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Metamath Proof Explorer

Theorem dihlsscpre

Description: Closure of isomorphism H for a lattice K when -. X .<_ W . (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 dihlsscpre ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 )

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 1 2 3 4 5 6 7 8 9 10 11 12 dihvalc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
14 simp1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → 𝑞 ∈ 𝐴 )
16 simp3ll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ¬ 𝑞 ≤ 𝑊 )
17 15 16 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) )
18 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → 𝑟 ∈ 𝐴 )
19 simp3rl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ¬ 𝑟 ≤ 𝑊 )
20 18 19 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) )
21 simp1rl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → 𝑋 ∈ 𝐵 )
22 simp3lr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
23 simp3rr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
24 22 23 eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) )
25 1 2 3 4 5 6 8 9 10 12 dihjust ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑟 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
26 14 17 20 21 24 25 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑟 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
27 26 3exp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ( 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) → ( ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑟 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
28 27 ralrimivv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∀ 𝑞 ∈ 𝐴 ∀ 𝑟 ∈ 𝐴 ( ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑟 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
29 1 2 3 4 5 6 lhpmcvr2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
30 simpll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
31 6 10 30 dvhlmod ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝑈 ∈ LMod )
32 2 5 6 10 9 11 diclss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( 𝐶 ‘ 𝑞 ) ∈ 𝑆 )
33 32 adantlr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( 𝐶 ‘ 𝑞 ) ∈ 𝑆 )
34 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
35 34 ad3antrrr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝐾 ∈ Lat )
36 simplrl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
37 1 6 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
38 37 ad3antlr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐵 )
39 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
40 35 36 38 39 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
41 1 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
42 35 36 38 41 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
43 1 2 6 10 8 11 diblss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝑆 )
44 30 40 42 43 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝑆 )
45 11 12 lsmcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐶 ‘ 𝑞 ) ∈ 𝑆 ∧ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝑆 ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ 𝑆 )
46 31 33 44 45 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ 𝑆 )
47 46 a1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ( ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ 𝑆 ) )
48 47 expr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑞 ∈ 𝐴 ) → ( ¬ 𝑞 ≤ 𝑊 → ( ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ 𝑆 ) ) )
49 48 impd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑞 ∈ 𝐴 ) → ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ 𝑆 ) )
50 49 ancld ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ 𝑞 ∈ 𝐴 ) → ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ 𝑆 ) ) )
51 50 reximdva ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ∃ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ 𝑆 ) ) )
52 29 51 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ 𝑆 ) )
53 breq1 ⊢ ( 𝑞 = 𝑟 → ( 𝑞 ≤ 𝑊 ↔ 𝑟 ≤ 𝑊 ) )
54 53 notbid ⊢ ( 𝑞 = 𝑟 → ( ¬ 𝑞 ≤ 𝑊 ↔ ¬ 𝑟 ≤ 𝑊 ) )
55 oveq1 ⊢ ( 𝑞 = 𝑟 → ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) )
56 55 eqeq1d ⊢ ( 𝑞 = 𝑟 → ( ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ↔ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
57 54 56 anbi12d ⊢ ( 𝑞 = 𝑟 → ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ↔ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) )
58 fveq2 ⊢ ( 𝑞 = 𝑟 → ( 𝐶 ‘ 𝑞 ) = ( 𝐶 ‘ 𝑟 ) )
59 58 oveq1d ⊢ ( 𝑞 = 𝑟 → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑟 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
60 57 59 reusv3 ⊢ ( ∃ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ 𝑆 ) → ( ∀ 𝑞 ∈ 𝐴 ∀ 𝑟 ∈ 𝐴 ( ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑟 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ↔ ∃ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
61 52 60 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ∀ 𝑞 ∈ 𝐴 ∀ 𝑟 ∈ 𝐴 ( ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑟 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ↔ ∃ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
62 28 61 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
63 reusv1 ⊢ ( ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ( ∃! 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ↔ ∃ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
64 29 63 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ∃! 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ↔ ∃ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
65 62 64 mpbird ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃! 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
66 riotacl ⊢ ( ∃! 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) → ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ∈ 𝑆 )
67 65 66 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ∈ 𝑆 )
68 13 67 eqeltrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 )