Step |
Hyp |
Ref |
Expression |
1 |
|
trlord.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
trlord.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
trlord.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
trlord.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
5 |
|
trlord.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
trlord.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) ) → 𝐾 ∈ HL ) |
8 |
7
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) ) → 𝐾 ∈ Lat ) |
9 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
10 |
|
simprlr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) ) → 𝑓 ∈ 𝑇 ) |
11 |
1 4 5 6
|
trlcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ 𝑇 ) → ( 𝑅 ‘ 𝑓 ) ∈ 𝐵 ) |
12 |
9 10 11
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑓 ) ∈ 𝐵 ) |
13 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
14 |
|
simpl3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) ) → 𝑌 ∈ 𝐵 ) |
15 |
|
simprr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) |
16 |
|
simprll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) ) → 𝑋 ≤ 𝑌 ) |
17 |
1 2 8 12 13 14 15 16
|
lattrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑓 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) |
18 |
17
|
exp44 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑓 ∈ 𝑇 → ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ) ) ) |
19 |
18
|
ralrimdv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝑋 ≤ 𝑌 → ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ) ) |
20 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → 𝐾 ∈ HL ) |
21 |
20
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → 𝐾 ∈ Lat ) |
22 |
|
simp2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → 𝑢 ∈ 𝐴 ) |
23 |
1 3
|
atbase |
⊢ ( 𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵 ) |
24 |
22 23
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → 𝑢 ∈ 𝐵 ) |
25 |
|
simp12l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → 𝑋 ∈ 𝐵 ) |
26 |
|
simp11r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → 𝑊 ∈ 𝐻 ) |
27 |
1 4
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
28 |
26 27
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → 𝑊 ∈ 𝐵 ) |
29 |
|
simp3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → 𝑢 ≤ 𝑋 ) |
30 |
|
simp12r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → 𝑋 ≤ 𝑊 ) |
31 |
1 2 21 24 25 28 29 30
|
lattrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → 𝑢 ≤ 𝑊 ) |
32 |
31 29
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑋 ) → ( 𝑢 ≤ 𝑊 ∧ 𝑢 ≤ 𝑋 ) ) |
33 |
32
|
3expia |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ) → ( 𝑢 ≤ 𝑋 → ( 𝑢 ≤ 𝑊 ∧ 𝑢 ≤ 𝑋 ) ) ) |
34 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
35 |
|
simp2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑊 ) → 𝑢 ∈ 𝐴 ) |
36 |
|
simp3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑊 ) → 𝑢 ≤ 𝑊 ) |
37 |
2 3 4 5 6
|
cdlemf |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊 ) ) → ∃ 𝑔 ∈ 𝑇 ( 𝑅 ‘ 𝑔 ) = 𝑢 ) |
38 |
34 35 36 37
|
syl12anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑊 ) → ∃ 𝑔 ∈ 𝑇 ( 𝑅 ‘ 𝑔 ) = 𝑢 ) |
39 |
|
simp2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑊 ) → ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ) |
40 |
|
fveq2 |
⊢ ( 𝑓 = 𝑔 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝑔 ) ) |
41 |
40
|
breq1d |
⊢ ( 𝑓 = 𝑔 → ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ↔ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) |
42 |
40
|
breq1d |
⊢ ( 𝑓 = 𝑔 → ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ↔ ( 𝑅 ‘ 𝑔 ) ≤ 𝑌 ) ) |
43 |
41 42
|
imbi12d |
⊢ ( 𝑓 = 𝑔 → ( ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ↔ ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑔 ) ≤ 𝑌 ) ) ) |
44 |
43
|
rspccv |
⊢ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) → ( 𝑔 ∈ 𝑇 → ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑔 ) ≤ 𝑌 ) ) ) |
45 |
39 44
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑊 ) → ( 𝑔 ∈ 𝑇 → ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑔 ) ≤ 𝑌 ) ) ) |
46 |
|
breq1 |
⊢ ( ( 𝑅 ‘ 𝑔 ) = 𝑢 → ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ↔ 𝑢 ≤ 𝑋 ) ) |
47 |
|
breq1 |
⊢ ( ( 𝑅 ‘ 𝑔 ) = 𝑢 → ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑌 ↔ 𝑢 ≤ 𝑌 ) ) |
48 |
46 47
|
imbi12d |
⊢ ( ( 𝑅 ‘ 𝑔 ) = 𝑢 → ( ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑔 ) ≤ 𝑌 ) ↔ ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) ) |
49 |
48
|
biimpcd |
⊢ ( ( ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑔 ) ≤ 𝑌 ) → ( ( 𝑅 ‘ 𝑔 ) = 𝑢 → ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) ) |
50 |
45 49
|
syl6 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑊 ) → ( 𝑔 ∈ 𝑇 → ( ( 𝑅 ‘ 𝑔 ) = 𝑢 → ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) ) ) |
51 |
50
|
rexlimdv |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑊 ) → ( ∃ 𝑔 ∈ 𝑇 ( 𝑅 ‘ 𝑔 ) = 𝑢 → ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) ) |
52 |
38 51
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ∧ 𝑢 ≤ 𝑊 ) → ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) |
53 |
52
|
3expia |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ) → ( 𝑢 ≤ 𝑊 → ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) ) |
54 |
53
|
impd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ) → ( ( 𝑢 ≤ 𝑊 ∧ 𝑢 ≤ 𝑋 ) → 𝑢 ≤ 𝑌 ) ) |
55 |
33 54
|
syld |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ∧ 𝑢 ∈ 𝐴 ) ) → ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) |
56 |
55
|
exp32 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) → ( 𝑢 ∈ 𝐴 → ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) ) ) |
57 |
56
|
ralrimdv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) → ∀ 𝑢 ∈ 𝐴 ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) ) |
58 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → 𝐾 ∈ HL ) |
59 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 ) |
60 |
|
simp3l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → 𝑌 ∈ 𝐵 ) |
61 |
1 2 3
|
hlatle |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ∀ 𝑢 ∈ 𝐴 ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) ) |
62 |
58 59 60 61
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝑋 ≤ 𝑌 ↔ ∀ 𝑢 ∈ 𝐴 ( 𝑢 ≤ 𝑋 → 𝑢 ≤ 𝑌 ) ) ) |
63 |
57 62
|
sylibrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) → 𝑋 ≤ 𝑌 ) ) |
64 |
19 63
|
impbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝑋 ≤ 𝑌 ↔ ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 → ( 𝑅 ‘ 𝑓 ) ≤ 𝑌 ) ) ) |