Step |
Hyp |
Ref |
Expression |
1 |
|
archiabllem.b |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
2 |
|
archiabllem.0 |
⊢ 0 = ( 0g ‘ 𝑊 ) |
3 |
|
archiabllem.e |
⊢ ≤ = ( le ‘ 𝑊 ) |
4 |
|
archiabllem.t |
⊢ < = ( lt ‘ 𝑊 ) |
5 |
|
archiabllem.m |
⊢ · = ( .g ‘ 𝑊 ) |
6 |
|
archiabllem.g |
⊢ ( 𝜑 → 𝑊 ∈ oGrp ) |
7 |
|
archiabllem.a |
⊢ ( 𝜑 → 𝑊 ∈ Archi ) |
8 |
|
archiabllem2.1 |
⊢ + = ( +g ‘ 𝑊 ) |
9 |
|
archiabllem2.2 |
⊢ ( 𝜑 → ( oppg ‘ 𝑊 ) ∈ oGrp ) |
10 |
|
archiabllem2.3 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎 ) → ∃ 𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎 ) ) |
11 |
|
archiabllem2a.4 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
12 |
|
archiabllem2a.5 |
⊢ ( 𝜑 → 0 < 𝑋 ) |
13 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ ( 𝑏 + 𝑏 ) ≤ 𝑋 ) → 𝑏 ∈ 𝐵 ) |
14 |
|
simplrl |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ ( 𝑏 + 𝑏 ) ≤ 𝑋 ) → 0 < 𝑏 ) |
15 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ ( 𝑏 + 𝑏 ) ≤ 𝑋 ) → ( 𝑏 + 𝑏 ) ≤ 𝑋 ) |
16 |
|
breq2 |
⊢ ( 𝑐 = 𝑏 → ( 0 < 𝑐 ↔ 0 < 𝑏 ) ) |
17 |
|
id |
⊢ ( 𝑐 = 𝑏 → 𝑐 = 𝑏 ) |
18 |
17 17
|
oveq12d |
⊢ ( 𝑐 = 𝑏 → ( 𝑐 + 𝑐 ) = ( 𝑏 + 𝑏 ) ) |
19 |
18
|
breq1d |
⊢ ( 𝑐 = 𝑏 → ( ( 𝑐 + 𝑐 ) ≤ 𝑋 ↔ ( 𝑏 + 𝑏 ) ≤ 𝑋 ) ) |
20 |
16 19
|
anbi12d |
⊢ ( 𝑐 = 𝑏 → ( ( 0 < 𝑐 ∧ ( 𝑐 + 𝑐 ) ≤ 𝑋 ) ↔ ( 0 < 𝑏 ∧ ( 𝑏 + 𝑏 ) ≤ 𝑋 ) ) ) |
21 |
20
|
rspcev |
⊢ ( ( 𝑏 ∈ 𝐵 ∧ ( 0 < 𝑏 ∧ ( 𝑏 + 𝑏 ) ≤ 𝑋 ) ) → ∃ 𝑐 ∈ 𝐵 ( 0 < 𝑐 ∧ ( 𝑐 + 𝑐 ) ≤ 𝑋 ) ) |
22 |
13 14 15 21
|
syl12anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ ( 𝑏 + 𝑏 ) ≤ 𝑋 ) → ∃ 𝑐 ∈ 𝐵 ( 0 < 𝑐 ∧ ( 𝑐 + 𝑐 ) ≤ 𝑋 ) ) |
23 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → 𝜑 ) |
24 |
|
ogrpgrp |
⊢ ( 𝑊 ∈ oGrp → 𝑊 ∈ Grp ) |
25 |
23 6 24
|
3syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → 𝑊 ∈ Grp ) |
26 |
23 11
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → 𝑋 ∈ 𝐵 ) |
27 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → 𝑏 ∈ 𝐵 ) |
28 |
|
eqid |
⊢ ( -g ‘ 𝑊 ) = ( -g ‘ 𝑊 ) |
29 |
1 28
|
grpsubcl |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ∈ 𝐵 ) |
30 |
25 26 27 29
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ∈ 𝐵 ) |
31 |
1 2 28
|
grpsubid |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑏 ∈ 𝐵 ) → ( 𝑏 ( -g ‘ 𝑊 ) 𝑏 ) = 0 ) |
32 |
25 27 31
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( 𝑏 ( -g ‘ 𝑊 ) 𝑏 ) = 0 ) |
33 |
23 6
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → 𝑊 ∈ oGrp ) |
34 |
|
simplrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → 𝑏 < 𝑋 ) |
35 |
1 4 28
|
ogrpsublt |
⊢ ( ( 𝑊 ∈ oGrp ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑏 < 𝑋 ) → ( 𝑏 ( -g ‘ 𝑊 ) 𝑏 ) < ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) |
36 |
33 27 26 27 34 35
|
syl131anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( 𝑏 ( -g ‘ 𝑊 ) 𝑏 ) < ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) |
37 |
32 36
|
eqbrtrrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → 0 < ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) |
38 |
23 9
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( oppg ‘ 𝑊 ) ∈ oGrp ) |
39 |
1 8
|
grpcl |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑏 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑏 + 𝑏 ) ∈ 𝐵 ) |
40 |
25 27 27 39
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( 𝑏 + 𝑏 ) ∈ 𝐵 ) |
41 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → 𝑋 < ( 𝑏 + 𝑏 ) ) |
42 |
1 4 28
|
ogrpsublt |
⊢ ( ( 𝑊 ∈ oGrp ∧ ( 𝑋 ∈ 𝐵 ∧ ( 𝑏 + 𝑏 ) ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) < ( ( 𝑏 + 𝑏 ) ( -g ‘ 𝑊 ) 𝑏 ) ) |
43 |
33 26 40 27 41 42
|
syl131anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) < ( ( 𝑏 + 𝑏 ) ( -g ‘ 𝑊 ) 𝑏 ) ) |
44 |
1 8 28
|
grpaddsubass |
⊢ ( ( 𝑊 ∈ Grp ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ( 𝑏 + 𝑏 ) ( -g ‘ 𝑊 ) 𝑏 ) = ( 𝑏 + ( 𝑏 ( -g ‘ 𝑊 ) 𝑏 ) ) ) |
45 |
25 27 27 27 44
|
syl13anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( ( 𝑏 + 𝑏 ) ( -g ‘ 𝑊 ) 𝑏 ) = ( 𝑏 + ( 𝑏 ( -g ‘ 𝑊 ) 𝑏 ) ) ) |
46 |
32
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( 𝑏 + ( 𝑏 ( -g ‘ 𝑊 ) 𝑏 ) ) = ( 𝑏 + 0 ) ) |
47 |
1 8 2
|
grprid |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑏 ∈ 𝐵 ) → ( 𝑏 + 0 ) = 𝑏 ) |
48 |
25 27 47
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( 𝑏 + 0 ) = 𝑏 ) |
49 |
45 46 48
|
3eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( ( 𝑏 + 𝑏 ) ( -g ‘ 𝑊 ) 𝑏 ) = 𝑏 ) |
50 |
43 49
|
breqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) < 𝑏 ) |
51 |
1 4 8 25 38 30 27 30 50
|
ogrpaddltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) < ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + 𝑏 ) ) |
52 |
1 8 28
|
grpnpcan |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + 𝑏 ) = 𝑋 ) |
53 |
25 26 27 52
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + 𝑏 ) = 𝑋 ) |
54 |
51 53
|
breqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) < 𝑋 ) |
55 |
|
ovexd |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) ∈ V ) |
56 |
3 4
|
pltle |
⊢ ( ( 𝑊 ∈ Grp ∧ ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) ∈ V ∧ 𝑋 ∈ 𝐵 ) → ( ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) < 𝑋 → ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) ≤ 𝑋 ) ) |
57 |
25 55 26 56
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) < 𝑋 → ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) ≤ 𝑋 ) ) |
58 |
54 57
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) ≤ 𝑋 ) |
59 |
|
breq2 |
⊢ ( 𝑐 = ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) → ( 0 < 𝑐 ↔ 0 < ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) ) |
60 |
|
id |
⊢ ( 𝑐 = ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) → 𝑐 = ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) |
61 |
60 60
|
oveq12d |
⊢ ( 𝑐 = ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) → ( 𝑐 + 𝑐 ) = ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) ) |
62 |
61
|
breq1d |
⊢ ( 𝑐 = ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) → ( ( 𝑐 + 𝑐 ) ≤ 𝑋 ↔ ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) ≤ 𝑋 ) ) |
63 |
59 62
|
anbi12d |
⊢ ( 𝑐 = ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) → ( ( 0 < 𝑐 ∧ ( 𝑐 + 𝑐 ) ≤ 𝑋 ) ↔ ( 0 < ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ∧ ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) ≤ 𝑋 ) ) ) |
64 |
63
|
rspcev |
⊢ ( ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ∈ 𝐵 ∧ ( 0 < ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ∧ ( ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) + ( 𝑋 ( -g ‘ 𝑊 ) 𝑏 ) ) ≤ 𝑋 ) ) → ∃ 𝑐 ∈ 𝐵 ( 0 < 𝑐 ∧ ( 𝑐 + 𝑐 ) ≤ 𝑋 ) ) |
65 |
30 37 58 64
|
syl12anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ∧ 𝑋 < ( 𝑏 + 𝑏 ) ) → ∃ 𝑐 ∈ 𝐵 ( 0 < 𝑐 ∧ ( 𝑐 + 𝑐 ) ≤ 𝑋 ) ) |
66 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) → 𝑊 ∈ oGrp ) |
67 |
|
isogrp |
⊢ ( 𝑊 ∈ oGrp ↔ ( 𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd ) ) |
68 |
67
|
simprbi |
⊢ ( 𝑊 ∈ oGrp → 𝑊 ∈ oMnd ) |
69 |
|
omndtos |
⊢ ( 𝑊 ∈ oMnd → 𝑊 ∈ Toset ) |
70 |
66 68 69
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) → 𝑊 ∈ Toset ) |
71 |
66 24
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) → 𝑊 ∈ Grp ) |
72 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) → 𝑏 ∈ 𝐵 ) |
73 |
71 72 72 39
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) → ( 𝑏 + 𝑏 ) ∈ 𝐵 ) |
74 |
11
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
75 |
1 3 4
|
tlt2 |
⊢ ( ( 𝑊 ∈ Toset ∧ ( 𝑏 + 𝑏 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑏 + 𝑏 ) ≤ 𝑋 ∨ 𝑋 < ( 𝑏 + 𝑏 ) ) ) |
76 |
70 73 74 75
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) → ( ( 𝑏 + 𝑏 ) ≤ 𝑋 ∨ 𝑋 < ( 𝑏 + 𝑏 ) ) ) |
77 |
22 65 76
|
mpjaodan |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) → ∃ 𝑐 ∈ 𝐵 ( 0 < 𝑐 ∧ ( 𝑐 + 𝑐 ) ≤ 𝑋 ) ) |
78 |
10
|
3expia |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐵 ) → ( 0 < 𝑎 → ∃ 𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎 ) ) ) |
79 |
78
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐵 ( 0 < 𝑎 → ∃ 𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎 ) ) ) |
80 |
|
breq2 |
⊢ ( 𝑎 = 𝑋 → ( 0 < 𝑎 ↔ 0 < 𝑋 ) ) |
81 |
|
breq2 |
⊢ ( 𝑎 = 𝑋 → ( 𝑏 < 𝑎 ↔ 𝑏 < 𝑋 ) ) |
82 |
81
|
anbi2d |
⊢ ( 𝑎 = 𝑋 → ( ( 0 < 𝑏 ∧ 𝑏 < 𝑎 ) ↔ ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ) |
83 |
82
|
rexbidv |
⊢ ( 𝑎 = 𝑋 → ( ∃ 𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎 ) ↔ ∃ 𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ) |
84 |
80 83
|
imbi12d |
⊢ ( 𝑎 = 𝑋 → ( ( 0 < 𝑎 → ∃ 𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎 ) ) ↔ ( 0 < 𝑋 → ∃ 𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ) ) |
85 |
84
|
rspcv |
⊢ ( 𝑋 ∈ 𝐵 → ( ∀ 𝑎 ∈ 𝐵 ( 0 < 𝑎 → ∃ 𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎 ) ) → ( 0 < 𝑋 → ∃ 𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) ) ) |
86 |
11 79 12 85
|
syl3c |
⊢ ( 𝜑 → ∃ 𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑋 ) ) |
87 |
77 86
|
r19.29a |
⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝐵 ( 0 < 𝑐 ∧ ( 𝑐 + 𝑐 ) ≤ 𝑋 ) ) |