Metamath Proof Explorer


Theorem wlkp1lem6

Description: Lemma for wlkp1 . (Contributed by AV, 6-Mar-2021)

Ref Expression
Hypotheses wlkp1.v V = Vtx G
wlkp1.i I = iEdg G
wlkp1.f φ Fun I
wlkp1.a φ I Fin
wlkp1.b φ B W
wlkp1.c φ C V
wlkp1.d φ ¬ B dom I
wlkp1.w φ F Walks G P
wlkp1.n N = F
wlkp1.e φ E Edg G
wlkp1.x φ P N C E
wlkp1.u φ iEdg S = I B E
wlkp1.h H = F N B
wlkp1.q Q = P N + 1 C
wlkp1.s φ Vtx S = V
Assertion wlkp1lem6 φ k 0 ..^ N Q k = P k Q k + 1 = P k + 1 iEdg S H k = I F k

Proof

Step Hyp Ref Expression
1 wlkp1.v V = Vtx G
2 wlkp1.i I = iEdg G
3 wlkp1.f φ Fun I
4 wlkp1.a φ I Fin
5 wlkp1.b φ B W
6 wlkp1.c φ C V
7 wlkp1.d φ ¬ B dom I
8 wlkp1.w φ F Walks G P
9 wlkp1.n N = F
10 wlkp1.e φ E Edg G
11 wlkp1.x φ P N C E
12 wlkp1.u φ iEdg S = I B E
13 wlkp1.h H = F N B
14 wlkp1.q Q = P N + 1 C
15 wlkp1.s φ Vtx S = V
16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 wlkp1lem5 φ x 0 N Q x = P x
17 elfzofz k 0 ..^ N k 0 N
18 17 adantl φ k 0 ..^ N k 0 N
19 fveq2 x = k Q x = Q k
20 fveq2 x = k P x = P k
21 19 20 eqeq12d x = k Q x = P x Q k = P k
22 21 rspcv k 0 N x 0 N Q x = P x Q k = P k
23 18 22 syl φ k 0 ..^ N x 0 N Q x = P x Q k = P k
24 23 imp φ k 0 ..^ N x 0 N Q x = P x Q k = P k
25 fzofzp1 k 0 ..^ N k + 1 0 N
26 25 adantl φ k 0 ..^ N k + 1 0 N
27 fveq2 x = k + 1 Q x = Q k + 1
28 fveq2 x = k + 1 P x = P k + 1
29 27 28 eqeq12d x = k + 1 Q x = P x Q k + 1 = P k + 1
30 29 rspcv k + 1 0 N x 0 N Q x = P x Q k + 1 = P k + 1
31 26 30 syl φ k 0 ..^ N x 0 N Q x = P x Q k + 1 = P k + 1
32 31 imp φ k 0 ..^ N x 0 N Q x = P x Q k + 1 = P k + 1
33 12 adantr φ k 0 ..^ N iEdg S = I B E
34 13 fveq1i H k = F N B k
35 fzonel ¬ N 0 ..^ N
36 eleq1 N = k N 0 ..^ N k 0 ..^ N
37 35 36 mtbii N = k ¬ k 0 ..^ N
38 37 a1i φ N = k ¬ k 0 ..^ N
39 38 con2d φ k 0 ..^ N ¬ N = k
40 39 imp φ k 0 ..^ N ¬ N = k
41 40 neqned φ k 0 ..^ N N k
42 fvunsn N k F N B k = F k
43 41 42 syl φ k 0 ..^ N F N B k = F k
44 34 43 eqtrid φ k 0 ..^ N H k = F k
45 33 44 fveq12d φ k 0 ..^ N iEdg S H k = I B E F k
46 9 oveq2i 0 ..^ N = 0 ..^ F
47 46 eleq2i k 0 ..^ N k 0 ..^ F
48 2 wlkf F Walks G P F Word dom I
49 8 48 syl φ F Word dom I
50 wrdsymbcl F Word dom I k 0 ..^ F F k dom I
51 50 ex F Word dom I k 0 ..^ F F k dom I
52 49 51 syl φ k 0 ..^ F F k dom I
53 47 52 syl5bi φ k 0 ..^ N F k dom I
54 53 imp φ k 0 ..^ N F k dom I
55 eleq1 B = F k B dom I F k dom I
56 54 55 syl5ibrcom φ k 0 ..^ N B = F k B dom I
57 56 con3d φ k 0 ..^ N ¬ B dom I ¬ B = F k
58 57 ex φ k 0 ..^ N ¬ B dom I ¬ B = F k
59 7 58 mpid φ k 0 ..^ N ¬ B = F k
60 59 imp φ k 0 ..^ N ¬ B = F k
61 60 neqned φ k 0 ..^ N B F k
62 fvunsn B F k I B E F k = I F k
63 61 62 syl φ k 0 ..^ N I B E F k = I F k
64 45 63 eqtrd φ k 0 ..^ N iEdg S H k = I F k
65 64 adantr φ k 0 ..^ N x 0 N Q x = P x iEdg S H k = I F k
66 24 32 65 3jca φ k 0 ..^ N x 0 N Q x = P x Q k = P k Q k + 1 = P k + 1 iEdg S H k = I F k
67 16 66 mpidan φ k 0 ..^ N Q k = P k Q k + 1 = P k + 1 iEdg S H k = I F k
68 67 ralrimiva φ k 0 ..^ N Q k = P k Q k + 1 = P k + 1 iEdg S H k = I F k