Step |
Hyp |
Ref |
Expression |
1 |
|
umgr2cycllem.1 |
⊢ 𝐹 = 〈“ 𝐽 𝐾 ”〉 |
2 |
|
umgr2cycllem.2 |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
umgr2cycllem.3 |
⊢ ( 𝜑 → 𝐺 ∈ UMGraph ) |
4 |
|
umgr2cycllem.4 |
⊢ ( 𝜑 → 𝐽 ∈ dom 𝐼 ) |
5 |
|
umgr2cycllem.5 |
⊢ ( 𝜑 → 𝐽 ≠ 𝐾 ) |
6 |
|
umgr2cycllem.6 |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝐽 ) = ( 𝐼 ‘ 𝐾 ) ) |
7 |
|
umgruhgr |
⊢ ( 𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph ) |
8 |
2
|
uhgrfun |
⊢ ( 𝐺 ∈ UHGraph → Fun 𝐼 ) |
9 |
3 7 8
|
3syl |
⊢ ( 𝜑 → Fun 𝐼 ) |
10 |
2
|
iedgedg |
⊢ ( ( Fun 𝐼 ∧ 𝐽 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝐽 ) ∈ ( Edg ‘ 𝐺 ) ) |
11 |
9 4 10
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝐽 ) ∈ ( Edg ‘ 𝐺 ) ) |
12 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
13 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
14 |
12 13
|
umgredg |
⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐼 ‘ 𝐽 ) ∈ ( Edg ‘ 𝐺 ) ) → ∃ 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∃ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) |
15 |
3 11 14
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∃ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) |
16 |
|
ax-5 |
⊢ ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) → ∀ 𝑏 𝑎 ∈ ( Vtx ‘ 𝐺 ) ) |
17 |
|
alral |
⊢ ( ∀ 𝑏 𝑎 ∈ ( Vtx ‘ 𝐺 ) → ∀ 𝑏 ∈ ( Vtx ‘ 𝐺 ) 𝑎 ∈ ( Vtx ‘ 𝐺 ) ) |
18 |
16 17
|
syl |
⊢ ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) → ∀ 𝑏 ∈ ( Vtx ‘ 𝐺 ) 𝑎 ∈ ( Vtx ‘ 𝐺 ) ) |
19 |
|
r19.29 |
⊢ ( ( ∀ 𝑏 ∈ ( Vtx ‘ 𝐺 ) 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ ∃ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → ∃ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) ) |
20 |
18 19
|
sylan |
⊢ ( ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ ∃ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → ∃ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) ) |
21 |
|
eqid |
⊢ 〈“ 𝑎 𝑏 𝑎 ”〉 = 〈“ 𝑎 𝑏 𝑎 ”〉 |
22 |
|
simp2 |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ) ) |
23 |
|
simp3l |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → 𝑎 ≠ 𝑏 ) |
24 |
|
eqimss2 |
⊢ ( ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } → { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐽 ) ) |
25 |
24
|
adantl |
⊢ ( ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) → { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐽 ) ) |
26 |
25
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐽 ) ) |
27 |
6
|
sseq2d |
⊢ ( 𝜑 → ( { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐽 ) ↔ { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐾 ) ) ) |
28 |
24 27
|
syl5ib |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } → { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐾 ) ) ) |
29 |
28
|
adantld |
⊢ ( 𝜑 → ( ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) → { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐾 ) ) ) |
30 |
29
|
adantld |
⊢ ( 𝜑 → ( ( ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐾 ) ) ) |
31 |
30
|
3impib |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐾 ) ) |
32 |
26 31
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → ( { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝑎 , 𝑏 } ⊆ ( 𝐼 ‘ 𝐾 ) ) ) |
33 |
5
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → 𝐽 ≠ 𝐾 ) |
34 |
21 1 22 23 32 12 2 33
|
2cycl2d |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → 𝐹 ( Cycles ‘ 𝐺 ) 〈“ 𝑎 𝑏 𝑎 ”〉 ) |
35 |
34
|
3expib |
⊢ ( 𝜑 → ( ( ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → 𝐹 ( Cycles ‘ 𝐺 ) 〈“ 𝑎 𝑏 𝑎 ”〉 ) ) |
36 |
35
|
exp4c |
⊢ ( 𝜑 → ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) → ( 𝑏 ∈ ( Vtx ‘ 𝐺 ) → ( ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) → 𝐹 ( Cycles ‘ 𝐺 ) 〈“ 𝑎 𝑏 𝑎 ”〉 ) ) ) ) |
37 |
36
|
com23 |
⊢ ( 𝜑 → ( 𝑏 ∈ ( Vtx ‘ 𝐺 ) → ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) → ( ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) → 𝐹 ( Cycles ‘ 𝐺 ) 〈“ 𝑎 𝑏 𝑎 ”〉 ) ) ) ) |
38 |
37
|
imp4a |
⊢ ( 𝜑 → ( 𝑏 ∈ ( Vtx ‘ 𝐺 ) → ( ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → 𝐹 ( Cycles ‘ 𝐺 ) 〈“ 𝑎 𝑏 𝑎 ”〉 ) ) ) |
39 |
|
s3cli |
⊢ 〈“ 𝑎 𝑏 𝑎 ”〉 ∈ Word V |
40 |
|
breq2 |
⊢ ( 𝑝 = 〈“ 𝑎 𝑏 𝑎 ”〉 → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ↔ 𝐹 ( Cycles ‘ 𝐺 ) 〈“ 𝑎 𝑏 𝑎 ”〉 ) ) |
41 |
40
|
rspcev |
⊢ ( ( 〈“ 𝑎 𝑏 𝑎 ”〉 ∈ Word V ∧ 𝐹 ( Cycles ‘ 𝐺 ) 〈“ 𝑎 𝑏 𝑎 ”〉 ) → ∃ 𝑝 ∈ Word V 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) |
42 |
39 41
|
mpan |
⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 〈“ 𝑎 𝑏 𝑎 ”〉 → ∃ 𝑝 ∈ Word V 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) |
43 |
|
rexex |
⊢ ( ∃ 𝑝 ∈ Word V 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 → ∃ 𝑝 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) |
44 |
42 43
|
syl |
⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 〈“ 𝑎 𝑏 𝑎 ”〉 → ∃ 𝑝 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) |
45 |
38 44
|
syl8 |
⊢ ( 𝜑 → ( 𝑏 ∈ ( Vtx ‘ 𝐺 ) → ( ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → ∃ 𝑝 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) ) ) |
46 |
45
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → ∃ 𝑝 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) ) |
47 |
20 46
|
syl5 |
⊢ ( 𝜑 → ( ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∧ ∃ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) ) → ∃ 𝑝 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) ) |
48 |
47
|
expd |
⊢ ( 𝜑 → ( 𝑎 ∈ ( Vtx ‘ 𝐺 ) → ( ∃ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) → ∃ 𝑝 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) ) ) |
49 |
48
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ ( Vtx ‘ 𝐺 ) ∃ 𝑏 ∈ ( Vtx ‘ 𝐺 ) ( 𝑎 ≠ 𝑏 ∧ ( 𝐼 ‘ 𝐽 ) = { 𝑎 , 𝑏 } ) → ∃ 𝑝 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) ) |
50 |
15 49
|
mpd |
⊢ ( 𝜑 → ∃ 𝑝 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) |