Theorem eupth2lem3fi 16700

Description: Lemma for eupth2fi 16703. (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 16679	. . . 4 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
7		wlkcl 16556	. . . 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 10577	. . 3 |- ( ( N e. NN0 /\ (♯ ` F ) e. NN0 /\ ( N + 1 ) <_ (♯ ` F ) ) -> N e. ( 0..^ (♯ ` F ) ) )
11	4, 8, 9, 10	syl3anc 1278	. 2 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
12		eupth2.u	. 2 |- ( ph -> U e. V )
13		eupthistrl 16678	. . 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 5696	. . 3 |- (Vtx ` H ) = (Vtx ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. )
17		eupth2fi.fi	. . . . 5 |- ( ph -> V e. Fin )
18	17	elexd 2835	. . . 4 |- ( ph -> V e. _V )
19		eupth2fi.g	. . . . . . 7 |- ( ph -> G e. UMGraph )
20		iedgex 16243	. . . . . . 7 |- ( G e. UMGraph -> (iEdg ` G ) e. _V )
21	19, 20	syl 14	. . . . . 6 |- ( ph -> (iEdg ` G ) e. _V )
22	2, 21	eqeltrid 2325	. . . . 5 |- ( ph -> I e. _V )
23		resexg 5101	. . . . 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 16246	. . . 4 |- ( ( V e. _V /\ ( I |` ( F " ( 0..^ N ) ) ) e. _V ) -> (Vtx ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. ) = V )
26	18, 24, 25	syl2anc 415	. . 3 |- ( ph -> (Vtx ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. ) = V )
27	16, 26	eqtrid 2283	. 2 |- ( ph -> (Vtx ` H ) = V )
28		eupthv 16670	. . . . . . . 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 1041	. . . . . 6 |- ( ph -> F e. _V )
31		fvexg 5712	. . . . . 6 |- ( ( F e. _V /\ N e. NN0 ) -> ( F ` N ) e. _V )
32	30, 4, 31	syl2anc 415	. . . . 5 |- ( ph -> ( F ` N ) e. _V )
33		fvexg 5712	. . . . . 6 |- ( ( I e. _V /\ ( F ` N ) e. _V ) -> ( I ` ( F ` N ) ) e. _V )
34	22, 32, 33	syl2anc 415	. . . . 5 |- ( ph -> ( I ` ( F ` N ) ) e. _V )
35		opexg 4366	. . . . 5 |- ( ( ( F ` N ) e. _V /\ ( I ` ( F ` N ) ) e. _V ) -> <. ( F ` N ) , ( I ` ( F ` N ) ) >. e. _V )
36	32, 34, 35	syl2anc 415	. . . 4 |- ( ph -> <. ( F ` N ) , ( I ` ( F ` N ) ) >. e. _V )
37		snexg 4319	. . . 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 16246	. . 3 |- ( ( V e. _V /\ { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } e. _V ) -> (Vtx ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = V )
40	18, 38, 39	syl2anc 415	. 2 |- ( ph -> (Vtx ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = V )
41		eupth2.x	. . . 4 |- X = <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >.
42	41	fveq2i 5696	. . 3 |- (Vtx ` X ) = (Vtx ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. )
43		resexg 5101	. . . . 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 16246	. . . 4 |- ( ( V e. _V /\ ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) e. _V ) -> (Vtx ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. ) = V )
46	18, 44, 45	syl2anc 415	. . 3 |- ( ph -> (Vtx ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. ) = V )
47	42, 46	eqtrid 2283	. 2 |- ( ph -> (Vtx ` X ) = V )
48	15	fveq2i 5696	. . 3 |- (iEdg ` H ) = (iEdg ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. )
49		opiedgfv 16249	. . . 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 415	. . 3 |- ( ph -> (iEdg ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. ) = ( I |` ( F " ( 0..^ N ) ) ) )
51	48, 50	eqtrid 2283	. 2 |- ( ph -> (iEdg ` H ) = ( I |` ( F " ( 0..^ N ) ) ) )
52		opiedgfv 16249	. . 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 415	. 2 |- ( ph -> (iEdg ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
54	41	fveq2i 5696	. . . 4 |- (iEdg ` X ) = (iEdg ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. )
55		opiedgfv 16249	. . . . 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 415	. . . 4 |- ( ph -> (iEdg ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. ) = ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) )
57	54, 56	eqtrid 2283	. . 3 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) )
58	4	nn0zd 9749	. . . . . 6 |- ( ph -> N e. ZZ )
59		fzval3 10605	. . . . . . 7 |- ( N e. ZZ -> ( 0 ... N ) = ( 0..^ ( N + 1 ) ) )
60	59	eqcomd 2244	. . . . . 6 |- ( N e. ZZ -> ( 0..^ ( N + 1 ) ) = ( 0 ... N ) )
61	58, 60	syl 14	. . . . 5 |- ( ph -> ( 0..^ ( N + 1 ) ) = ( 0 ... N ) )
62	61	imaeq2d 5124	. . . 4 |- ( ph -> ( F " ( 0..^ ( N + 1 ) ) ) = ( F " ( 0 ... N ) ) )
63	62	reseq2d 5061	. . 3 |- ( ph -> ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) = ( I |` ( F " ( 0 ... N ) ) ) )
64	57, 63	eqtrd 2271	. 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 5698	. . . 4 |- ( k = N -> ( I ` ( F ` k ) ) = ( I ` ( F ` N ) ) )
67		fveq2 5693	. . . . 5 |- ( k = N -> ( P ` k ) = ( P ` N ) )
68		fvoveq1 6102	. . . . 5 |- ( k = N -> ( P ` ( k + 1 ) ) = ( P ` ( N + 1 ) ) )
69	67, 68	preq12d 3795	. . . 4 |- ( k = N -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
70	66, 69	eqeq12d 2253	. . 3 |- ( k = N -> ( ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } ) )
71		umgrupgr 16336	. . . . 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 16585	. . . 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 415	. . 3 |- ( ph -> A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
76	70, 75, 11	rspcdva 2934	. 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 16698	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 1009   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   (/)c0 3520   ifcif 3638   {csn 3708   {cpr 3709   <.cop 3711   class class class wbr 4128   |` cres 4774   "cima 4775   Fun wfun 5369   ` cfv 5375  (class class class)co 6079   Fincfn 7016   0cc0 8173   1c1 8174   + caddc 8176   <_ cle 8355   2c2 9338   NN0cn0 9546   ZZcz 9627   ...cfz 10394  ..^cfzo 10532  ♯chash 11197   || cdvds 12537  Vtxcvtx 16236  iEdgciedg 16237  UPGraphcupgr 16315  UMGraphcumgr 16316  VtxDegcvtxdg 16510  Walkscwlks 16541  Trailsctrls 16604  EulerPathsceupth 16666

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-map 6918  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-q 10003  df-rp 10038  df-xadd 10158  df-fz 10395  df-fzo 10533  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-word 11288  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-dvds 12538  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-edg 16282  df-uhgrm 16293  df-ushgrm 16294  df-upgren 16317  df-umgren 16318  df-uspgren 16379  df-subgr 16478  df-vtxdg 16511  df-wlks 16542  df-trls 16605  df-eupth 16667

This theorem is referenced by:  eupth2lemsfi  16702