Metamath Proof Explorer


Theorem cdleme26fALTN

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph, 6th and 7th lines on p. 115. F , N represent f(t), f_t(s) respectively. If t <_ t \/ v, then f_t(s) <_ f(t) \/ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013) (New usage is discouraged.)

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 )
cdleme26f.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme26f.f 𝐹 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdleme26f.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑆 𝑡 ) 𝑊 ) ) )
cdleme26f.i 𝐼 = ( 𝑢𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑁 ) )
Assertion cdleme26fALTN ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝐼 ( 𝐹 𝑉 ) )

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 cdleme26f.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdleme26f.f 𝐹 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
9 cdleme26f.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑆 𝑡 ) 𝑊 ) ) )
10 cdleme26f.i 𝐼 = ( 𝑢𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑁 ) )
11 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
12 simp21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
13 simp22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
14 simp23l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑆𝐴 )
15 simp23r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ¬ 𝑆 𝑊 )
16 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑃𝑄 )
17 simp12r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑆 ( 𝑃 𝑄 ) )
18 1 2 3 4 5 6 7 8 9 10 cdleme25cl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ) → 𝐼𝐵 )
19 11 12 13 14 15 16 17 18 syl322anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝐼𝐵 )
20 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑡𝐴 )
21 simp31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) )
22 1 fvexi 𝐵 ∈ V
23 22 10 riotasv ( ( 𝐼𝐵𝑡𝐴 ∧ ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ) → 𝐼 = 𝑁 )
24 19 20 21 23 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝐼 = 𝑁 )
25 simp23 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
26 simp33 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑉𝐴𝑉 𝑊 ) )
27 simp32 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) )
28 2 3 4 5 6 7 8 9 cdleme22f ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ 𝑡𝐴 ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ) → 𝑁 ( 𝐹 𝑉 ) )
29 11 12 13 25 20 26 27 28 syl331anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑁 ( 𝐹 𝑉 ) )
30 24 29 eqbrtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝑄𝑆 ( 𝑃 𝑄 ) ) ∧ ( 𝑡𝐴 ∧ ¬ 𝑡 𝑊 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) ∧ ( 𝑆𝑡𝑆 ( 𝑡 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝐼 ( 𝐹 𝑉 ) )