Metamath Proof Explorer


Theorem usgr2pthlem

Description: Lemma for usgr2pth . (Contributed by Alexander van der Vekens, 27-Jan-2018) (Revised by AV, 5-Jun-2021)

Ref Expression
Hypotheses usgr2pthlem.v V = Vtx G
usgr2pthlem.i I = iEdg G
Assertion usgr2pthlem F : 0 ..^ F 1-1 dom I P : 0 F V i 0 ..^ F I F i = P i P i + 1 G USGraph F = 2 x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z

Proof

Step Hyp Ref Expression
1 usgr2pthlem.v V = Vtx G
2 usgr2pthlem.i I = iEdg G
3 0nn0 0 0
4 2nn0 2 0
5 0le2 0 2
6 elfz2nn0 0 0 2 0 0 2 0 0 2
7 3 4 5 6 mpbir3an 0 0 2
8 ffvelrn P : 0 2 V 0 0 2 P 0 V
9 7 8 mpan2 P : 0 2 V P 0 V
10 9 adantl F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 0 V
11 1nn0 1 0
12 1le2 1 2
13 elfz2nn0 1 0 2 1 0 2 0 1 2
14 11 4 12 13 mpbir3an 1 0 2
15 ffvelrn P : 0 2 V 1 0 2 P 1 V
16 14 15 mpan2 P : 0 2 V P 1 V
17 16 adantl F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 1 V
18 simpr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph G USGraph
19 fvex P 1 V
20 18 19 jctir F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph G USGraph P 1 V
21 prcom P 0 P 1 = P 1 P 0
22 21 eqeq2i I F 0 = P 0 P 1 I F 0 = P 1 P 0
23 22 biimpi I F 0 = P 0 P 1 I F 0 = P 1 P 0
24 23 adantr I F 0 = P 0 P 1 I F 1 = P 1 P 2 I F 0 = P 1 P 0
25 24 ad2antlr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph I F 0 = P 1 P 0
26 2 usgrnloopv G USGraph P 1 V I F 0 = P 1 P 0 P 1 P 0
27 20 25 26 sylc F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P 1 P 0
28 27 adantr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 1 P 0
29 19 elsn P 1 P 0 P 1 = P 0
30 29 necon3bbii ¬ P 1 P 0 P 1 P 0
31 28 30 sylibr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V ¬ P 1 P 0
32 17 31 eldifd F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 1 V P 0
33 32 adantr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V x = P 0 P 1 V P 0
34 sneq x = P 0 x = P 0
35 34 difeq2d x = P 0 V x = V P 0
36 35 eleq2d x = P 0 P 1 V x P 1 V P 0
37 36 adantl F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V x = P 0 P 1 V x P 1 V P 0
38 33 37 mpbird F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V x = P 0 P 1 V x
39 2re 2
40 39 leidi 2 2
41 elfz2nn0 2 0 2 2 0 2 0 2 2
42 4 4 40 41 mpbir3an 2 0 2
43 ffvelrn P : 0 2 V 2 0 2 P 2 V
44 42 43 mpan2 P : 0 2 V P 2 V
45 44 adantl F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 2 V
46 2 usgrf1 G USGraph I : dom I 1-1 ran I
47 46 ad2antlr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V I : dom I 1-1 ran I
48 simpl F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 F : 0 ..^ 2 1-1 dom I
49 48 ad2antrr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V F : 0 ..^ 2 1-1 dom I
50 47 49 jca F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V I : dom I 1-1 ran I F : 0 ..^ 2 1-1 dom I
51 2nn 2
52 lbfzo0 0 0 ..^ 2 2
53 51 52 mpbir 0 0 ..^ 2
54 1lt2 1 < 2
55 elfzo0 1 0 ..^ 2 1 0 2 1 < 2
56 11 51 54 55 mpbir3an 1 0 ..^ 2
57 53 56 pm3.2i 0 0 ..^ 2 1 0 ..^ 2
58 57 a1i F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V 0 0 ..^ 2 1 0 ..^ 2
59 0ne1 0 1
60 59 a1i F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V 0 1
61 50 58 60 3jca F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V I : dom I 1-1 ran I F : 0 ..^ 2 1-1 dom I 0 0 ..^ 2 1 0 ..^ 2 0 1
62 simpr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 I F 0 = P 0 P 1 I F 1 = P 1 P 2
63 62 ad2antrr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V I F 0 = P 0 P 1 I F 1 = P 1 P 2
64 2f1fvneq I : dom I 1-1 ran I F : 0 ..^ 2 1-1 dom I 0 0 ..^ 2 1 0 ..^ 2 0 1 I F 0 = P 0 P 1 I F 1 = P 1 P 2 P 0 P 1 P 1 P 2
65 61 63 64 sylc F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 0 P 1 P 1 P 2
66 necom P 2 P 0 P 0 P 2
67 fvex P 0 V
68 fvex P 2 V
69 67 19 68 3pm3.2i P 0 V P 1 V P 2 V
70 fvexd F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P 0 V
71 simpl I F 0 = P 0 P 1 I F 1 = P 1 P 2 I F 0 = P 0 P 1
72 71 ad2antlr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph I F 0 = P 0 P 1
73 18 70 72 jca31 F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph G USGraph P 0 V I F 0 = P 0 P 1
74 73 adantr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V G USGraph P 0 V I F 0 = P 0 P 1
75 2 usgrnloopv G USGraph P 0 V I F 0 = P 0 P 1 P 0 P 1
76 75 imp G USGraph P 0 V I F 0 = P 0 P 1 P 0 P 1
77 74 76 syl F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 0 P 1
78 pr1nebg P 0 V P 1 V P 2 V P 0 P 1 P 0 P 2 P 0 P 1 P 1 P 2
79 69 77 78 sylancr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 0 P 2 P 0 P 1 P 1 P 2
80 66 79 syl5bb F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 2 P 0 P 0 P 1 P 1 P 2
81 65 80 mpbird F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 2 P 0
82 fvexd F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P 2 V
83 prcom P 1 P 2 = P 2 P 1
84 83 eqeq2i I F 1 = P 1 P 2 I F 1 = P 2 P 1
85 84 biimpi I F 1 = P 1 P 2 I F 1 = P 2 P 1
86 85 adantl I F 0 = P 0 P 1 I F 1 = P 1 P 2 I F 1 = P 2 P 1
87 86 ad2antlr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph I F 1 = P 2 P 1
88 18 82 87 jca31 F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph G USGraph P 2 V I F 1 = P 2 P 1
89 88 adantr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V G USGraph P 2 V I F 1 = P 2 P 1
90 2 usgrnloopv G USGraph P 2 V I F 1 = P 2 P 1 P 2 P 1
91 90 imp G USGraph P 2 V I F 1 = P 2 P 1 P 2 P 1
92 89 91 syl F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 2 P 1
93 81 92 nelprd F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V ¬ P 2 P 0 P 1
94 45 93 eldifd F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V P 2 V P 0 P 1
95 94 ad2antrr F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V x = P 0 y = P 1 P 2 V P 0 P 1
96 preq12 x = P 0 y = P 1 x y = P 0 P 1
97 96 difeq2d x = P 0 y = P 1 V x y = V P 0 P 1
98 97 eleq2d x = P 0 y = P 1 P 2 V x y P 2 V P 0 P 1
99 98 adantll F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V x = P 0 y = P 1 P 2 V x y P 2 V P 0 P 1
100 95 99 mpbird F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V x = P 0 y = P 1 P 2 V x y
101 eqcom x = P 0 P 0 = x
102 eqcom y = P 1 P 1 = y
103 eqcom z = P 2 P 2 = z
104 101 102 103 3anbi123i x = P 0 y = P 1 z = P 2 P 0 = x P 1 = y P 2 = z
105 104 biimpi x = P 0 y = P 1 z = P 2 P 0 = x P 1 = y P 2 = z
106 105 ad4ant123 x = P 0 y = P 1 z = P 2 I F 0 = P 0 P 1 I F 1 = P 1 P 2 P 0 = x P 1 = y P 2 = z
107 101 biimpi x = P 0 P 0 = x
108 107 ad2antrr x = P 0 y = P 1 z = P 2 P 0 = x
109 102 biimpi y = P 1 P 1 = y
110 109 ad2antlr x = P 0 y = P 1 z = P 2 P 1 = y
111 108 110 preq12d x = P 0 y = P 1 z = P 2 P 0 P 1 = x y
112 111 eqeq2d x = P 0 y = P 1 z = P 2 I F 0 = P 0 P 1 I F 0 = x y
113 103 biimpi z = P 2 P 2 = z
114 113 adantl x = P 0 y = P 1 z = P 2 P 2 = z
115 110 114 preq12d x = P 0 y = P 1 z = P 2 P 1 P 2 = y z
116 115 eqeq2d x = P 0 y = P 1 z = P 2 I F 1 = P 1 P 2 I F 1 = y z
117 112 116 anbi12d x = P 0 y = P 1 z = P 2 I F 0 = P 0 P 1 I F 1 = P 1 P 2 I F 0 = x y I F 1 = y z
118 117 biimpa x = P 0 y = P 1 z = P 2 I F 0 = P 0 P 1 I F 1 = P 1 P 2 I F 0 = x y I F 1 = y z
119 106 118 jca x = P 0 y = P 1 z = P 2 I F 0 = P 0 P 1 I F 1 = P 1 P 2 P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
120 119 exp41 x = P 0 y = P 1 z = P 2 I F 0 = P 0 P 1 I F 1 = P 1 P 2 P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
121 120 adantl F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V x = P 0 y = P 1 z = P 2 I F 0 = P 0 P 1 I F 1 = P 1 P 2 P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
122 121 imp31 F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V x = P 0 y = P 1 z = P 2 I F 0 = P 0 P 1 I F 1 = P 1 P 2 P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
123 100 122 rspcimedv F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V x = P 0 y = P 1 I F 0 = P 0 P 1 I F 1 = P 1 P 2 z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
124 38 123 rspcimedv F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V x = P 0 I F 0 = P 0 P 1 I F 1 = P 1 P 2 y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
125 10 124 rspcimedv F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V I F 0 = P 0 P 1 I F 1 = P 1 P 2 x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
126 125 exp41 F : 0 ..^ 2 1-1 dom I I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V I F 0 = P 0 P 1 I F 1 = P 1 P 2 x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
127 126 com15 I F 0 = P 0 P 1 I F 1 = P 1 P 2 I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V F : 0 ..^ 2 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
128 127 pm2.43i I F 0 = P 0 P 1 I F 1 = P 1 P 2 G USGraph P : 0 2 V F : 0 ..^ 2 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
129 128 com12 G USGraph I F 0 = P 0 P 1 I F 1 = P 1 P 2 P : 0 2 V F : 0 ..^ 2 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
130 129 adantr G USGraph F = 2 I F 0 = P 0 P 1 I F 1 = P 1 P 2 P : 0 2 V F : 0 ..^ 2 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
131 oveq2 F = 2 0 ..^ F = 0 ..^ 2
132 131 raleqdv F = 2 i 0 ..^ F I F i = P i P i + 1 i 0 ..^ 2 I F i = P i P i + 1
133 fzo0to2pr 0 ..^ 2 = 0 1
134 133 raleqi i 0 ..^ 2 I F i = P i P i + 1 i 0 1 I F i = P i P i + 1
135 2wlklem i 0 1 I F i = P i P i + 1 I F 0 = P 0 P 1 I F 1 = P 1 P 2
136 134 135 bitri i 0 ..^ 2 I F i = P i P i + 1 I F 0 = P 0 P 1 I F 1 = P 1 P 2
137 132 136 bitrdi F = 2 i 0 ..^ F I F i = P i P i + 1 I F 0 = P 0 P 1 I F 1 = P 1 P 2
138 137 adantl G USGraph F = 2 i 0 ..^ F I F i = P i P i + 1 I F 0 = P 0 P 1 I F 1 = P 1 P 2
139 oveq2 F = 2 0 F = 0 2
140 139 feq2d F = 2 P : 0 F V P : 0 2 V
141 140 adantl G USGraph F = 2 P : 0 F V P : 0 2 V
142 f1eq2 0 ..^ F = 0 ..^ 2 F : 0 ..^ F 1-1 dom I F : 0 ..^ 2 1-1 dom I
143 131 142 syl F = 2 F : 0 ..^ F 1-1 dom I F : 0 ..^ 2 1-1 dom I
144 143 imbi1d F = 2 F : 0 ..^ F 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z F : 0 ..^ 2 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
145 144 adantl G USGraph F = 2 F : 0 ..^ F 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z F : 0 ..^ 2 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
146 141 145 imbi12d G USGraph F = 2 P : 0 F V F : 0 ..^ F 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z P : 0 2 V F : 0 ..^ 2 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
147 130 138 146 3imtr4d G USGraph F = 2 i 0 ..^ F I F i = P i P i + 1 P : 0 F V F : 0 ..^ F 1-1 dom I x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
148 147 com14 F : 0 ..^ F 1-1 dom I i 0 ..^ F I F i = P i P i + 1 P : 0 F V G USGraph F = 2 x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
149 148 com23 F : 0 ..^ F 1-1 dom I P : 0 F V i 0 ..^ F I F i = P i P i + 1 G USGraph F = 2 x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z
150 149 3imp F : 0 ..^ F 1-1 dom I P : 0 F V i 0 ..^ F I F i = P i P i + 1 G USGraph F = 2 x V y V x z V x y P 0 = x P 1 = y P 2 = z I F 0 = x y I F 1 = y z