Theorem eupth2lem3lem3 27090

Description: Lemma for eupth2lem3 27096, formerly part of proof of eupth2lem3 27096: If a loop { ( P ` N ) , ( P ` ( N + 1 ) ) } is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-Feb-2021.)

Hypotheses

Ref	Expression
trlsegvdeg.v	|- V = (Vtx ` G )
trlsegvdeg.i	|- I = (iEdg ` G )
trlsegvdeg.f	|- ( ph -> Fun I )
trlsegvdeg.n	|- ( ph -> N e. ( 0..^ ( # ` F ) ) )
trlsegvdeg.u	|- ( ph -> U e. V )
trlsegvdeg.w	|- ( ph -> F (Trails ` G ) P )
trlsegvdeg.vx	|- ( ph -> (Vtx ` X ) = V )
trlsegvdeg.vy	|- ( ph -> (Vtx ` Y ) = V )
trlsegvdeg.vz	|- ( ph -> (Vtx ` Z ) = V )
trlsegvdeg.ix	|- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
trlsegvdeg.iy	|- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
trlsegvdeg.iz	|- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
eupth2lem3.o	|- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
eupth2lem3lem3.e	|- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )

Assertion

Ref	Expression
eupth2lem3lem3	|- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

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

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

Proof of Theorem eupth2lem3lem3

Step	Hyp	Ref	Expression
1		trlsegvdeg.u	. . . . 5 |- ( ph -> U e. V )
2		fveq2 6191	. . . . . . . 8 |- ( x = U -> ( (VtxDeg ` X ) ` x ) = ( (VtxDeg ` X ) ` U ) )
3	2	breq2d 4665	. . . . . . 7 |- ( x = U -> ( 2 || ( (VtxDeg ` X ) ` x ) <-> 2 || ( (VtxDeg ` X ) ` U ) ) )
4	3	notbid 308	. . . . . 6 |- ( x = U -> ( -. 2 || ( (VtxDeg ` X ) ` x ) <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
5	4	elrab3 3364	. . . . 5 |- ( U e. V -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
6	1, 5	syl 17	. . . 4 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
7		eupth2lem3.o	. . . . 5 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
8	7	eleq2d 2687	. . . 4 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
9	6, 8	bitr3d 270	. . 3 |- ( ph -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
10	9	adantr 481	. 2 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
11		2z 11409	. . . . . 6 |- 2 e. ZZ
12	11	a1i 11	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> 2 e. ZZ )
13		trlsegvdeg.v	. . . . . . . 8 |- V = (Vtx ` G )
14		trlsegvdeg.i	. . . . . . . 8 |- I = (iEdg ` G )
15		trlsegvdeg.f	. . . . . . . 8 |- ( ph -> Fun I )
16		trlsegvdeg.n	. . . . . . . 8 |- ( ph -> N e. ( 0..^ ( # ` F ) ) )
17		trlsegvdeg.w	. . . . . . . 8 |- ( ph -> F (Trails ` G ) P )
18		trlsegvdeg.vx	. . . . . . . 8 |- ( ph -> (Vtx ` X ) = V )
19		trlsegvdeg.vy	. . . . . . . 8 |- ( ph -> (Vtx ` Y ) = V )
20		trlsegvdeg.vz	. . . . . . . 8 |- ( ph -> (Vtx ` Z ) = V )
21		trlsegvdeg.ix	. . . . . . . 8 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
22		trlsegvdeg.iy	. . . . . . . 8 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
23		trlsegvdeg.iz	. . . . . . . 8 |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
24	13, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23	eupth2lem3lem1 27088	. . . . . . 7 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )
25	24	nn0zd 11480	. . . . . 6 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. ZZ )
26	25	adantr 481	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( (VtxDeg ` X ) ` U ) e. ZZ )
27	13, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23	eupth2lem3lem2 27089	. . . . . . 7 |- ( ph -> ( (VtxDeg ` Y ) ` U ) e. NN0 )
28	27	nn0zd 11480	. . . . . 6 |- ( ph -> ( (VtxDeg ` Y ) ` U ) e. ZZ )
29	28	adantr 481	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( (VtxDeg ` Y ) ` U ) e. ZZ )
30		iddvds 14995	. . . . . . . 8 |- ( 2 e. ZZ -> 2 || 2 )
31	11, 30	ax-mp 5	. . . . . . 7 |- 2 || 2
32	19	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> (Vtx ` Y ) = V )
33		fvexd 6203	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> ( F ` N ) e. _V )
34	1	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> U e. V )
35	22	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
36		eupth2lem3lem3.e	. . . . . . . . . . . . . 14 |- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
37	36	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
38		ifptru 1023	. . . . . . . . . . . . . 14 |- ( ( P ` N ) = ( P ` ( N + 1 ) ) -> (if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) <-> ( I ` ( F ` N ) ) = { ( P ` N ) } ) )
39	38	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> (if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) <-> ( I ` ( F ` N ) ) = { ( P ` N ) } ) )
40	37, 39	mpbid 222	. . . . . . . . . . . 12 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( I ` ( F ` N ) ) = { ( P ` N ) } )
41		sneq 4187	. . . . . . . . . . . . 13 |- ( ( P ` N ) = U -> { ( P ` N ) } = { U } )
42	41	eqcoms 2630	. . . . . . . . . . . 12 |- ( U = ( P ` N ) -> { ( P ` N ) } = { U } )
43	40, 42	sylan9eq 2676	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> ( I ` ( F ` N ) ) = { U } )
44	43	opeq2d 4409	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> <. ( F ` N ) , ( I ` ( F ` N ) ) >. = <. ( F ` N ) , { U } >. )
45	44	sneqd 4189	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } = { <. ( F ` N ) , { U } >. } )
46	35, 45	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , { U } >. } )
47	32, 33, 34, 46	1loopgrvd2 26399	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> ( (VtxDeg ` Y ) ` U ) = 2 )
48	31, 47	syl5breqr 4691	. . . . . 6 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> 2 || ( (VtxDeg ` Y ) ` U ) )
49		dvds0 14997	. . . . . . . 8 |- ( 2 e. ZZ -> 2 || 0 )
50	11, 49	ax-mp 5	. . . . . . 7 |- 2 || 0
51	19	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> (Vtx ` Y ) = V )
52		fvexd 6203	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> ( F ` N ) e. _V )
53	13, 14, 15, 16, 1, 17	trlsegvdeglem1 27080	. . . . . . . . . 10 |- ( ph -> ( ( P ` N ) e. V /\ ( P ` ( N + 1 ) ) e. V ) )
54	53	simpld 475	. . . . . . . . 9 |- ( ph -> ( P ` N ) e. V )
55	54	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> ( P ` N ) e. V )
56	22	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
57	40	opeq2d 4409	. . . . . . . . . . 11 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> <. ( F ` N ) , ( I ` ( F ` N ) ) >. = <. ( F ` N ) , { ( P ` N ) } >. )
58	57	sneqd 4189	. . . . . . . . . 10 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } = { <. ( F ` N ) , { ( P ` N ) } >. } )
59	56, 58	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , { ( P ` N ) } >. } )
60	59	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , { ( P ` N ) } >. } )
61	1	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> U e. V )
62	61	anim1i 592	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> ( U e. V /\ U =/= ( P ` N ) ) )
63		eldifsn 4317	. . . . . . . . 9 |- ( U e. ( V \ { ( P ` N ) } ) <-> ( U e. V /\ U =/= ( P ` N ) ) )
64	62, 63	sylibr 224	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> U e. ( V \ { ( P ` N ) } ) )
65	51, 52, 55, 60, 64	1loopgrvd0 26400	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> ( (VtxDeg ` Y ) ` U ) = 0 )
66	50, 65	syl5breqr 4691	. . . . . 6 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> 2 || ( (VtxDeg ` Y ) ` U ) )
67	48, 66	pm2.61dane 2881	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> 2 || ( (VtxDeg ` Y ) ` U ) )
68		dvdsadd2b 15028	. . . . 5 |- ( ( 2 e. ZZ /\ ( (VtxDeg ` X ) ` U ) e. ZZ /\ ( ( (VtxDeg ` Y ) ` U ) e. ZZ /\ 2 || ( (VtxDeg ` Y ) ` U ) ) ) -> ( 2 || ( (VtxDeg ` X ) ` U ) <-> 2 || ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) ) )
69	12, 26, 29, 67, 68	syl112anc 1330	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( 2 || ( (VtxDeg ` X ) ` U ) <-> 2 || ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) ) )
70	27	nn0cnd 11353	. . . . . . 7 |- ( ph -> ( (VtxDeg ` Y ) ` U ) e. CC )
71	24	nn0cnd 11353	. . . . . . 7 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. CC )
72	70, 71	addcomd 10238	. . . . . 6 |- ( ph -> ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )
73	72	breq2d 4665	. . . . 5 |- ( ph -> ( 2 || ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
74	73	adantr 481	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( 2 || ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
75	69, 74	bitrd 268	. . 3 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( 2 || ( (VtxDeg ` X ) ` U ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
76	75	notbid 308	. 2 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
77		simpr 477	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( P ` N ) = ( P ` ( N + 1 ) ) )
78	77	eqeq2d 2632	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( ( P ` 0 ) = ( P ` N ) <-> ( P ` 0 ) = ( P ` ( N + 1 ) ) ) )
79	77	preq2d 4275	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> { ( P ` 0 ) , ( P ` N ) } = { ( P ` 0 ) , ( P ` ( N + 1 ) ) } )
80	78, 79	ifbieq2d 4111	. . 3 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) = if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) )
81	80	eleq2d 2687	. 2 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
82	10, 76, 81	3bitr3d 298	1 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384  if-wif 1012   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)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   2c2 11070   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117   || cdvds 14983  Vtxcvtx 25874  iEdgciedg 25875  VtxDegcvtxdg 26361  Trailsctrls 26587

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

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-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-xadd 11947  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-dvds 14984  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-uspgr 26045  df-vtxdg 26362  df-wlks 26495  df-trls 26589

This theorem is referenced by:  eupth2lem3lem7  27094