Step |
Hyp |
Ref |
Expression |
1 |
|
dihopelvalcp.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihopelvalcp.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihopelvalcp.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dihopelvalcp.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
dihopelvalcp.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
dihopelvalcp.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
dihopelvalcp.p |
⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihopelvalcp.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dihopelvalcp.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
dihopelvalcp.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
dihopelvalcp.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
|
dihopelvalcp.g |
⊢ 𝐺 = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) |
13 |
|
dihopelvalcp.f |
⊢ 𝐹 ∈ V |
14 |
|
dihopelvalcp.s |
⊢ 𝑆 ∈ V |
15 |
|
dihopelvalcp.z |
⊢ 𝑍 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
16 |
|
dihopelvalcp.n |
⊢ 𝑁 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
17 |
|
dihopelvalcp.c |
⊢ 𝐶 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
18 |
|
dihopelvalcp.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
19 |
|
dihopelvalcp.d |
⊢ + = ( +g ‘ 𝑈 ) |
20 |
|
dihopelvalcp.v |
⊢ 𝑉 = ( LSubSp ‘ 𝑈 ) |
21 |
|
dihopelvalcp.y |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
22 |
|
dihopelvalcp.o |
⊢ 𝑂 = ( 𝑎 ∈ 𝐸 , 𝑏 ∈ 𝐸 ↦ ( ℎ ∈ 𝑇 ↦ ( ( 𝑎 ‘ ℎ ) ∘ ( 𝑏 ‘ ℎ ) ) ) ) |
23 |
1 2 3 4 5 6 11 16 17 18 21
|
dihvalcq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |
24 |
23
|
eleq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑋 ) ↔ 〈 𝐹 , 𝑆 〉 ∈ ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
25 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
26 |
|
simp3l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
27 |
2 5 6 18 17 20
|
diclss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐶 ‘ 𝑄 ) ∈ 𝑉 ) |
28 |
25 26 27
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐶 ‘ 𝑄 ) ∈ 𝑉 ) |
29 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝐾 ∈ HL ) |
30 |
29
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝐾 ∈ Lat ) |
31 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
32 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑊 ∈ 𝐻 ) |
33 |
1 6
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
34 |
32 33
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑊 ∈ 𝐵 ) |
35 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
36 |
30 31 34 35
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
37 |
1 2 4
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) |
38 |
30 31 34 37
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) |
39 |
1 2 6 18 16 20
|
diblss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝑉 ) |
40 |
25 36 38 39
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝑉 ) |
41 |
6 18 19 20 21
|
dvhopellsm |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐶 ‘ 𝑄 ) ∈ 𝑉 ∧ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝑉 ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 ‘ 𝑄 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ) ) ) |
42 |
25 28 40 41
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 ‘ 𝑄 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ) ) ) |
43 |
|
vex |
⊢ 𝑥 ∈ V |
44 |
|
vex |
⊢ 𝑦 ∈ V |
45 |
2 5 6 7 8 10 17 12 43 44
|
dicopelval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 ‘ 𝑄 ) ↔ ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ) ) |
46 |
25 26 45
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 ‘ 𝑄 ) ↔ ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ) ) |
47 |
1 2 6 8 9 15 16
|
dibopelval3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 〈 𝑧 , 𝑤 〉 ∈ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ↔ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) |
48 |
25 36 38 47
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 〈 𝑧 , 𝑤 〉 ∈ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ↔ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) |
49 |
46 48
|
anbi12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 ‘ 𝑄 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ↔ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) ) |
50 |
49
|
anbi1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 ‘ 𝑄 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ) ↔ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ) ) ) |
51 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
52 |
|
simprll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑥 = ( 𝑦 ‘ 𝐺 ) ) |
53 |
|
simprlr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑦 ∈ 𝐸 ) |
54 |
2 5 6 7
|
lhpocnel2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
55 |
51 54
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
56 |
|
simpl3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
57 |
2 5 6 8 12
|
ltrniotacl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐺 ∈ 𝑇 ) |
58 |
51 55 56 57
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝐺 ∈ 𝑇 ) |
59 |
6 8 10
|
tendocl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑦 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑦 ‘ 𝐺 ) ∈ 𝑇 ) |
60 |
51 53 58 59
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 𝑦 ‘ 𝐺 ) ∈ 𝑇 ) |
61 |
52 60
|
eqeltrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑥 ∈ 𝑇 ) |
62 |
|
simprll |
⊢ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) → 𝑧 ∈ 𝑇 ) |
63 |
62
|
adantl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑧 ∈ 𝑇 ) |
64 |
|
simprrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑤 = 𝑍 ) |
65 |
1 6 8 10 15
|
tendo0cl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑍 ∈ 𝐸 ) |
66 |
51 65
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑍 ∈ 𝐸 ) |
67 |
64 66
|
eqeltrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑤 ∈ 𝐸 ) |
68 |
|
eqid |
⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 ) |
69 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( +g ‘ ( Scalar ‘ 𝑈 ) ) |
70 |
6 8 10 18 68 19 69
|
dvhopvadd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑧 ∈ 𝑇 ∧ 𝑤 ∈ 𝐸 ) ) → ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑦 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑤 ) 〉 ) |
71 |
51 61 53 63 67 70
|
syl122anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑦 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑤 ) 〉 ) |
72 |
6 8 10 18 68 22 69
|
dvhfplusr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ ( Scalar ‘ 𝑈 ) ) = 𝑂 ) |
73 |
51 72
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( +g ‘ ( Scalar ‘ 𝑈 ) ) = 𝑂 ) |
74 |
73
|
oveqd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 𝑦 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑤 ) = ( 𝑦 𝑂 𝑤 ) ) |
75 |
74
|
opeq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑦 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑤 ) 〉 = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑦 𝑂 𝑤 ) 〉 ) |
76 |
71 75
|
eqtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑦 𝑂 𝑤 ) 〉 ) |
77 |
76
|
eqeq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ↔ 〈 𝐹 , 𝑆 〉 = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑦 𝑂 𝑤 ) 〉 ) ) |
78 |
13 14
|
opth |
⊢ ( 〈 𝐹 , 𝑆 〉 = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑦 𝑂 𝑤 ) 〉 ↔ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = ( 𝑦 𝑂 𝑤 ) ) ) |
79 |
64
|
oveq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 𝑦 𝑂 𝑤 ) = ( 𝑦 𝑂 𝑍 ) ) |
80 |
1 6 8 10 15 22
|
tendo0plr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑦 ∈ 𝐸 ) → ( 𝑦 𝑂 𝑍 ) = 𝑦 ) |
81 |
51 53 80
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 𝑦 𝑂 𝑍 ) = 𝑦 ) |
82 |
79 81
|
eqtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 𝑦 𝑂 𝑤 ) = 𝑦 ) |
83 |
82
|
eqeq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 𝑆 = ( 𝑦 𝑂 𝑤 ) ↔ 𝑆 = 𝑦 ) ) |
84 |
83
|
anbi2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = ( 𝑦 𝑂 𝑤 ) ) ↔ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) |
85 |
78 84
|
syl5bb |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 〈 𝐹 , 𝑆 〉 = 〈 ( 𝑥 ∘ 𝑧 ) , ( 𝑦 𝑂 𝑤 ) 〉 ↔ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) |
86 |
77 85
|
bitrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ↔ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) |
87 |
86
|
pm5.32da |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ) ↔ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) ) |
88 |
|
simplll |
⊢ ( ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) → 𝑥 = ( 𝑦 ‘ 𝐺 ) ) |
89 |
88
|
adantl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑥 = ( 𝑦 ‘ 𝐺 ) ) |
90 |
|
simprrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑆 = 𝑦 ) |
91 |
90
|
fveq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑆 ‘ 𝐺 ) = ( 𝑦 ‘ 𝐺 ) ) |
92 |
89 91
|
eqtr4d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑥 = ( 𝑆 ‘ 𝐺 ) ) |
93 |
90
|
eqcomd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑦 = 𝑆 ) |
94 |
|
coass |
⊢ ( ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) ∘ 𝑧 ) = ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ) |
95 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
96 |
|
simpllr |
⊢ ( ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) → 𝑦 ∈ 𝐸 ) |
97 |
96
|
adantl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑦 ∈ 𝐸 ) |
98 |
90 97
|
eqeltrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑆 ∈ 𝐸 ) |
99 |
58
|
adantrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝐺 ∈ 𝑇 ) |
100 |
6 8 10
|
tendocl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ) |
101 |
95 98 99 100
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ) |
102 |
1 6 8
|
ltrn1o |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ) → ( 𝑆 ‘ 𝐺 ) : 𝐵 –1-1-onto→ 𝐵 ) |
103 |
95 101 102
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑆 ‘ 𝐺 ) : 𝐵 –1-1-onto→ 𝐵 ) |
104 |
|
f1ococnv1 |
⊢ ( ( 𝑆 ‘ 𝐺 ) : 𝐵 –1-1-onto→ 𝐵 → ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) = ( I ↾ 𝐵 ) ) |
105 |
103 104
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) = ( I ↾ 𝐵 ) ) |
106 |
105
|
coeq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) ∘ 𝑧 ) = ( ( I ↾ 𝐵 ) ∘ 𝑧 ) ) |
107 |
62
|
ad2antrl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑧 ∈ 𝑇 ) |
108 |
1 6 8
|
ltrn1o |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑧 ∈ 𝑇 ) → 𝑧 : 𝐵 –1-1-onto→ 𝐵 ) |
109 |
95 107 108
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑧 : 𝐵 –1-1-onto→ 𝐵 ) |
110 |
|
f1of |
⊢ ( 𝑧 : 𝐵 –1-1-onto→ 𝐵 → 𝑧 : 𝐵 ⟶ 𝐵 ) |
111 |
|
fcoi2 |
⊢ ( 𝑧 : 𝐵 ⟶ 𝐵 → ( ( I ↾ 𝐵 ) ∘ 𝑧 ) = 𝑧 ) |
112 |
109 110 111
|
3syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( ( I ↾ 𝐵 ) ∘ 𝑧 ) = 𝑧 ) |
113 |
106 112
|
eqtr2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑧 = ( ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) ∘ 𝑧 ) ) |
114 |
|
simprrl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝐹 = ( 𝑥 ∘ 𝑧 ) ) |
115 |
92
|
coeq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑥 ∘ 𝑧 ) = ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ) |
116 |
114 115
|
eqtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝐹 = ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ) |
117 |
116
|
coeq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) = ( ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) |
118 |
6 8
|
ltrncnv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ) → ◡ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ) |
119 |
95 101 118
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ◡ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ) |
120 |
6 8
|
ltrnco |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ∧ 𝑧 ∈ 𝑇 ) → ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ∈ 𝑇 ) |
121 |
95 101 107 120
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ∈ 𝑇 ) |
122 |
6 8
|
ltrncom |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ◡ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ∧ ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ∈ 𝑇 ) → ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ) = ( ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) |
123 |
95 119 121 122
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ) = ( ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) |
124 |
117 123
|
eqtr4d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) = ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( ( 𝑆 ‘ 𝐺 ) ∘ 𝑧 ) ) ) |
125 |
94 113 124
|
3eqtr4a |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) |
126 |
|
simplrr |
⊢ ( ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) → 𝑤 = 𝑍 ) |
127 |
126
|
adantl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑤 = 𝑍 ) |
128 |
125 127
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) |
129 |
92 93 128
|
jca31 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) |
130 |
129
|
ex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) → ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ) |
131 |
130
|
pm4.71rd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ↔ ( ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) ) ) |
132 |
87 131
|
bitrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ) ↔ ( ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) ) ) |
133 |
|
simprrl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝐹 = ( 𝑥 ∘ 𝑧 ) ) |
134 |
|
simpll1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
135 |
88
|
adantl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑥 = ( 𝑦 ‘ 𝐺 ) ) |
136 |
96
|
adantl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑦 ∈ 𝐸 ) |
137 |
134 54
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
138 |
|
simpl3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
139 |
138
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
140 |
134 137 139 57
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝐺 ∈ 𝑇 ) |
141 |
134 136 140 59
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑦 ‘ 𝐺 ) ∈ 𝑇 ) |
142 |
135 141
|
eqeltrd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑥 ∈ 𝑇 ) |
143 |
62
|
ad2antrl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑧 ∈ 𝑇 ) |
144 |
6 8
|
ltrnco |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇 ) → ( 𝑥 ∘ 𝑧 ) ∈ 𝑇 ) |
145 |
134 142 143 144
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑥 ∘ 𝑧 ) ∈ 𝑇 ) |
146 |
133 145
|
eqeltrd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝐹 ∈ 𝑇 ) |
147 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝐾 ∈ HL ) |
148 |
147
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝐾 ∈ HL ) |
149 |
148
|
hllatd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝐾 ∈ Lat ) |
150 |
1 6 8 9
|
trlcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑧 ∈ 𝑇 ) → ( 𝑅 ‘ 𝑧 ) ∈ 𝐵 ) |
151 |
134 143 150
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑅 ‘ 𝑧 ) ∈ 𝐵 ) |
152 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑋 ∈ 𝐵 ) |
153 |
152
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑋 ∈ 𝐵 ) |
154 |
|
simpl1r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑊 ∈ 𝐻 ) |
155 |
154
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑊 ∈ 𝐻 ) |
156 |
155 33
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → 𝑊 ∈ 𝐵 ) |
157 |
149 153 156 35
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
158 |
|
simprlr |
⊢ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) → ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) |
159 |
158
|
ad2antrl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) |
160 |
1 2 4
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑋 ) |
161 |
149 153 156 160
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑋 ) |
162 |
1 2 149 151 157 153 159 161
|
lattrd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) |
163 |
146 136 162
|
jca31 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) → ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) |
164 |
|
simprll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑥 = ( 𝑆 ‘ 𝐺 ) ) |
165 |
164
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑥 = ( 𝑆 ‘ 𝐺 ) ) |
166 |
|
simprlr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑦 = 𝑆 ) |
167 |
166
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑦 = 𝑆 ) |
168 |
167
|
fveq1d |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑦 ‘ 𝐺 ) = ( 𝑆 ‘ 𝐺 ) ) |
169 |
165 168
|
eqtr4d |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑥 = ( 𝑦 ‘ 𝐺 ) ) |
170 |
|
simprlr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑦 ∈ 𝐸 ) |
171 |
169 170
|
jca |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ) |
172 |
|
simprrl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) |
173 |
172
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) |
174 |
|
simpll1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
175 |
|
simprll |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝐹 ∈ 𝑇 ) |
176 |
167 170
|
eqeltrrd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑆 ∈ 𝐸 ) |
177 |
174 54
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
178 |
138
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
179 |
174 177 178 57
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝐺 ∈ 𝑇 ) |
180 |
174 176 179 100
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ) |
181 |
174 180 118
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ◡ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ) |
182 |
6 8
|
ltrnco |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ◡ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ) → ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∈ 𝑇 ) |
183 |
174 175 181 182
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∈ 𝑇 ) |
184 |
173 183
|
eqeltrd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑧 ∈ 𝑇 ) |
185 |
|
simprr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) |
186 |
2 6 8 9
|
trlle |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑧 ∈ 𝑇 ) → ( 𝑅 ‘ 𝑧 ) ≤ 𝑊 ) |
187 |
174 184 186
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑧 ) ≤ 𝑊 ) |
188 |
147
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝐾 ∈ HL ) |
189 |
188
|
hllatd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝐾 ∈ Lat ) |
190 |
174 184 150
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑧 ) ∈ 𝐵 ) |
191 |
152
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
192 |
154
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑊 ∈ 𝐻 ) |
193 |
192 33
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑊 ∈ 𝐵 ) |
194 |
1 2 4
|
latlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑅 ‘ 𝑧 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑊 ) ↔ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) |
195 |
189 190 191 193 194
|
syl13anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( ( ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑊 ) ↔ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) |
196 |
185 187 195
|
mpbi2and |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) |
197 |
|
simprrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) → 𝑤 = 𝑍 ) |
198 |
197
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑤 = 𝑍 ) |
199 |
184 196 198
|
jca31 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) |
200 |
174 180 102
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑆 ‘ 𝐺 ) : 𝐵 –1-1-onto→ 𝐵 ) |
201 |
200 104
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) = ( I ↾ 𝐵 ) ) |
202 |
201
|
coeq2d |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝐹 ∘ ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) ) = ( 𝐹 ∘ ( I ↾ 𝐵 ) ) ) |
203 |
1 6 8
|
ltrn1o |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
204 |
174 175 203
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
205 |
|
f1of |
⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → 𝐹 : 𝐵 ⟶ 𝐵 ) |
206 |
|
fcoi1 |
⊢ ( 𝐹 : 𝐵 ⟶ 𝐵 → ( 𝐹 ∘ ( I ↾ 𝐵 ) ) = 𝐹 ) |
207 |
204 205 206
|
3syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝐹 ∘ ( I ↾ 𝐵 ) ) = 𝐹 ) |
208 |
202 207
|
eqtr2d |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝐹 = ( 𝐹 ∘ ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) ) ) |
209 |
|
coass |
⊢ ( ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∘ ( 𝑆 ‘ 𝐺 ) ) = ( 𝐹 ∘ ( ◡ ( 𝑆 ‘ 𝐺 ) ∘ ( 𝑆 ‘ 𝐺 ) ) ) |
210 |
208 209
|
eqtr4di |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝐹 = ( ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∘ ( 𝑆 ‘ 𝐺 ) ) ) |
211 |
6 8
|
ltrncom |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ∧ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∈ 𝑇 ) → ( ( 𝑆 ‘ 𝐺 ) ∘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) = ( ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∘ ( 𝑆 ‘ 𝐺 ) ) ) |
212 |
174 180 183 211
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( ( 𝑆 ‘ 𝐺 ) ∘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) = ( ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∘ ( 𝑆 ‘ 𝐺 ) ) ) |
213 |
210 212
|
eqtr4d |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝐹 = ( ( 𝑆 ‘ 𝐺 ) ∘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) ) |
214 |
165 173
|
coeq12d |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝑥 ∘ 𝑧 ) = ( ( 𝑆 ‘ 𝐺 ) ∘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) ) |
215 |
213 214
|
eqtr4d |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝐹 = ( 𝑥 ∘ 𝑧 ) ) |
216 |
167
|
eqcomd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → 𝑆 = 𝑦 ) |
217 |
215 216
|
jca |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) |
218 |
171 199 217
|
jca31 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) → ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) |
219 |
163 218
|
impbida |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ) → ( ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) ) |
220 |
219
|
pm5.32da |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) ↔ ( ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) ) ) |
221 |
|
df-3an |
⊢ ( ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) ↔ ( ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) ) |
222 |
220 221
|
bitr4di |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( ( ( 𝑥 = ( 𝑦 ‘ 𝐺 ) ∧ 𝑦 ∈ 𝐸 ) ∧ ( ( 𝑧 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑧 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑤 = 𝑍 ) ) ∧ ( 𝐹 = ( 𝑥 ∘ 𝑧 ) ∧ 𝑆 = 𝑦 ) ) ) ↔ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) ) ) |
223 |
50 132 222
|
3bitrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 ‘ 𝑄 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ) ↔ ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) ) ) |
224 |
223
|
4exbidv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 ‘ 𝑄 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) ) ) |
225 |
|
fvex |
⊢ ( 𝑆 ‘ 𝐺 ) ∈ V |
226 |
225
|
cnvex |
⊢ ◡ ( 𝑆 ‘ 𝐺 ) ∈ V |
227 |
13 226
|
coex |
⊢ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∈ V |
228 |
8
|
fvexi |
⊢ 𝑇 ∈ V |
229 |
228
|
mptex |
⊢ ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) ∈ V |
230 |
15 229
|
eqeltri |
⊢ 𝑍 ∈ V |
231 |
|
biidd |
⊢ ( 𝑥 = ( 𝑆 ‘ 𝐺 ) → ( ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) ) |
232 |
|
eleq1 |
⊢ ( 𝑦 = 𝑆 → ( 𝑦 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸 ) ) |
233 |
232
|
anbi2d |
⊢ ( 𝑦 = 𝑆 → ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ↔ ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) ) ) |
234 |
233
|
anbi1d |
⊢ ( 𝑦 = 𝑆 → ( ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) ) |
235 |
|
fveq2 |
⊢ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) → ( 𝑅 ‘ 𝑧 ) = ( 𝑅 ‘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) ) |
236 |
235
|
breq1d |
⊢ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) → ( ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ↔ ( 𝑅 ‘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) ≤ 𝑋 ) ) |
237 |
236
|
anbi2d |
⊢ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) → ( ( ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) ∧ ( 𝑅 ‘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) ≤ 𝑋 ) ) ) |
238 |
|
biidd |
⊢ ( 𝑤 = 𝑍 → ( ( ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) ∧ ( 𝑅 ‘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) ≤ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) ∧ ( 𝑅 ‘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) ≤ 𝑋 ) ) ) |
239 |
225 14 227 230 231 234 237 238
|
ceqsex4v |
⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑦 = 𝑆 ) ∧ ( 𝑧 = ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ∧ 𝑤 = 𝑍 ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸 ) ∧ ( 𝑅 ‘ 𝑧 ) ≤ 𝑋 ) ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) ∧ ( 𝑅 ‘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) ≤ 𝑋 ) ) |
240 |
224 239
|
bitrdi |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 ‘ 𝑄 ) ∧ 〈 𝑧 , 𝑤 〉 ∈ ( 𝑁 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∧ 〈 𝐹 , 𝑆 〉 = ( 〈 𝑥 , 𝑦 〉 + 〈 𝑧 , 𝑤 〉 ) ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) ∧ ( 𝑅 ‘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) ≤ 𝑋 ) ) ) |
241 |
24 42 240
|
3bitrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 〈 𝐹 , 𝑆 〉 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸 ) ∧ ( 𝑅 ‘ ( 𝐹 ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) ) ≤ 𝑋 ) ) ) |