌 = ( ( 𝑄 ( 𝑅𝑔 ) ) ( 𝑍 ( 𝑅 ‘ ( 𝑔 𝑏 ) ) ) )
cdleml6.x 𝑋 = ( 𝑧𝑇𝑏𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅 ‘ ( 𝑠 ) ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅𝑔 ) ) → ( 𝑧𝑄 ) = 𝑌 ) )
cdleml6.u 𝑈 = ( 𝑔𝑇 ↦ if ( ( 𝑠 ) = , 𝑔 , 𝑋 ) )
cdleml6.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
cdleml6.o 0 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
Assertion cdleml6 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑇 ∧ ( 𝑠𝐸𝑠0 ) ) → ( 𝑈𝐸 ∧ ( 𝑈 ‘ ( 𝑠 ) ) = ) )

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 ( 𝑅𝑏 ) ) ( ( 𝑄 ) ( 𝑅 ‘ ( 𝑏 ( 𝑠 ) ) ) ) )
9 cdleml6.y 𝑌 = ( ( 𝑄 ( 𝑅𝑔 ) ) ( 𝑍 ( 𝑅 ‘ ( 𝑔 𝑏 ) ) ) )
10 cdleml6.x 𝑋 = ( 𝑧𝑇𝑏𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅 ‘ ( 𝑠 ) ) ∧ ( 𝑅𝑏 ) ≠ ( 𝑅𝑔 ) ) → ( 𝑧𝑄 ) = 𝑌 ) )
11 cdleml6.u 𝑈 = ( 𝑔𝑇 ↦ if ( ( 𝑠 ) = , 𝑔 , 𝑋 ) )
12 cdleml6.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
13 cdleml6.o 0 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
14 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑇 ∧ ( 𝑠𝐸𝑠0 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
15 simp3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑇 ∧ ( 𝑠𝐸𝑠0 ) ) → 𝑠𝐸 )
16 simp2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑇 ∧ ( 𝑠𝐸𝑠0 ) ) → 𝑇 )
17 4 5 12 tendocl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑠𝐸𝑇 ) → ( 𝑠 ) ∈ 𝑇 )
18 14 15 16 17 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑇 ∧ ( 𝑠𝐸𝑠0 ) ) → ( 𝑠 ) ∈ 𝑇 )
19 1 4 5 6 12 13 tendotr ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑠𝐸𝑠0 ) ∧ 𝑇 ) → ( 𝑅 ‘ ( 𝑠 ) ) = ( 𝑅 ) )
20 19 3com23 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑇 ∧ ( 𝑠𝐸𝑠0 ) ) → ( 𝑅 ‘ ( 𝑠 ) ) = ( 𝑅 ) )
21 eqid ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
22 eqid ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
23 1 2 3 21 22 4 5 6 7 8 9 10 11 12 cdlemk56w ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑠 ) ∈ 𝑇𝑇 ) ∧ ( 𝑅 ‘ ( 𝑠 ) ) = ( 𝑅 ) ) → ( 𝑈𝐸 ∧ ( 𝑈 ‘ ( 𝑠 ) ) = ) )
24 14 18 16 20 23 syl121anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑇 ∧ ( 𝑠𝐸𝑠0 ) ) → ( 𝑈𝐸 ∧ ( 𝑈 ‘ ( 𝑠 ) ) = ) )