Theorem eupth2lem3fi 16330

Description: Lemma for eupth2fi 16333. (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 16309	. . . 4 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
7		wlkcl 16186	. . . 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 10422	. . 3 |- ( ( N e. NN0 /\ (♯ ` F ) e. NN0 /\ ( N + 1 ) <_ (♯ ` F ) ) -> N e. ( 0..^ (♯ ` F ) ) )
11	4, 8, 9, 10	syl3anc 1273	. 2 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
12		eupth2.u	. 2 |- ( ph -> U e. V )
13		eupthistrl 16308	. . 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 5642	. . 3 |- (Vtx ` H ) = (Vtx ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. )
17		eupth2fi.fi	. . . . 5 |- ( ph -> V e. Fin )
18	17	elexd 2816	. . . 4 |- ( ph -> V e. _V )
19		eupth2fi.g	. . . . . . 7 |- ( ph -> G e. UMGraph )
20		iedgex 15873	. . . . . . 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 5053	. . . . 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 15876	. . . 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 16300	. . . . . . . 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 1036	. . . . . 6 |- ( ph -> F e. _V )
31		fvexg 5658	. . . . . 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 5658	. . . . . 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 4320	. . . . 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 4274	. . . 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 15876	. . 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 5642	. . 3 |- (Vtx ` X ) = (Vtx ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. )
43		resexg 5053	. . . . 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 15876	. . . 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 5642	. . 3 |- (iEdg ` H ) = (iEdg ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. )
49		opiedgfv 15879	. . . 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 15879	. . 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 5642	. . . 4 |- (iEdg ` X ) = (iEdg ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. )
55		opiedgfv 15879	. . . . 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 9600	. . . . . 6 |- ( ph -> N e. ZZ )
59		fzval3 10450	. . . . . . 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 5076	. . . 4 |- ( ph -> ( F " ( 0..^ ( N + 1 ) ) ) = ( F " ( 0 ... N ) ) )
63	62	reseq2d 5013	. . 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 5644	. . . 4 |- ( k = N -> ( I ` ( F ` k ) ) = ( I ` ( F ` N ) ) )
67		fveq2 5639	. . . . 5 |- ( k = N -> ( P ` k ) = ( P ` N ) )
68		fvoveq1 6041	. . . . 5 |- ( k = N -> ( P ` ( k + 1 ) ) = ( P ` ( N + 1 ) ) )
69	67, 68	preq12d 3756	. . . 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 15966	. . . . 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 16215	. . . 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 2915	. 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 16328	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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   {crab 2514   _Vcvv 2802   (/)c0 3494   ifcif 3605   {csn 3669   {cpr 3670   <.cop 3672   class class class wbr 4088   |` cres 4727   "cima 4728   Fun wfun 5320   ` cfv 5326  (class class class)co 6018   Fincfn 6909   0cc0 8032   1c1 8033   + caddc 8035   <_ cle 8215   2c2 9194   NN0cn0 9402   ZZcz 9479   ...cfz 10243  ..^cfzo 10377  ♯chash 11038   || cdvds 12350  Vtxcvtx 15866  iEdgciedg 15867  UPGraphcupgr 15945  UMGraphcumgr 15946  VtxDegcvtxdg 16140  Walkscwlks 16171  Trailsctrls 16234  EulerPathsceupth 16296

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-ifp 986  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  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-rmo 2518  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-po 4393  df-iso 4394  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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-map 6819  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-q 9854  df-rp 9889  df-xadd 10008  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-ihash 11039  df-word 11115  df-cj 11404  df-re 11405  df-im 11406  df-rsqrt 11560  df-abs 11561  df-dvds 12351  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-edg 15912  df-uhgrm 15923  df-ushgrm 15924  df-upgren 15947  df-umgren 15948  df-uspgren 16009  df-subgr 16108  df-vtxdg 16141  df-wlks 16172  df-trls 16235  df-eupth 16297

This theorem is referenced by:  eupth2lemsfi  16332