Theorem upgrwlkvtxedg 16376

Description: The pairs of connected vertices of a walk are edges in a pseudograph. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)

Hypothesis

Ref	Expression
wlkvtxedg.e	|- E = (Edg ` G )

Assertion

Ref	Expression
upgrwlkvtxedg	|- ( ( G e. UPGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E )

Distinct variable groups:   k, F   k, G   P, k

Allowed substitution hint:   E( k)

Proof of Theorem upgrwlkvtxedg

Step	Hyp	Ref	Expression
1		eqid 2234	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2234	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	upgriswlkdc 16372	. . . 4 |- ( G e. UPGraph -> ( F (Walks ` G ) P <-> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) )
4		simpr 110	. . . . . 6 |- ( (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
5	4	ralimi 2607	. . . . 5 |- ( A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> A. k e. ( 0..^ (♯ ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
6	5	3anim3i 1214	. . . 4 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ (♯ ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
7	3, 6	biimtrdi 163	. . 3 |- ( G e. UPGraph -> ( F (Walks ` G ) P -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ (♯ ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
8		wlkvtxedg.e	. . . . . . . . . . 11 |- E = (Edg ` G )
9	2, 8	upgredginwlk 16368	. . . . . . . . . 10 |- ( ( G e. UPGraph /\ F e. Word dom (iEdg ` G ) ) -> ( k e. ( 0..^ (♯ ` F ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) e. E ) )
10	9	ancoms 268	. . . . . . . . 9 |- ( ( F e. Word dom (iEdg ` G ) /\ G e. UPGraph ) -> ( k e. ( 0..^ (♯ ` F ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) e. E ) )
11	10	imp 124	. . . . . . . 8 |- ( ( ( F e. Word dom (iEdg ` G ) /\ G e. UPGraph ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) e. E )
12		eleq1 2297	. . . . . . . . 9 |- ( { ( P ` k ) , ( P ` ( k + 1 ) ) } = ( (iEdg ` G ) ` ( F ` k ) ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> ( (iEdg ` G ) ` ( F ` k ) ) e. E ) )
13	12	eqcoms 2237	. . . . . . . 8 |- ( ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> ( (iEdg ` G ) ` ( F ` k ) ) e. E ) )
14	11, 13	syl5ibrcom 157	. . . . . . 7 |- ( ( ( F e. Word dom (iEdg ` G ) /\ G e. UPGraph ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) )
15	14	ralimdva 2611	. . . . . 6 |- ( ( F e. Word dom (iEdg ` G ) /\ G e. UPGraph ) -> ( A. k e. ( 0..^ (♯ ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) )
16	15	impancom 260	. . . . 5 |- ( ( F e. Word dom (iEdg ` G ) /\ A. k e. ( 0..^ (♯ ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( G e. UPGraph -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) )
17	16	3adant2 1043	. . . 4 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ (♯ ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( G e. UPGraph -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) )
18	17	com12 30	. . 3 |- ( G e. UPGraph -> ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ (♯ ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) )
19	7, 18	syld 45	. 2 |- ( G e. UPGraph -> ( F (Walks ` G ) P -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) )
20	19	imp 124	1 |- ( ( G e. UPGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   {cpr 3692   class class class wbr 4111   dom cdm 4751   -->wf 5350   ` cfv 5354  (class class class)co 6052   0cc0 8129   1c1 8130   + caddc 8132   ...cfz 10345  ..^cfzo 10480  ♯chash 11142  Word cword 11228  Vtxcvtx 16024  iEdgciedg 16025  Edgcedg 16069  UPGraphcupgr 16103  Walkscwlks 16329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-2o 6650  df-er 6769  df-map 6886  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-z 9580  df-dec 9713  df-uz 9857  df-fz 10346  df-fzo 10481  df-ihash 11143  df-word 11229  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-edg 16070  df-uhgrm 16081  df-upgren 16105  df-wlks 16330

This theorem is referenced by:  umgrwlknloop  16380