Metamath Proof Explorer


Theorem cdleme28c

Description: Part of proof of Lemma E in Crawley p. 113. Eliminate the s =/= t antecedent in cdleme28b . TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013)

Ref Expression
Hypotheses cdleme26.b 𝐵 = ( Base ‘ 𝐾 )
cdleme26.l = ( le ‘ 𝐾 )
cdleme26.j = ( join ‘ 𝐾 )
cdleme26.m = ( meet ‘ 𝐾 )
cdleme26.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme26.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme27.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme27.f 𝐹 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
cdleme27.z 𝑍 = ( ( 𝑧 𝑈 ) ( 𝑄 ( ( 𝑃 𝑧 ) 𝑊 ) ) )
cdleme27.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝑍 ( ( 𝑠 𝑧 ) 𝑊 ) ) )
cdleme27.d 𝐷 = ( 𝑢𝐵𝑧𝐴 ( ( ¬ 𝑧 𝑊 ∧ ¬ 𝑧 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑁 ) )
cdleme27.c 𝐶 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐷 , 𝐹 )
cdleme27.g 𝐺 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdleme27.o 𝑂 = ( ( 𝑃 𝑄 ) ( 𝑍 ( ( 𝑡 𝑧 ) 𝑊 ) ) )
cdleme27.e 𝐸 = ( 𝑢𝐵𝑧𝐴 ( ( ¬ 𝑧 𝑊 ∧ ¬ 𝑧 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑂 ) )
cdleme27.y 𝑌 = if ( 𝑡 ( 𝑃 𝑄 ) , 𝐸 , 𝐺 )
Assertion cdleme28c ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) → ( 𝐶 ( 𝑋 𝑊 ) ) = ( 𝑌 ( 𝑋 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 cdleme26.b 𝐵 = ( Base ‘ 𝐾 )
2 cdleme26.l = ( le ‘ 𝐾 )
3 cdleme26.j = ( join ‘ 𝐾 )
4 cdleme26.m = ( meet ‘ 𝐾 )
5 cdleme26.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme26.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme27.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdleme27.f 𝐹 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
9 cdleme27.z 𝑍 = ( ( 𝑧 𝑈 ) ( 𝑄 ( ( 𝑃 𝑧 ) 𝑊 ) ) )
10 cdleme27.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝑍 ( ( 𝑠 𝑧 ) 𝑊 ) ) )
11 cdleme27.d 𝐷 = ( 𝑢𝐵𝑧𝐴 ( ( ¬ 𝑧 𝑊 ∧ ¬ 𝑧 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑁 ) )
12 cdleme27.c 𝐶 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐷 , 𝐹 )
13 cdleme27.g 𝐺 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
14 cdleme27.o 𝑂 = ( ( 𝑃 𝑄 ) ( 𝑍 ( ( 𝑡 𝑧 ) 𝑊 ) ) )
15 cdleme27.e 𝐸 = ( 𝑢𝐵𝑧𝐴 ( ( ¬ 𝑧 𝑊 ∧ ¬ 𝑧 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑂 ) )
16 cdleme27.y 𝑌 = if ( 𝑡 ( 𝑃 𝑄 ) , 𝐸 , 𝐺 )
17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cdleme27b ( 𝑠 = 𝑡𝐶 = 𝑌 )
18 17 oveq1d ( 𝑠 = 𝑡 → ( 𝐶 ( 𝑋 𝑊 ) ) = ( 𝑌 ( 𝑋 𝑊 ) ) )
19 18 adantl ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠 = 𝑡 ) → ( 𝐶 ( 𝑋 𝑊 ) ) = ( 𝑌 ( 𝑋 𝑊 ) ) )
20 simpl11 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
21 simpl12 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
22 simpl13 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
23 simpl21 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → 𝑃𝑄 )
24 simpl22 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) )
25 simpl23 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) )
26 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → 𝑠𝑡 )
27 simpl31 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 )
28 simpl32 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 )
29 27 28 jca ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ) )
30 simpl33 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) )
31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cdleme28b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( 𝑠𝑡 ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) → ( 𝐶 ( 𝑋 𝑊 ) ) = ( 𝑌 ( 𝑋 𝑊 ) ) )
32 20 21 22 23 24 25 26 29 30 31 syl333anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) ∧ 𝑠𝑡 ) → ( 𝐶 ( 𝑋 𝑊 ) ) = ( 𝑌 ( 𝑋 𝑊 ) ) )
33 19 32 pm2.61dane ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑠 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) ) → ( 𝐶 ( 𝑋 𝑊 ) ) = ( 𝑌 ( 𝑋 𝑊 ) ) )