Theorem eupth2lem3fi 16397

Description: Lemma for eupth2fi 16400. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

Hypotheses

Ref	Expression
eupth2.v	|- V = (Vtx ` G )
eupth2.i	|- I = (iEdg ` G )
eupth2fi.g	|- ( ph -> G e. UMGraph )
eupth2.f	|- ( ph -> Fun I )
eupth2.p	|- ( ph -> F (EulerPaths ` G ) P )
eupth2fi.fi	|- ( ph -> V e. Fin )
eupth2.h	|- H = <. V , ( I |` ( F " ( 0..^ N ) ) ) >.
eupth2.x	|- X = <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >.
eupth2.n	|- ( ph -> N e. NN0 )
eupth2.l	|- ( ph -> ( N + 1 ) <_ (♯ ` F ) )
eupth2.u	|- ( ph -> U e. V )
eupth2.o	|- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` H ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )

Assertion

Ref	Expression
eupth2lem3fi	|- ( ph -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Distinct variable groups:   x, H   x, U   x, V

Allowed substitution hints:   ph( x)   P( x)   F( x)   G( x)   I( x)   N( x)   X( x)

Proof of Theorem eupth2lem3fi

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eupth2.v	. 2 |- V = (Vtx ` G )
2		eupth2.i	. 2 |- I = (iEdg ` G )
3		eupth2.f	. 2 |- ( ph -> Fun I )
4		eupth2.n	. . 3 |- ( ph -> N e. NN0 )
5		eupth2.p	. . . 4 |- ( ph -> F (EulerPaths ` G ) P )
6		eupthiswlk 16376	. . . 4 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
7		wlkcl 16253	. . . 4 |- ( F (Walks ` G ) P -> (♯ ` F ) e. NN0 )
8	5, 6, 7	3syl 17	. . 3 |- ( ph -> (♯ ` F ) e. NN0 )
9		eupth2.l	. . 3 |- ( ph -> ( N + 1 ) <_ (♯ ` F ) )
10		nn0p1elfzo 10465	. . 3 |- ( ( N e. NN0 /\ (♯ ` F ) e. NN0 /\ ( N + 1 ) <_ (♯ ` F ) ) -> N e. ( 0..^ (♯ ` F ) ) )
11	4, 8, 9, 10	syl3anc 1274	. 2 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
12		eupth2.u	. 2 |- ( ph -> U e. V )
13		eupthistrl 16375	. . 3 |- ( F (EulerPaths ` G ) P -> F (Trails ` G ) P )
14	5, 13	syl 14	. 2 |- ( ph -> F (Trails ` G ) P )
15		eupth2.h	. . . 4 |- H = <. V , ( I |` ( F " ( 0..^ N ) ) ) >.
16	15	fveq2i 5651	. . 3 |- (Vtx ` H ) = (Vtx ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. )
17		eupth2fi.fi	. . . . 5 |- ( ph -> V e. Fin )
18	17	elexd 2817	. . . 4 |- ( ph -> V e. _V )
19		eupth2fi.g	. . . . . . 7 |- ( ph -> G e. UMGraph )
20		iedgex 15940	. . . . . . 7 |- ( G e. UMGraph -> (iEdg ` G ) e. _V )
21	19, 20	syl 14	. . . . . 6 |- ( ph -> (iEdg ` G ) e. _V )
22	2, 21	eqeltrid 2318	. . . . 5 |- ( ph -> I e. _V )
23		resexg 5059	. . . . 5 |- ( I e. _V -> ( I |` ( F " ( 0..^ N ) ) ) e. _V )
24	22, 23	syl 14	. . . 4 |- ( ph -> ( I |` ( F " ( 0..^ N ) ) ) e. _V )
25		opvtxfv 15943	. . . 4 |- ( ( V e. _V /\ ( I |` ( F " ( 0..^ N ) ) ) e. _V ) -> (Vtx ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. ) = V )
26	18, 24, 25	syl2anc 411	. . 3 |- ( ph -> (Vtx ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. ) = V )
27	16, 26	eqtrid 2276	. 2 |- ( ph -> (Vtx ` H ) = V )
28		eupthv 16367	. . . . . . . 8 |- ( F (EulerPaths ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
29	5, 28	syl 14	. . . . . . 7 |- ( ph -> ( G e. _V /\ F e. _V /\ P e. _V ) )
30	29	simp2d 1037	. . . . . 6 |- ( ph -> F e. _V )
31		fvexg 5667	. . . . . 6 |- ( ( F e. _V /\ N e. NN0 ) -> ( F ` N ) e. _V )
32	30, 4, 31	syl2anc 411	. . . . 5 |- ( ph -> ( F ` N ) e. _V )
33		fvexg 5667	. . . . . 6 |- ( ( I e. _V /\ ( F ` N ) e. _V ) -> ( I ` ( F ` N ) ) e. _V )
34	22, 32, 33	syl2anc 411	. . . . 5 |- ( ph -> ( I ` ( F ` N ) ) e. _V )
35		opexg 4326	. . . . 5 |- ( ( ( F ` N ) e. _V /\ ( I ` ( F ` N ) ) e. _V ) -> <. ( F ` N ) , ( I ` ( F ` N ) ) >. e. _V )
36	32, 34, 35	syl2anc 411	. . . 4 |- ( ph -> <. ( F ` N ) , ( I ` ( F ` N ) ) >. e. _V )
37		snexg 4280	. . . 4 |- ( <. ( F ` N ) , ( I ` ( F ` N ) ) >. e. _V -> { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } e. _V )
38	36, 37	syl 14	. . 3 |- ( ph -> { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } e. _V )
39		opvtxfv 15943	. . 3 |- ( ( V e. _V /\ { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } e. _V ) -> (Vtx ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = V )
40	18, 38, 39	syl2anc 411	. 2 |- ( ph -> (Vtx ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = V )
41		eupth2.x	. . . 4 |- X = <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >.
42	41	fveq2i 5651	. . 3 |- (Vtx ` X ) = (Vtx ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. )
43		resexg 5059	. . . . 5 |- ( I e. _V -> ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) e. _V )
44	22, 43	syl 14	. . . 4 |- ( ph -> ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) e. _V )
45		opvtxfv 15943	. . . 4 |- ( ( V e. _V /\ ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) e. _V ) -> (Vtx ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. ) = V )
46	18, 44, 45	syl2anc 411	. . 3 |- ( ph -> (Vtx ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. ) = V )
47	42, 46	eqtrid 2276	. 2 |- ( ph -> (Vtx ` X ) = V )
48	15	fveq2i 5651	. . 3 |- (iEdg ` H ) = (iEdg ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. )
49		opiedgfv 15946	. . . 4 |- ( ( V e. _V /\ ( I |` ( F " ( 0..^ N ) ) ) e. _V ) -> (iEdg ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. ) = ( I |` ( F " ( 0..^ N ) ) ) )
50	18, 24, 49	syl2anc 411	. . 3 |- ( ph -> (iEdg ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. ) = ( I |` ( F " ( 0..^ N ) ) ) )
51	48, 50	eqtrid 2276	. 2 |- ( ph -> (iEdg ` H ) = ( I |` ( F " ( 0..^ N ) ) ) )
52		opiedgfv 15946	. . 3 |- ( ( V e. _V /\ { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } e. _V ) -> (iEdg ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
53	18, 38, 52	syl2anc 411	. 2 |- ( ph -> (iEdg ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
54	41	fveq2i 5651	. . . 4 |- (iEdg ` X ) = (iEdg ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. )
55		opiedgfv 15946	. . . . 5 |- ( ( V e. _V /\ ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) e. _V ) -> (iEdg ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. ) = ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) )
56	18, 44, 55	syl2anc 411	. . . 4 |- ( ph -> (iEdg ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. ) = ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) )
57	54, 56	eqtrid 2276	. . 3 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) )
58	4	nn0zd 9643	. . . . . 6 |- ( ph -> N e. ZZ )
59		fzval3 10493	. . . . . . 7 |- ( N e. ZZ -> ( 0 ... N ) = ( 0..^ ( N + 1 ) ) )
60	59	eqcomd 2237	. . . . . 6 |- ( N e. ZZ -> ( 0..^ ( N + 1 ) ) = ( 0 ... N ) )
61	58, 60	syl 14	. . . . 5 |- ( ph -> ( 0..^ ( N + 1 ) ) = ( 0 ... N ) )
62	61	imaeq2d 5082	. . . 4 |- ( ph -> ( F " ( 0..^ ( N + 1 ) ) ) = ( F " ( 0 ... N ) ) )
63	62	reseq2d 5019	. . 3 |- ( ph -> ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) = ( I |` ( F " ( 0 ... N ) ) ) )
64	57, 63	eqtrd 2264	. 2 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0 ... N ) ) ) )
65		eupth2.o	. 2 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` H ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
66		2fveq3 5653	. . . 4 |- ( k = N -> ( I ` ( F ` k ) ) = ( I ` ( F ` N ) ) )
67		fveq2 5648	. . . . 5 |- ( k = N -> ( P ` k ) = ( P ` N ) )
68		fvoveq1 6051	. . . . 5 |- ( k = N -> ( P ` ( k + 1 ) ) = ( P ` ( N + 1 ) ) )
69	67, 68	preq12d 3760	. . . 4 |- ( k = N -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
70	66, 69	eqeq12d 2246	. . 3 |- ( k = N -> ( ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } ) )
71		umgrupgr 16033	. . . . 5 |- ( G e. UMGraph -> G e. UPGraph )
72	19, 71	syl 14	. . . 4 |- ( ph -> G e. UPGraph )
73	5, 6	syl 14	. . . 4 |- ( ph -> F (Walks ` G ) P )
74	2	upgrwlkedg 16282	. . . 4 |- ( ( G e. UPGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
75	72, 73, 74	syl2anc 411	. . 3 |- ( ph -> A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
76	70, 75, 11	rspcdva 2916	. 2 |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
77	1, 2, 3, 11, 12, 14, 27, 40, 47, 51, 53, 64, 19, 17, 65, 76	eupth2lem3lem7fi 16395	1 |- ( ph -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   {crab 2515   _Vcvv 2803   (/)c0 3496   ifcif 3607   {csn 3673   {cpr 3674   <.cop 3676   class class class wbr 4093   |` cres 4733   "cima 4734   Fun wfun 5327   ` cfv 5333  (class class class)co 6028   Fincfn 6952   0cc0 8075   1c1 8076   + caddc 8078   <_ cle 8258   2c2 9237   NN0cn0 9445   ZZcz 9522   ...cfz 10286  ..^cfzo 10420  ♯chash 11081   || cdvds 12409  Vtxcvtx 15933  iEdgciedg 15934  UPGraphcupgr 16012  UMGraphcumgr 16013  VtxDegcvtxdg 16207  Walkscwlks 16238  Trailsctrls 16301  EulerPathsceupth 16363

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-2o 6626  df-oadd 6629  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-dec 9655  df-uz 9799  df-q 9897  df-rp 9932  df-xadd 10051  df-fz 10287  df-fzo 10421  df-fl 10574  df-mod 10629  df-seqfrec 10754  df-exp 10845  df-ihash 11082  df-word 11161  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-dvds 12410  df-ndx 13146  df-slot 13147  df-base 13149  df-edgf 15926  df-vtx 15935  df-iedg 15936  df-edg 15979  df-uhgrm 15990  df-ushgrm 15991  df-upgren 16014  df-umgren 16015  df-uspgren 16076  df-subgr 16175  df-vtxdg 16208  df-wlks 16239  df-trls 16302  df-eupth 16364

This theorem is referenced by:  eupth2lemsfi  16399