Theorem eupth2lem3fi 16458

Description: Lemma for eupth2fi 16461. (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 16437	. . . 4 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
7		wlkcl 16314	. . . 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 10517	. . 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 16436	. . 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 5672	. . 3 |- (Vtx ` H ) = (Vtx ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. )
17		eupth2fi.fi	. . . . 5 |- ( ph -> V e. Fin )
18	17	elexd 2826	. . . 4 |- ( ph -> V e. _V )
19		eupth2fi.g	. . . . . . 7 |- ( ph -> G e. UMGraph )
20		iedgex 16001	. . . . . . 7 |- ( G e. UMGraph -> (iEdg ` G ) e. _V )
21	19, 20	syl 14	. . . . . 6 |- ( ph -> (iEdg ` G ) e. _V )
22	2, 21	eqeltrid 2319	. . . . 5 |- ( ph -> I e. _V )
23		resexg 5077	. . . . 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 16004	. . . 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 2277	. 2 |- ( ph -> (Vtx ` H ) = V )
28		eupthv 16428	. . . . . . . 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 5688	. . . . . 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 5688	. . . . . 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 4343	. . . . 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 4296	. . . 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 16004	. . 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 5672	. . 3 |- (Vtx ` X ) = (Vtx ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. )
43		resexg 5077	. . . . 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 16004	. . . 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 2277	. 2 |- ( ph -> (Vtx ` X ) = V )
48	15	fveq2i 5672	. . 3 |- (iEdg ` H ) = (iEdg ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. )
49		opiedgfv 16007	. . . 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 2277	. 2 |- ( ph -> (iEdg ` H ) = ( I |` ( F " ( 0..^ N ) ) ) )
52		opiedgfv 16007	. . 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 5672	. . . 4 |- (iEdg ` X ) = (iEdg ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. )
55		opiedgfv 16007	. . . . 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 2277	. . 3 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) )
58	4	nn0zd 9694	. . . . . 6 |- ( ph -> N e. ZZ )
59		fzval3 10545	. . . . . . 7 |- ( N e. ZZ -> ( 0 ... N ) = ( 0..^ ( N + 1 ) ) )
60	59	eqcomd 2238	. . . . . 6 |- ( N e. ZZ -> ( 0..^ ( N + 1 ) ) = ( 0 ... N ) )
61	58, 60	syl 14	. . . . 5 |- ( ph -> ( 0..^ ( N + 1 ) ) = ( 0 ... N ) )
62	61	imaeq2d 5100	. . . 4 |- ( ph -> ( F " ( 0..^ ( N + 1 ) ) ) = ( F " ( 0 ... N ) ) )
63	62	reseq2d 5037	. . 3 |- ( ph -> ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) = ( I |` ( F " ( 0 ... N ) ) ) )
64	57, 63	eqtrd 2265	. 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 5674	. . . 4 |- ( k = N -> ( I ` ( F ` k ) ) = ( I ` ( F ` N ) ) )
67		fveq2 5669	. . . . 5 |- ( k = N -> ( P ` k ) = ( P ` N ) )
68		fvoveq1 6072	. . . . 5 |- ( k = N -> ( P ` ( k + 1 ) ) = ( P ` ( N + 1 ) ) )
69	67, 68	preq12d 3775	. . . 4 |- ( k = N -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
70	66, 69	eqeq12d 2247	. . 3 |- ( k = N -> ( ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } ) )
71		umgrupgr 16094	. . . . 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 16343	. . . 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 2925	. 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 16456	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 2203   A.wral 2520   {crab 2524   _Vcvv 2812   (/)c0 3507   ifcif 3619   {csn 3688   {cpr 3689   <.cop 3691   class class class wbr 4108   |` cres 4750   "cima 4751   Fun wfun 5345   ` cfv 5351  (class class class)co 6049   Fincfn 6974   0cc0 8123   1c1 8124   + caddc 8126   <_ cle 8305   2c2 9284   NN0cn0 9492   ZZcz 9573   ...cfz 10338  ..^cfzo 10472  ♯chash 11133   || cdvds 12466  Vtxcvtx 15994  iEdgciedg 15995  UPGraphcupgr 16073  UMGraphcumgr 16074  VtxDegcvtxdg 16268  Walkscwlks 16299  Trailsctrls 16362  EulerPathsceupth 16424

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-map 6883  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-z 9574  df-dec 9706  df-uz 9850  df-q 9948  df-rp 9983  df-xadd 10102  df-fz 10339  df-fzo 10473  df-fl 10626  df-mod 10681  df-seqfrec 10806  df-exp 10897  df-ihash 11134  df-word 11218  df-cj 11520  df-re 11521  df-im 11522  df-rsqrt 11676  df-abs 11677  df-dvds 12467  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-edg 16040  df-uhgrm 16051  df-ushgrm 16052  df-upgren 16075  df-umgren 16076  df-uspgren 16137  df-subgr 16236  df-vtxdg 16269  df-wlks 16300  df-trls 16363  df-eupth 16425

This theorem is referenced by:  eupth2lemsfi  16460