Theorem eupth2lem3 27096

Description: Lemma for eupth2 27099. (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 )
eupth2.g	|- ( ph -> G e. UPGraph )
eupth2.f	|- ( ph -> Fun I )
eupth2.p	|- ( ph -> F (EulerPaths ` G ) P )
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
eupth2lem3	|- ( 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 eupth2lem3

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 27072	. . . 4 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
7		wlkcl 26511	. . . 4 |- ( F (Walks ` G ) P -> ( # ` F ) e. NN0 )
8	5, 6, 7	3syl 18	. . 3 |- ( ph -> ( # ` F ) e. NN0 )
9		eupth2.l	. . 3 |- ( ph -> ( N + 1 ) <_ ( # ` F ) )
10		nn0p1elfzo 12510	. . 3 |- ( ( N e. NN0 /\ ( # ` F ) e. NN0 /\ ( N + 1 ) <_ ( # ` F ) ) -> N e. ( 0..^ ( # ` F ) ) )
11	4, 8, 9, 10	syl3anc 1326	. 2 |- ( ph -> N e. ( 0..^ ( # ` F ) ) )
12		eupth2.u	. 2 |- ( ph -> U e. V )
13		eupthistrl 27071	. . 3 |- ( F (EulerPaths ` G ) P -> F (Trails ` G ) P )
14	5, 13	syl 17	. 2 |- ( ph -> F (Trails ` G ) P )
15		eupth2.h	. . . . 5 |- H = <. V , ( I |` ( F " ( 0..^ N ) ) ) >.
16	15	fveq2i 6194	. . . 4 |- (Vtx ` H ) = (Vtx ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. )
17		fvex 6201	. . . . . 6 |- (Vtx ` G ) e. _V
18	1, 17	eqeltri 2697	. . . . 5 |- V e. _V
19		fvex 6201	. . . . . . 7 |- (iEdg ` G ) e. _V
20	2, 19	eqeltri 2697	. . . . . 6 |- I e. _V
21	20	resex 5443	. . . . 5 |- ( I |` ( F " ( 0..^ N ) ) ) e. _V
22	18, 21	opvtxfvi 25889	. . . 4 |- (Vtx ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. ) = V
23	16, 22	eqtri 2644	. . 3 |- (Vtx ` H ) = V
24	23	a1i 11	. 2 |- ( ph -> (Vtx ` H ) = V )
25		snex 4908	. . . 4 |- { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } e. _V
26	18, 25	opvtxfvi 25889	. . 3 |- (Vtx ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = V
27	26	a1i 11	. 2 |- ( ph -> (Vtx ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = V )
28		eupth2.x	. . . . 5 |- X = <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >.
29	28	fveq2i 6194	. . . 4 |- (Vtx ` X ) = (Vtx ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. )
30	20	resex 5443	. . . . 5 |- ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) e. _V
31	18, 30	opvtxfvi 25889	. . . 4 |- (Vtx ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. ) = V
32	29, 31	eqtri 2644	. . 3 |- (Vtx ` X ) = V
33	32	a1i 11	. 2 |- ( ph -> (Vtx ` X ) = V )
34	15	fveq2i 6194	. . . 4 |- (iEdg ` H ) = (iEdg ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. )
35	18, 21	opiedgfvi 25890	. . . 4 |- (iEdg ` <. V , ( I |` ( F " ( 0..^ N ) ) ) >. ) = ( I |` ( F " ( 0..^ N ) ) )
36	34, 35	eqtri 2644	. . 3 |- (iEdg ` H ) = ( I |` ( F " ( 0..^ N ) ) )
37	36	a1i 11	. 2 |- ( ph -> (iEdg ` H ) = ( I |` ( F " ( 0..^ N ) ) ) )
38	18, 25	opiedgfvi 25890	. . 3 |- (iEdg ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. }
39	38	a1i 11	. 2 |- ( ph -> (iEdg ` <. V , { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } >. ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
40	28	fveq2i 6194	. . . 4 |- (iEdg ` X ) = (iEdg ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. )
41	18, 30	opiedgfvi 25890	. . . 4 |- (iEdg ` <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >. ) = ( I |` ( F " ( 0..^ ( N + 1 ) ) ) )
42	40, 41	eqtri 2644	. . 3 |- (iEdg ` X ) = ( I |` ( F " ( 0..^ ( N + 1 ) ) ) )
43	4	nn0zd 11480	. . . . . 6 |- ( ph -> N e. ZZ )
44		fzval3 12536	. . . . . . 7 |- ( N e. ZZ -> ( 0 ... N ) = ( 0..^ ( N + 1 ) ) )
45	44	eqcomd 2628	. . . . . 6 |- ( N e. ZZ -> ( 0..^ ( N + 1 ) ) = ( 0 ... N ) )
46	43, 45	syl 17	. . . . 5 |- ( ph -> ( 0..^ ( N + 1 ) ) = ( 0 ... N ) )
47	46	imaeq2d 5466	. . . 4 |- ( ph -> ( F " ( 0..^ ( N + 1 ) ) ) = ( F " ( 0 ... N ) ) )
48	47	reseq2d 5396	. . 3 |- ( ph -> ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) = ( I |` ( F " ( 0 ... N ) ) ) )
49	42, 48	syl5eq 2668	. 2 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0 ... N ) ) ) )
50		eupth2.o	. 2 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` H ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
51		eupth2.g	. . . 4 |- ( ph -> G e. UPGraph )
52	5, 6	syl 17	. . . 4 |- ( ph -> F (Walks ` G ) P )
53	2	upgrwlkedg 26538	. . . 4 |- ( ( G e. UPGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
54	51, 52, 53	syl2anc 693	. . 3 |- ( ph -> A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
55		fveq2 6191	. . . . . 6 |- ( k = N -> ( F ` k ) = ( F ` N ) )
56	55	fveq2d 6195	. . . . 5 |- ( k = N -> ( I ` ( F ` k ) ) = ( I ` ( F ` N ) ) )
57		fveq2 6191	. . . . . 6 |- ( k = N -> ( P ` k ) = ( P ` N ) )
58		oveq1 6657	. . . . . . 7 |- ( k = N -> ( k + 1 ) = ( N + 1 ) )
59	58	fveq2d 6195	. . . . . 6 |- ( k = N -> ( P ` ( k + 1 ) ) = ( P ` ( N + 1 ) ) )
60	57, 59	preq12d 4276	. . . . 5 |- ( k = N -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
61	56, 60	eqeq12d 2637	. . . 4 |- ( k = N -> ( ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } ) )
62	61	rspcv 3305	. . 3 |- ( N e. ( 0..^ ( # ` F ) ) -> ( A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } ) )
63	11, 54, 62	sylc 65	. 2 |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
64	1, 2, 3, 11, 12, 14, 24, 27, 33, 37, 39, 49, 50, 63	eupth2lem3lem7 27094	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 196   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   (/)c0 3915   ifcif 4086   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   |` cres 5116   "cima 5117   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   2c2 11070   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117   || cdvds 14983  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975  VtxDegcvtxdg 26361  Walkscwlks 26492  Trailsctrls 26587  EulerPathsceupth 27057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-uspgr 26045  df-vtxdg 26362  df-wlks 26495  df-trls 26589  df-eupth 27058

This theorem is referenced by:  eupth2lems  27098