Metamath Proof Explorer


Theorem dihvalcqpre

Description: Value of isomorphism H for a lattice K when -. X .<_ W , given auxiliary atom Q . (Contributed by NM, 6-Mar-2014)

Ref Expression
Hypotheses dihval.b 𝐵 = ( Base ‘ 𝐾 )
dihval.l = ( le ‘ 𝐾 )
dihval.j = ( join ‘ 𝐾 )
dihval.m = ( meet ‘ 𝐾 )
dihval.a 𝐴 = ( Atoms ‘ 𝐾 )
dihval.h 𝐻 = ( LHyp ‘ 𝐾 )
dihval.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihval.d 𝐷 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
dihval.c 𝐶 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
dihval.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihval.s 𝑆 = ( LSubSp ‘ 𝑈 )
dihval.p = ( LSSum ‘ 𝑈 )
Assertion dihvalcqpre ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐼𝑋 ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) )

Proof

Step Hyp Ref Expression
1 dihval.b 𝐵 = ( Base ‘ 𝐾 )
2 dihval.l = ( le ‘ 𝐾 )
3 dihval.j = ( join ‘ 𝐾 )
4 dihval.m = ( meet ‘ 𝐾 )
5 dihval.a 𝐴 = ( Atoms ‘ 𝐾 )
6 dihval.h 𝐻 = ( LHyp ‘ 𝐾 )
7 dihval.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
8 dihval.d 𝐷 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
9 dihval.c 𝐶 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
10 dihval.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
11 dihval.s 𝑆 = ( LSubSp ‘ 𝑈 )
12 dihval.p = ( LSSum ‘ 𝑈 )
13 11 fvexi 𝑆 ∈ V
14 nfv 𝑞 ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) )
15 nfvd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → Ⅎ 𝑞 ( 𝐼𝑋 ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) )
16 1 2 3 4 5 6 7 8 9 10 11 12 dihvalc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → ( 𝐼𝑋 ) = ( 𝑢𝑆𝑞𝐴 ( ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶𝑞 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) ) ) )
17 16 3adant3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐼𝑋 ) = ( 𝑢𝑆𝑞𝐴 ( ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶𝑞 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) ) ) )
18 eqeq1 ( ( ( 𝐶𝑞 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) = ( 𝐼𝑋 ) → ( ( ( 𝐶𝑞 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) ↔ ( 𝐼𝑋 ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) ) )
19 18 adantl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝐶𝑞 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) = ( 𝐼𝑋 ) ) → ( ( ( 𝐶𝑞 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) ↔ ( 𝐼𝑋 ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) ) )
20 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
21 simprl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ) → 𝑞𝐴 )
22 simprrl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ) → ¬ 𝑞 𝑊 )
23 21 22 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞𝐴 ∧ ¬ 𝑞 𝑊 ) )
24 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
25 simpl2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ) → 𝑋𝐵 )
26 simprrr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 )
27 simpl3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 )
28 26 27 eqtr4d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ( 𝑋 𝑊 ) ) = ( 𝑄 ( 𝑋 𝑊 ) ) )
29 1 2 3 4 5 6 8 9 10 12 dihjust ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑞𝐴 ∧ ¬ 𝑞 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = ( 𝑄 ( 𝑋 𝑊 ) ) ) → ( ( 𝐶𝑞 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) )
30 20 23 24 25 28 29 syl131anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ) → ( ( 𝐶𝑞 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) )
31 30 ex ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( ( 𝑞𝐴 ∧ ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐶𝑞 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) ) )
32 1 2 3 4 5 6 7 8 9 10 11 12 dihlsscpre ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → ( 𝐼𝑋 ) ∈ 𝑆 )
33 32 3adant3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐼𝑋 ) ∈ 𝑆 )
34 1 2 3 4 5 6 lhpmcvr2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ) → ∃ 𝑞𝐴 ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) )
35 34 3adant3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ∃ 𝑞𝐴 ( ¬ 𝑞 𝑊 ∧ ( 𝑞 ( 𝑋 𝑊 ) ) = 𝑋 ) )
36 14 15 17 19 31 33 35 riotasv3d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) ∧ 𝑆 ∈ V ) → ( 𝐼𝑋 ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) )
37 13 36 mpan2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵 ∧ ¬ 𝑋 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ) ) → ( 𝐼𝑋 ) = ( ( 𝐶𝑄 ) ( 𝐷 ‘ ( 𝑋 𝑊 ) ) ) )