Metamath Proof Explorer

Theorem gpgprismgr4cycllem9

Description: Lemma 9 for gpgprismgr4cycl0 . (Contributed by AV, 3-Nov-2025)

Ref Expression
Hypotheses gpgprismgr4cycl.p 
|- P = <" <. 0 , 0 >. <. 0 , 1 >. <. 1 , 1 >. <. 1 , 0 >. <. 0 , 0 >. ">
gpgprismgr4cycl.f 
|- F = <" { <. 0 , 0 >. , <. 0 , 1 >. } { <. 0 , 1 >. , <. 1 , 1 >. } { <. 1 , 1 >. , <. 1 , 0 >. } { <. 1 , 0 >. , <. 0 , 0 >. } ">
gpgprismgr4cycl.g 
|- G = ( N gPetersenGr 1 )
Assertion gpgprismgr4cycllem9 
|- ( N e. ( ZZ>= ` 3 ) -> P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) )

Proof

Step Hyp Ref Expression
1 gpgprismgr4cycl.p 
 |-  P = <" <. 0 , 0 >. <. 0 , 1 >. <. 1 , 1 >. <. 1 , 0 >. <. 0 , 0 >. ">
2 gpgprismgr4cycl.f 
 |-  F = <" { <. 0 , 0 >. , <. 0 , 1 >. } { <. 0 , 1 >. , <. 1 , 1 >. } { <. 1 , 1 >. , <. 1 , 0 >. } { <. 1 , 0 >. , <. 0 , 0 >. } ">
3 gpgprismgr4cycl.g 
 |-  G = ( N gPetersenGr 1 )
4 eluz3nn 
 |-  ( N e. ( ZZ>= ` 3 ) -> N e. NN )
5 lbfzo0 
 |-  ( 0 e. ( 0 ..^ N ) <-> N e. NN )
6 4 5 sylibr 
 |-  ( N e. ( ZZ>= ` 3 ) -> 0 e. ( 0 ..^ N ) )
7 1nn0 
 |-  1 e. NN0
8 7 a1i 
 |-  ( N e. ( ZZ>= ` 3 ) -> 1 e. NN0 )
9 eluzelz 
 |-  ( N e. ( ZZ>= ` 3 ) -> N e. ZZ )
10 uzuzle23 
 |-  ( N e. ( ZZ>= ` 3 ) -> N e. ( ZZ>= ` 2 ) )
11 eluz2gt1 
 |-  ( N e. ( ZZ>= ` 2 ) -> 1 < N )
12 10 11 syl 
 |-  ( N e. ( ZZ>= ` 3 ) -> 1 < N )
13 elfzo0z 
 |-  ( 1 e. ( 0 ..^ N ) <-> ( 1 e. NN0 /\ N e. ZZ /\ 1 < N ) )
14 8 9 12 13 syl3anbrc 
 |-  ( N e. ( ZZ>= ` 3 ) -> 1 e. ( 0 ..^ N ) )
15 0elpr01 
 |-  0 e. { 0 , 1 }
16 15 a1i 
 |-  ( ( 0 e. ( 0 ..^ N ) /\ 1 e. ( 0 ..^ N ) ) -> 0 e. { 0 , 1 } )
17 simpl 
 |-  ( ( 0 e. ( 0 ..^ N ) /\ 1 e. ( 0 ..^ N ) ) -> 0 e. ( 0 ..^ N ) )
18 16 17 opelxpd 
 |-  ( ( 0 e. ( 0 ..^ N ) /\ 1 e. ( 0 ..^ N ) ) -> <. 0 , 0 >. e. ( { 0 , 1 } X. ( 0 ..^ N ) ) )
19 simpr 
 |-  ( ( 0 e. ( 0 ..^ N ) /\ 1 e. ( 0 ..^ N ) ) -> 1 e. ( 0 ..^ N ) )
20 16 19 opelxpd 
 |-  ( ( 0 e. ( 0 ..^ N ) /\ 1 e. ( 0 ..^ N ) ) -> <. 0 , 1 >. e. ( { 0 , 1 } X. ( 0 ..^ N ) ) )
21 1elpr01 
 |-  1 e. { 0 , 1 }
22 21 a1i 
 |-  ( ( 0 e. ( 0 ..^ N ) /\ 1 e. ( 0 ..^ N ) ) -> 1 e. { 0 , 1 } )
23 22 19 opelxpd 
 |-  ( ( 0 e. ( 0 ..^ N ) /\ 1 e. ( 0 ..^ N ) ) -> <. 1 , 1 >. e. ( { 0 , 1 } X. ( 0 ..^ N ) ) )
24 22 17 opelxpd 
 |-  ( ( 0 e. ( 0 ..^ N ) /\ 1 e. ( 0 ..^ N ) ) -> <. 1 , 0 >. e. ( { 0 , 1 } X. ( 0 ..^ N ) ) )
25 18 20 23 24 18 s5cld 
 |-  ( ( 0 e. ( 0 ..^ N ) /\ 1 e. ( 0 ..^ N ) ) -> <" <. 0 , 0 >. <. 0 , 1 >. <. 1 , 1 >. <. 1 , 0 >. <. 0 , 0 >. "> e. Word ( { 0 , 1 } X. ( 0 ..^ N ) ) )
26 6 14 25 syl2anc 
 |-  ( N e. ( ZZ>= ` 3 ) -> <" <. 0 , 0 >. <. 0 , 1 >. <. 1 , 1 >. <. 1 , 0 >. <. 0 , 0 >. "> e. Word ( { 0 , 1 } X. ( 0 ..^ N ) ) )
27 3 fveq2i 
 |-  ( Vtx ` G ) = ( Vtx ` ( N gPetersenGr 1 ) )
28 1elfzo1ceilhalf1 
 |-  ( N e. ( ZZ>= ` 3 ) -> 1 e. ( 1 ..^ ( |^ ` ( N / 2 ) ) ) )
29 eqid 
 |-  ( 1 ..^ ( |^ ` ( N / 2 ) ) ) = ( 1 ..^ ( |^ ` ( N / 2 ) ) )
30 eqid 
 |-  ( 0 ..^ N ) = ( 0 ..^ N )
31 29 30 gpgvtx 
 |-  ( ( N e. NN /\ 1 e. ( 1 ..^ ( |^ ` ( N / 2 ) ) ) ) -> ( Vtx ` ( N gPetersenGr 1 ) ) = ( { 0 , 1 } X. ( 0 ..^ N ) ) )
32 4 28 31 syl2anc 
 |-  ( N e. ( ZZ>= ` 3 ) -> ( Vtx ` ( N gPetersenGr 1 ) ) = ( { 0 , 1 } X. ( 0 ..^ N ) ) )
33 27 32 eqtrid 
 |-  ( N e. ( ZZ>= ` 3 ) -> ( Vtx ` G ) = ( { 0 , 1 } X. ( 0 ..^ N ) ) )
34 wrdeq 
 |-  ( ( Vtx ` G ) = ( { 0 , 1 } X. ( 0 ..^ N ) ) -> Word ( Vtx ` G ) = Word ( { 0 , 1 } X. ( 0 ..^ N ) ) )
35 33 34 syl 
 |-  ( N e. ( ZZ>= ` 3 ) -> Word ( Vtx ` G ) = Word ( { 0 , 1 } X. ( 0 ..^ N ) ) )
36 26 35 eleqtrrd 
 |-  ( N e. ( ZZ>= ` 3 ) -> <" <. 0 , 0 >. <. 0 , 1 >. <. 1 , 1 >. <. 1 , 0 >. <. 0 , 0 >. "> e. Word ( Vtx ` G ) )
37 1 36 eqeltrid 
 |-  ( N e. ( ZZ>= ` 3 ) -> P e. Word ( Vtx ` G ) )
38 wrdf 
 |-  ( P e. Word ( Vtx ` G ) -> P : ( 0 ..^ ( # ` P ) ) --> ( Vtx ` G ) )
39 37 38 syl 
 |-  ( N e. ( ZZ>= ` 3 ) -> P : ( 0 ..^ ( # ` P ) ) --> ( Vtx ` G ) )
40 4z 
 |-  4 e. ZZ
41 fzval3 
 |-  ( 4 e. ZZ -> ( 0 ... 4 ) = ( 0 ..^ ( 4 + 1 ) ) )
42 40 41 ax-mp 
 |-  ( 0 ... 4 ) = ( 0 ..^ ( 4 + 1 ) )
43 2 gpgprismgr4cycllem1 
 |-  ( # ` F ) = 4
44 43 oveq2i 
 |-  ( 0 ... ( # ` F ) ) = ( 0 ... 4 )
45 1 gpgprismgr4cycllem4 
 |-  ( # ` P ) = 5
46 df-5 
 |-  5 = ( 4 + 1 )
47 45 46 eqtri 
 |-  ( # ` P ) = ( 4 + 1 )
48 47 oveq2i 
 |-  ( 0 ..^ ( # ` P ) ) = ( 0 ..^ ( 4 + 1 ) )
49 42 44 48 3eqtr4i 
 |-  ( 0 ... ( # ` F ) ) = ( 0 ..^ ( # ` P ) )
50 49 a1i 
 |-  ( N e. ( ZZ>= ` 3 ) -> ( 0 ... ( # ` F ) ) = ( 0 ..^ ( # ` P ) ) )
51 50 feq2d 
 |-  ( N e. ( ZZ>= ` 3 ) -> ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) <-> P : ( 0 ..^ ( # ` P ) ) --> ( Vtx ` G ) ) )
52 39 51 mpbird 
 |-  ( N e. ( ZZ>= ` 3 ) -> P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) )