Metamath Proof Explorer

Theorem cdlemg1a

Description: Shorter expression for G . TODO: fix comment. TODO: shorten using cdleme or vice-versa? Also, if not shortened with cdleme , then it can be moved up to save repeating hypotheses. (Contributed by NM, 15-Apr-2013)

Ref Expression
Hypotheses cdlemg1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemg1.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg1.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg1.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemg1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg1.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdlemg1.d ⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdlemg1.e ⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdlemg1.g ⊢ 𝐺 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) , 𝑥 ) )
cdlemg1.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg1a ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐺 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 ) )

Proof

Step Hyp Ref Expression
1 cdlemg1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemg1.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemg1.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemg1.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemg1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemg1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemg1.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdlemg1.d ⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
9 cdlemg1.e ⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
10 cdlemg1.g ⊢ 𝐺 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) , 𝑥 ) )
11 cdlemg1.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
12 1 2 3 4 5 6 7 8 9 10 11 cdleme50ltrn ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐺 ∈ 𝑇 )
13 simpll1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑓 ‘ 𝑃 ) = 𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14 simplr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑓 ‘ 𝑃 ) = 𝑄 ) → 𝑓 ∈ 𝑇 )
15 12 ad2antrr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑓 ‘ 𝑃 ) = 𝑄 ) → 𝐺 ∈ 𝑇 )
16 simpll2 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑓 ‘ 𝑃 ) = 𝑄 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
17 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑓 ‘ 𝑃 ) = 𝑄 ) → ( 𝑓 ‘ 𝑃 ) = 𝑄 )
18 1 2 3 4 5 6 7 8 9 10 cdleme17d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐺 ‘ 𝑃 ) = 𝑄 )
19 18 ad2antrr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑓 ‘ 𝑃 ) = 𝑄 ) → ( 𝐺 ‘ 𝑃 ) = 𝑄 )
20 17 19 eqtr4d ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑓 ‘ 𝑃 ) = 𝑄 ) → ( 𝑓 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
21 2 5 6 11 cdlemd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑓 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝑓 = 𝐺 )
22 13 14 15 16 20 21 syl311anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑓 ‘ 𝑃 ) = 𝑄 ) → 𝑓 = 𝐺 )
23 22 ex ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) → ( ( 𝑓 ‘ 𝑃 ) = 𝑄 → 𝑓 = 𝐺 ) )
24 18 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) → ( 𝐺 ‘ 𝑃 ) = 𝑄 )
25 fveq1 ⊢ ( 𝑓 = 𝐺 → ( 𝑓 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
26 25 eqeq1d ⊢ ( 𝑓 = 𝐺 → ( ( 𝑓 ‘ 𝑃 ) = 𝑄 ↔ ( 𝐺 ‘ 𝑃 ) = 𝑄 ) )
27 24 26 syl5ibrcom ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) → ( 𝑓 = 𝐺 → ( 𝑓 ‘ 𝑃 ) = 𝑄 ) )
28 23 27 impbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑓 ∈ 𝑇 ) → ( ( 𝑓 ‘ 𝑃 ) = 𝑄 ↔ 𝑓 = 𝐺 ) )
29 12 28 riota5 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 ) = 𝐺 )
30 29 eqcomd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐺 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 ) )