Step |
Hyp |
Ref |
Expression |
1 |
|
trlsegvdeg.v |
|- V = ( Vtx ` G ) |
2 |
|
trlsegvdeg.i |
|- I = ( iEdg ` G ) |
3 |
|
trlsegvdeg.f |
|- ( ph -> Fun I ) |
4 |
|
trlsegvdeg.n |
|- ( ph -> N e. ( 0 ..^ ( # ` F ) ) ) |
5 |
|
trlsegvdeg.u |
|- ( ph -> U e. V ) |
6 |
|
trlsegvdeg.w |
|- ( ph -> F ( Trails ` G ) P ) |
7 |
|
trlsegvdeg.vx |
|- ( ph -> ( Vtx ` X ) = V ) |
8 |
|
trlsegvdeg.vy |
|- ( ph -> ( Vtx ` Y ) = V ) |
9 |
|
trlsegvdeg.vz |
|- ( ph -> ( Vtx ` Z ) = V ) |
10 |
|
trlsegvdeg.ix |
|- ( ph -> ( iEdg ` X ) = ( I |` ( F " ( 0 ..^ N ) ) ) ) |
11 |
|
trlsegvdeg.iy |
|- ( ph -> ( iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) |
12 |
|
trlsegvdeg.iz |
|- ( ph -> ( iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) ) |
13 |
|
eupth2lem3.o |
|- ( ph -> { x e. V | -. 2 || ( ( VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) |
14 |
|
eupth2lem3lem3.e |
|- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) |
15 |
|
eupth2lem3lem4.i |
|- ( ph -> ( I ` ( F ` N ) ) e. ~P V ) |
16 |
|
fvexd |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( F ` N ) e. _V ) |
17 |
5
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> U e. V ) |
18 |
1 2 3 4 5 6
|
trlsegvdeglem1 |
|- ( ph -> ( ( P ` N ) e. V /\ ( P ` ( N + 1 ) ) e. V ) ) |
19 |
18
|
simprd |
|- ( ph -> ( P ` ( N + 1 ) ) e. V ) |
20 |
19
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( P ` ( N + 1 ) ) e. V ) |
21 |
|
neeq1 |
|- ( ( P ` N ) = U -> ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> U =/= ( P ` ( N + 1 ) ) ) ) |
22 |
21
|
biimpcd |
|- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( ( P ` N ) = U -> U =/= ( P ` ( N + 1 ) ) ) ) |
23 |
22
|
adantl |
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` N ) = U -> U =/= ( P ` ( N + 1 ) ) ) ) |
24 |
23
|
imp |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> U =/= ( P ` ( N + 1 ) ) ) |
25 |
15
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( I ` ( F ` N ) ) e. ~P V ) |
26 |
11
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) |
27 |
14
|
adantr |
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) |
28 |
|
df-ne |
|- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> -. ( P ` N ) = ( P ` ( N + 1 ) ) ) |
29 |
|
ifpfal |
|- ( -. ( P ` N ) = ( P ` ( N + 1 ) ) -> ( if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) <-> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) |
30 |
28 29
|
sylbi |
|- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) <-> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) |
31 |
30
|
adantl |
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) <-> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) |
32 |
|
preq1 |
|- ( ( P ` N ) = U -> { ( P ` N ) , ( P ` ( N + 1 ) ) } = { U , ( P ` ( N + 1 ) ) } ) |
33 |
32
|
sseq1d |
|- ( ( P ` N ) = U -> ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) <-> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) |
34 |
33
|
biimpcd |
|- ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) -> ( ( P ` N ) = U -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) |
35 |
31 34
|
syl6bi |
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) -> ( ( P ` N ) = U -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) ) |
36 |
27 35
|
mpd |
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` N ) = U -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) |
37 |
36
|
imp |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) |
38 |
8
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( Vtx ` Y ) = V ) |
39 |
16 17 20 24 25 26 37 38
|
1hegrvtxdg1 |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( ( VtxDeg ` Y ) ` U ) = 1 ) |
40 |
39
|
oveq2d |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) = ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) |
41 |
40
|
breq2d |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) ) |
42 |
41
|
notbid |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> -. 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) ) |
43 |
1 2 3 4 5 6 7 8 9 10 11 12
|
eupth2lem3lem1 |
|- ( ph -> ( ( VtxDeg ` X ) ` U ) e. NN0 ) |
44 |
43
|
nn0zd |
|- ( ph -> ( ( VtxDeg ` X ) ` U ) e. ZZ ) |
45 |
|
2nn |
|- 2 e. NN |
46 |
45
|
a1i |
|- ( ph -> 2 e. NN ) |
47 |
|
1lt2 |
|- 1 < 2 |
48 |
47
|
a1i |
|- ( ph -> 1 < 2 ) |
49 |
|
ndvdsp1 |
|- ( ( ( ( VtxDeg ` X ) ` U ) e. ZZ /\ 2 e. NN /\ 1 < 2 ) -> ( 2 || ( ( VtxDeg ` X ) ` U ) -> -. 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) ) |
50 |
44 46 48 49
|
syl3anc |
|- ( ph -> ( 2 || ( ( VtxDeg ` X ) ` U ) -> -. 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) ) |
51 |
50
|
con2d |
|- ( ph -> ( 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) -> -. 2 || ( ( VtxDeg ` X ) ` U ) ) ) |
52 |
|
1z |
|- 1 e. ZZ |
53 |
|
n2dvds1 |
|- -. 2 || 1 |
54 |
|
opoe |
|- ( ( ( ( ( VtxDeg ` X ) ` U ) e. ZZ /\ -. 2 || ( ( VtxDeg ` X ) ` U ) ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) |
55 |
52 53 54
|
mpanr12 |
|- ( ( ( ( VtxDeg ` X ) ` U ) e. ZZ /\ -. 2 || ( ( VtxDeg ` X ) ` U ) ) -> 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) |
56 |
55
|
ex |
|- ( ( ( VtxDeg ` X ) ` U ) e. ZZ -> ( -. 2 || ( ( VtxDeg ` X ) ` U ) -> 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) ) |
57 |
44 56
|
syl |
|- ( ph -> ( -. 2 || ( ( VtxDeg ` X ) ` U ) -> 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) ) |
58 |
51 57
|
impbid |
|- ( ph -> ( 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) <-> -. 2 || ( ( VtxDeg ` X ) ` U ) ) ) |
59 |
|
fveq2 |
|- ( x = U -> ( ( VtxDeg ` X ) ` x ) = ( ( VtxDeg ` X ) ` U ) ) |
60 |
59
|
breq2d |
|- ( x = U -> ( 2 || ( ( VtxDeg ` X ) ` x ) <-> 2 || ( ( VtxDeg ` X ) ` U ) ) ) |
61 |
60
|
notbid |
|- ( x = U -> ( -. 2 || ( ( VtxDeg ` X ) ` x ) <-> -. 2 || ( ( VtxDeg ` X ) ` U ) ) ) |
62 |
61
|
elrab3 |
|- ( U e. V -> ( U e. { x e. V | -. 2 || ( ( VtxDeg ` X ) ` x ) } <-> -. 2 || ( ( VtxDeg ` X ) ` U ) ) ) |
63 |
5 62
|
syl |
|- ( ph -> ( U e. { x e. V | -. 2 || ( ( VtxDeg ` X ) ` x ) } <-> -. 2 || ( ( VtxDeg ` X ) ` U ) ) ) |
64 |
13
|
eleq2d |
|- ( ph -> ( U e. { x e. V | -. 2 || ( ( VtxDeg ` X ) ` x ) } <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) ) |
65 |
58 63 64
|
3bitr2d |
|- ( ph -> ( 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) ) |
66 |
65
|
notbid |
|- ( ph -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) <-> -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) ) |
67 |
66
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) <-> -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) ) |
68 |
|
fvex |
|- ( P ` N ) e. _V |
69 |
68
|
eupth2lem2 |
|- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( P ` N ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) |
70 |
69
|
adantll |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) |
71 |
42 67 70
|
3bitrd |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) |
72 |
71
|
expcom |
|- ( ( P ` N ) = U -> ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) ) |
73 |
72
|
eqcoms |
|- ( U = ( P ` N ) -> ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) ) |
74 |
|
fvexd |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( F ` N ) e. _V ) |
75 |
18
|
simpld |
|- ( ph -> ( P ` N ) e. V ) |
76 |
75
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` N ) e. V ) |
77 |
5
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> U e. V ) |
78 |
|
neeq2 |
|- ( ( P ` ( N + 1 ) ) = U -> ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> ( P ` N ) =/= U ) ) |
79 |
78
|
biimpcd |
|- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( ( P ` ( N + 1 ) ) = U -> ( P ` N ) =/= U ) ) |
80 |
79
|
adantl |
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` ( N + 1 ) ) = U -> ( P ` N ) =/= U ) ) |
81 |
80
|
imp |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` N ) =/= U ) |
82 |
15
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( I ` ( F ` N ) ) e. ~P V ) |
83 |
11
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) |
84 |
|
preq2 |
|- ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , ( P ` ( N + 1 ) ) } = { ( P ` N ) , U } ) |
85 |
84
|
sseq1d |
|- ( ( P ` ( N + 1 ) ) = U -> ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) <-> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) ) |
86 |
85
|
biimpcd |
|- ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) -> ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) ) |
87 |
31 86
|
syl6bi |
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) -> ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) ) ) |
88 |
27 87
|
mpd |
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) ) |
89 |
88
|
imp |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) |
90 |
8
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( Vtx ` Y ) = V ) |
91 |
74 76 77 81 82 83 89 90
|
1hegrvtxdg1r |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( ( VtxDeg ` Y ) ` U ) = 1 ) |
92 |
91
|
oveq2d |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) = ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) |
93 |
92
|
breq2d |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) ) |
94 |
93
|
notbid |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> -. 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) ) ) |
95 |
66
|
ad2antrr |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + 1 ) <-> -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) ) |
96 |
|
necom |
|- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> ( P ` ( N + 1 ) ) =/= ( P ` N ) ) |
97 |
|
fvex |
|- ( P ` ( N + 1 ) ) e. _V |
98 |
97
|
eupth2lem2 |
|- ( ( ( P ` ( N + 1 ) ) =/= ( P ` N ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) ) |
99 |
96 98
|
sylanb |
|- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) ) |
100 |
99
|
con1bid |
|- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) |
101 |
100
|
adantll |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) |
102 |
94 95 101
|
3bitrd |
|- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) |
103 |
102
|
expcom |
|- ( ( P ` ( N + 1 ) ) = U -> ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) ) |
104 |
103
|
eqcoms |
|- ( U = ( P ` ( N + 1 ) ) -> ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) ) |
105 |
73 104
|
jaoi |
|- ( ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) -> ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) ) |
106 |
105
|
com12 |
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) ) |
107 |
106
|
3impia |
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) |