Theorem upgrwlkvtxedg 16214

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 2231	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2231	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	upgriswlkdc 16210	. . . 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 2595	. . . . 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 1213	. . . 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 16206	. . . . . . . . . 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 2294	. . . . . . . . 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 2234	. . . . . . . 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 2599	. . . . . 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 1042	. . . 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 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   {cpr 3670   class class class wbr 4088   dom cdm 4725   -->wf 5322   ` cfv 5326  (class class class)co 6017   0cc0 8031   1c1 8032   + caddc 8034   ...cfz 10242  ..^cfzo 10376  ♯chash 11036  Word cword 11112  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907  UPGraphcupgr 15941  Walkscwlks 16167

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-ifp 986  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-2o 6582  df-er 6701  df-map 6818  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-dec 9611  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-edg 15908  df-uhgrm 15919  df-upgren 15943  df-wlks 16168

This theorem is referenced by:  umgrwlknloop  16218