Theorem wlkp1lem6 26575

Description: Lemma for wlkp1 26578. (Contributed by AV, 6-Mar-2021.)

Hypotheses

Ref	Expression
wlkp1.v	|- V = (Vtx ` G )
wlkp1.i	|- I = (iEdg ` G )
wlkp1.f	|- ( ph -> Fun I )
wlkp1.a	|- ( ph -> I e. Fin )
wlkp1.b	|- ( ph -> B e. _V )
wlkp1.c	|- ( ph -> C e. V )
wlkp1.d	|- ( ph -> -. B e. dom I )
wlkp1.w	|- ( ph -> F (Walks ` G ) P )
wlkp1.n	|- N = ( # ` F )
wlkp1.e	|- ( ph -> E e. (Edg ` G ) )
wlkp1.x	|- ( ph -> { ( P ` N ) , C } C_ E )
wlkp1.u	|- ( ph -> (iEdg ` S ) = ( I u. { <. B , E >. } ) )
wlkp1.h	|- H = ( F u. { <. N , B >. } )
wlkp1.q	|- Q = ( P u. { <. ( N + 1 ) , C >. } )
wlkp1.s	|- ( ph -> (Vtx ` S ) = V )

Assertion

Ref	Expression
wlkp1lem6	|- ( ph -> A. k e. ( 0..^ N ) ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) )

Distinct variable group:   ph, k

Allowed substitution hints:   B( k)   C( k)   P( k)   Q( k)   S( k)   E( k)   F( k)   G( k)   H( k)   I( k)   N( k)   V( k)

Proof of Theorem wlkp1lem6

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkp1.v	. . . 4 |- V = (Vtx ` G )
2		wlkp1.i	. . . 4 |- I = (iEdg ` G )
3		wlkp1.f	. . . 4 |- ( ph -> Fun I )
4		wlkp1.a	. . . 4 |- ( ph -> I e. Fin )
5		wlkp1.b	. . . 4 |- ( ph -> B e. _V )
6		wlkp1.c	. . . 4 |- ( ph -> C e. V )
7		wlkp1.d	. . . 4 |- ( ph -> -. B e. dom I )
8		wlkp1.w	. . . 4 |- ( ph -> F (Walks ` G ) P )
9		wlkp1.n	. . . 4 |- N = ( # ` F )
10		wlkp1.e	. . . 4 |- ( ph -> E e. (Edg ` G ) )
11		wlkp1.x	. . . 4 |- ( ph -> { ( P ` N ) , C } C_ E )
12		wlkp1.u	. . . 4 |- ( ph -> (iEdg ` S ) = ( I u. { <. B , E >. } ) )
13		wlkp1.h	. . . 4 |- H = ( F u. { <. N , B >. } )
14		wlkp1.q	. . . 4 |- Q = ( P u. { <. ( N + 1 ) , C >. } )
15		wlkp1.s	. . . 4 |- ( ph -> (Vtx ` S ) = V )
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	wlkp1lem5 26574	. . 3 |- ( ph -> A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) )
17		elfzofz 12485	. . . . . . 7 |- ( k e. ( 0..^ N ) -> k e. ( 0 ... N ) )
18	17	adantl 482	. . . . . 6 |- ( ( ph /\ k e. ( 0..^ N ) ) -> k e. ( 0 ... N ) )
19		fveq2 6191	. . . . . . . 8 |- ( x = k -> ( Q ` x ) = ( Q ` k ) )
20		fveq2 6191	. . . . . . . 8 |- ( x = k -> ( P ` x ) = ( P ` k ) )
21	19, 20	eqeq12d 2637	. . . . . . 7 |- ( x = k -> ( ( Q ` x ) = ( P ` x ) <-> ( Q ` k ) = ( P ` k ) ) )
22	21	rspcv 3305	. . . . . 6 |- ( k e. ( 0 ... N ) -> ( A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) -> ( Q ` k ) = ( P ` k ) ) )
23	18, 22	syl 17	. . . . 5 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) -> ( Q ` k ) = ( P ` k ) ) )
24	23	imp 445	. . . 4 |- ( ( ( ph /\ k e. ( 0..^ N ) ) /\ A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) ) -> ( Q ` k ) = ( P ` k ) )
25		fzofzp1 12565	. . . . . . 7 |- ( k e. ( 0..^ N ) -> ( k + 1 ) e. ( 0 ... N ) )
26	25	adantl 482	. . . . . 6 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( k + 1 ) e. ( 0 ... N ) )
27		fveq2 6191	. . . . . . . 8 |- ( x = ( k + 1 ) -> ( Q ` x ) = ( Q ` ( k + 1 ) ) )
28		fveq2 6191	. . . . . . . 8 |- ( x = ( k + 1 ) -> ( P ` x ) = ( P ` ( k + 1 ) ) )
29	27, 28	eqeq12d 2637	. . . . . . 7 |- ( x = ( k + 1 ) -> ( ( Q ` x ) = ( P ` x ) <-> ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) ) )
30	29	rspcv 3305	. . . . . 6 |- ( ( k + 1 ) e. ( 0 ... N ) -> ( A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) -> ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) ) )
31	26, 30	syl 17	. . . . 5 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) -> ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) ) )
32	31	imp 445	. . . 4 |- ( ( ( ph /\ k e. ( 0..^ N ) ) /\ A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) ) -> ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) )
33	12	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. ( 0..^ N ) ) -> (iEdg ` S ) = ( I u. { <. B , E >. } ) )
34	13	fveq1i 6192	. . . . . . . 8 |- ( H ` k ) = ( ( F u. { <. N , B >. } ) ` k )
35		fzonel 12483	. . . . . . . . . . . . . 14 |- -. N e. ( 0..^ N )
36		eleq1 2689	. . . . . . . . . . . . . 14 |- ( N = k -> ( N e. ( 0..^ N ) <-> k e. ( 0..^ N ) ) )
37	35, 36	mtbii 316	. . . . . . . . . . . . 13 |- ( N = k -> -. k e. ( 0..^ N ) )
38	37	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( N = k -> -. k e. ( 0..^ N ) ) )
39	38	con2d 129	. . . . . . . . . . 11 |- ( ph -> ( k e. ( 0..^ N ) -> -. N = k ) )
40	39	imp 445	. . . . . . . . . 10 |- ( ( ph /\ k e. ( 0..^ N ) ) -> -. N = k )
41	40	neqned 2801	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0..^ N ) ) -> N =/= k )
42		fvunsn 6445	. . . . . . . . 9 |- ( N =/= k -> ( ( F u. { <. N , B >. } ) ` k ) = ( F ` k ) )
43	41, 42	syl 17	. . . . . . . 8 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( ( F u. { <. N , B >. } ) ` k ) = ( F ` k ) )
44	34, 43	syl5eq 2668	. . . . . . 7 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( H ` k ) = ( F ` k ) )
45	33, 44	fveq12d 6197	. . . . . 6 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( (iEdg ` S ) ` ( H ` k ) ) = ( ( I u. { <. B , E >. } ) ` ( F ` k ) ) )
46	9	oveq2i 6661	. . . . . . . . . . . . . . . 16 |- ( 0..^ N ) = ( 0..^ ( # ` F ) )
47	46	eleq2i 2693	. . . . . . . . . . . . . . 15 |- ( k e. ( 0..^ N ) <-> k e. ( 0..^ ( # ` F ) ) )
48	2	wlkf 26510	. . . . . . . . . . . . . . . . 17 |- ( F (Walks ` G ) P -> F e. Word dom I )
49	8, 48	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> F e. Word dom I )
50		wrdsymbcl 13318	. . . . . . . . . . . . . . . . 17 |- ( ( F e. Word dom I /\ k e. ( 0..^ ( # ` F ) ) ) -> ( F ` k ) e. dom I )
51	50	ex 450	. . . . . . . . . . . . . . . 16 |- ( F e. Word dom I -> ( k e. ( 0..^ ( # ` F ) ) -> ( F ` k ) e. dom I ) )
52	49, 51	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( k e. ( 0..^ ( # ` F ) ) -> ( F ` k ) e. dom I ) )
53	47, 52	syl5bi 232	. . . . . . . . . . . . . 14 |- ( ph -> ( k e. ( 0..^ N ) -> ( F ` k ) e. dom I ) )
54	53	imp 445	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( F ` k ) e. dom I )
55		eleq1 2689	. . . . . . . . . . . . 13 |- ( B = ( F ` k ) -> ( B e. dom I <-> ( F ` k ) e. dom I ) )
56	54, 55	syl5ibrcom 237	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( B = ( F ` k ) -> B e. dom I ) )
57	56	con3d 148	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( -. B e. dom I -> -. B = ( F ` k ) ) )
58	57	ex 450	. . . . . . . . . 10 |- ( ph -> ( k e. ( 0..^ N ) -> ( -. B e. dom I -> -. B = ( F ` k ) ) ) )
59	7, 58	mpid 44	. . . . . . . . 9 |- ( ph -> ( k e. ( 0..^ N ) -> -. B = ( F ` k ) ) )
60	59	imp 445	. . . . . . . 8 |- ( ( ph /\ k e. ( 0..^ N ) ) -> -. B = ( F ` k ) )
61	60	neqned 2801	. . . . . . 7 |- ( ( ph /\ k e. ( 0..^ N ) ) -> B =/= ( F ` k ) )
62		fvunsn 6445	. . . . . . 7 |- ( B =/= ( F ` k ) -> ( ( I u. { <. B , E >. } ) ` ( F ` k ) ) = ( I ` ( F ` k ) ) )
63	61, 62	syl 17	. . . . . 6 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( ( I u. { <. B , E >. } ) ` ( F ` k ) ) = ( I ` ( F ` k ) ) )
64	45, 63	eqtrd 2656	. . . . 5 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) )
65	64	adantr 481	. . . 4 |- ( ( ( ph /\ k e. ( 0..^ N ) ) /\ A. x e. ( 0 ... N ) ( Q ` x ) = ( P ` x ) ) -> ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) )
66	24, 32, 65	3jca 1242	. . 3 |- ( ( ( ph /\ k e. ( 0..^ N ) ) /\ A. x e. ( 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 704	. 2 |- ( ( ph /\ k e. ( 0..^ N ) ) -> ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) )
68	67	ralrimiva 2966	1 |- ( ph -> A. k e. ( 0..^ N ) ( ( Q ` k ) = ( P ` k ) /\ ( Q ` ( k + 1 ) ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` S ) ` ( H ` k ) ) = ( I ` ( F ` k ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   dom cdm 5114   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939  Walkscwlks 26492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495

This theorem is referenced by:  wlkp1lem8  26577