Theorem eupth2lem3lem4fi 16327

Description: Lemma for eupth2lem3fi 16330. If an edge (not a loop) is added to a trail, the degree of the end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-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 ) ) ) )
eupth2lem3lem4fi.g	|- ( ph -> G e. UMGraph )
eupth2lem3lem4fi.v	|- ( ph -> V e. Fin )
eupth2lem3lem4fi.o	|- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
eupth2lem3lem4fi.e	|- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
eupth2lem3lem4.i	|- ( ph -> ( I ` ( F ` N ) ) e. ~P V )

Assertion

Ref	Expression
eupth2lem3lem4fi	|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U = ( P ` N ) \/ U = ( 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 eupth2lem3lem4fi

Step	Hyp	Ref	Expression
1		trlsegvdeg.w	. . . . . . . . . . . . . 14 |- ( ph -> F (Trails ` G ) P )
2		trlsegvdeg.i	. . . . . . . . . . . . . . 15 |- I = (iEdg ` G )
3	2	trlf1 16242	. . . . . . . . . . . . . 14 |- ( F (Trails ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )
4		f1f 5542	. . . . . . . . . . . . . 14 |- ( F : ( 0..^ (♯ ` F ) ) -1-1-> dom I -> F : ( 0..^ (♯ ` F ) ) --> dom I )
5	1, 3, 4	3syl 17	. . . . . . . . . . . . 13 |- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )
6		trlsegvdeg.n	. . . . . . . . . . . . 13 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
7	5, 6	ffvelcdmd 5783	. . . . . . . . . . . 12 |- ( ph -> ( F ` N ) e. dom I )
8	7	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( F ` N ) e. dom I )
9		trlsegvdeg.u	. . . . . . . . . . . 12 |- ( ph -> U e. V )
10	9	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> U e. V )
11		trlsegvdeg.v	. . . . . . . . . . . . . 14 |- V = (Vtx ` G )
12		trlsegvdeg.f	. . . . . . . . . . . . . 14 |- ( ph -> Fun I )
13	11, 2, 12, 6, 9, 1	trlsegvdeglem1 16314	. . . . . . . . . . . . 13 |- ( ph -> ( ( P ` N ) e. V /\ ( P ` ( N + 1 ) ) e. V ) )
14	13	simprd 114	. . . . . . . . . . . 12 |- ( ph -> ( P ` ( N + 1 ) ) e. V )
15	14	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( P ` ( N + 1 ) ) e. V )
16		neeq1 2415	. . . . . . . . . . . . . 14 |- ( ( P ` N ) = U -> ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> U =/= ( P ` ( N + 1 ) ) ) )
17	16	biimpcd 159	. . . . . . . . . . . . 13 |- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( ( P ` N ) = U -> U =/= ( P ` ( N + 1 ) ) ) )
18	17	adantl 277	. . . . . . . . . . . 12 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` N ) = U -> U =/= ( P ` ( N + 1 ) ) ) )
19	18	imp 124	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> U =/= ( P ` ( N + 1 ) ) )
20		eupth2lem3lem4.i	. . . . . . . . . . . 12 |- ( ph -> ( I ` ( F ` N ) ) e. ~P V )
21	20	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( I ` ( F ` N ) ) e. ~P V )
22		trlsegvdeg.iy	. . . . . . . . . . . 12 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
23	22	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
24		eupth2lem3lem4fi.e	. . . . . . . . . . . . . 14 |- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
25	24	adantr 276	. . . . . . . . . . . . 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 ) ) ) )
26		df-ne 2403	. . . . . . . . . . . . . . . 16 |- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> -. ( P ` N ) = ( P ` ( N + 1 ) ) )
27		ifpfal 998	. . . . . . . . . . . . . . . 16 |- ( -. ( 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 ) ) ) <-> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
28	26, 27	sylbi 121	. . . . . . . . . . . . . . 15 |- ( ( 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 ) ) ) <-> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
29	28	adantl 277	. . . . . . . . . . . . . 14 |- ( ( 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 ) ) ) <-> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
30		preq1 3748	. . . . . . . . . . . . . . . 16 |- ( ( P ` N ) = U -> { ( P ` N ) , ( P ` ( N + 1 ) ) } = { U , ( P ` ( N + 1 ) ) } )
31	30	sseq1d 3256	. . . . . . . . . . . . . . 15 |- ( ( P ` N ) = U -> ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) <-> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
32	31	biimpcd 159	. . . . . . . . . . . . . 14 |- ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) -> ( ( P ` N ) = U -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
33	29, 32	biimtrdi 163	. . . . . . . . . . . . 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 ) ) ) -> ( ( P ` N ) = U -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) )
34	25, 33	mpd 13	. . . . . . . . . . . 12 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` N ) = U -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
35	34	imp 124	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) )
36		trlsegvdeg.vy	. . . . . . . . . . . 12 |- ( ph -> (Vtx ` Y ) = V )
37	36	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> (Vtx ` Y ) = V )
38		eupth2lem3lem4fi.v	. . . . . . . . . . . 12 |- ( ph -> V e. Fin )
39	38	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> V e. Fin )
40	9, 36	eleqtrrd 2311	. . . . . . . . . . . . . 14 |- ( ph -> U e. (Vtx ` Y ) )
41		df-vtx 15868	. . . . . . . . . . . . . . 15 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
42	41	mptrcl 5729	. . . . . . . . . . . . . 14 |- ( U e. (Vtx ` Y ) -> Y e. _V )
43	40, 42	syl 14	. . . . . . . . . . . . 13 |- ( ph -> Y e. _V )
44	12	funfnd 5357	. . . . . . . . . . . . . . 15 |- ( ph -> I Fn dom I )
45		fnressn 5840	. . . . . . . . . . . . . . 15 |- ( ( I Fn dom I /\ ( F ` N ) e. dom I ) -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
46	44, 7, 45	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ph -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
47	22, 46	eqtr4d 2267	. . . . . . . . . . . . 13 |- ( ph -> (iEdg ` Y ) = ( I |` { ( F ` N ) } ) )
48		eupth2lem3lem4fi.g	. . . . . . . . . . . . 13 |- ( ph -> G e. UMGraph )
49	11, 2, 43, 36, 47, 48	umgrspan 16134	. . . . . . . . . . . 12 |- ( ph -> Y e. UMGraph )
50	49	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> Y e. UMGraph )
51	8, 10, 15, 19, 21, 23, 35, 37, 39, 50	1hegrvtxdg1fi 16163	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( (VtxDeg ` Y ) ` U ) = 1 )
52	51	oveq2d 6034	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + 1 ) )
53	52	breq2d 4100	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
54	53	notbid 673	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
55		trlsegvdeg.vx	. . . . . . . . . . . . . . 15 |- ( ph -> (Vtx ` X ) = V )
56		trlsegvdeg.vz	. . . . . . . . . . . . . . 15 |- ( ph -> (Vtx ` Z ) = V )
57		trlsegvdeg.ix	. . . . . . . . . . . . . . 15 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
58		trlsegvdeg.iz	. . . . . . . . . . . . . . 15 |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
59		umgrupgr 15966	. . . . . . . . . . . . . . . 16 |- ( G e. UMGraph -> G e. UPGraph )
60	48, 59	syl 14	. . . . . . . . . . . . . . 15 |- ( ph -> G e. UPGraph )
61	11, 2, 12, 6, 9, 1, 55, 36, 56, 57, 22, 58, 60, 38	eupth2lem3lem1fi 16322	. . . . . . . . . . . . . 14 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )
62	61	nn0zd 9600	. . . . . . . . . . . . 13 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. ZZ )
63		2nn 9305	. . . . . . . . . . . . . 14 |- 2 e. NN
64	63	a1i 9	. . . . . . . . . . . . 13 |- ( ph -> 2 e. NN )
65		1lt2 9313	. . . . . . . . . . . . . 14 |- 1 < 2
66	65	a1i 9	. . . . . . . . . . . . 13 |- ( ph -> 1 < 2 )
67		ndvdsp1 12495	. . . . . . . . . . . . 13 |- ( ( ( (VtxDeg ` X ) ` U ) e. ZZ /\ 2 e. NN /\ 1 < 2 ) -> ( 2 || ( (VtxDeg ` X ) ` U ) -> -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
68	62, 64, 66, 67	syl3anc 1273	. . . . . . . . . . . 12 |- ( ph -> ( 2 || ( (VtxDeg ` X ) ` U ) -> -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
69	68	con2d 629	. . . . . . . . . . 11 |- ( ph -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) -> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
70		1z 9505	. . . . . . . . . . . . . 14 |- 1 e. ZZ
71		n2dvds1 12475	. . . . . . . . . . . . . 14 |- -. 2 || 1
72		opoe 12458	. . . . . . . . . . . . . 14 |- ( ( ( ( (VtxDeg ` X ) ` U ) e. ZZ /\ -. 2 || ( (VtxDeg ` X ) ` U ) ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) )
73	70, 71, 72	mpanr12 439	. . . . . . . . . . . . 13 |- ( ( ( (VtxDeg ` X ) ` U ) e. ZZ /\ -. 2 || ( (VtxDeg ` X ) ` U ) ) -> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) )
74	73	ex 115	. . . . . . . . . . . 12 |- ( ( (VtxDeg ` X ) ` U ) e. ZZ -> ( -. 2 || ( (VtxDeg ` X ) ` U ) -> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
75	62, 74	syl 14	. . . . . . . . . . 11 |- ( ph -> ( -. 2 || ( (VtxDeg ` X ) ` U ) -> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
76	69, 75	impbid 129	. . . . . . . . . 10 |- ( ph -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
77		fveq2 5639	. . . . . . . . . . . . . 14 |- ( x = U -> ( (VtxDeg ` X ) ` x ) = ( (VtxDeg ` X ) ` U ) )
78	77	breq2d 4100	. . . . . . . . . . . . 13 |- ( x = U -> ( 2 || ( (VtxDeg ` X ) ` x ) <-> 2 || ( (VtxDeg ` X ) ` U ) ) )
79	78	notbid 673	. . . . . . . . . . . 12 |- ( x = U -> ( -. 2 || ( (VtxDeg ` X ) ` x ) <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
80	79	elrab3 2963	. . . . . . . . . . 11 |- ( U e. V -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
81	9, 80	syl 14	. . . . . . . . . 10 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
82		eupth2lem3lem4fi.o	. . . . . . . . . . 11 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
83	82	eleq2d 2301	. . . . . . . . . 10 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
84	76, 81, 83	3bitr2d 216	. . . . . . . . 9 |- ( ph -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
85	84	notbid 673	. . . . . . . 8 |- ( ph -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) <-> -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
86	85	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) <-> -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
87		trliswlk 16240	. . . . . . . . . . 11 |- ( F (Trails ` G ) P -> F (Walks ` G ) P )
88	11	wlkp 16188	. . . . . . . . . . 11 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
89	1, 87, 88	3syl 17	. . . . . . . . . 10 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
90		elfzofz 10398	. . . . . . . . . . 11 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. ( 0 ... (♯ ` F ) ) )
91	6, 90	syl 14	. . . . . . . . . 10 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
92	89, 91	ffvelcdmd 5783	. . . . . . . . 9 |- ( ph -> ( P ` N ) e. V )
93	92	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( P ` N ) e. V )
94		elfzuz2 10264	. . . . . . . . . . . 12 |- ( N e. ( 0 ... (♯ ` F ) ) -> (♯ ` F ) e. ( ZZ>= ` 0 ) )
95		eluzfz1 10266	. . . . . . . . . . . 12 |- ( (♯ ` F ) e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... (♯ ` F ) ) )
96	91, 94, 95	3syl 17	. . . . . . . . . . 11 |- ( ph -> 0 e. ( 0 ... (♯ ` F ) ) )
97	89, 96	ffvelcdmd 5783	. . . . . . . . . 10 |- ( ph -> ( P ` 0 ) e. V )
98	97	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( P ` 0 ) e. V )
99		fidceq 7056	. . . . . . . . 9 |- ( ( V e. Fin /\ ( P ` 0 ) e. V /\ ( P ` N ) e. V ) -> DECID ( P ` 0 ) = ( P ` N ) )
100	39, 98, 93, 99	syl3anc 1273	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> DECID ( P ` 0 ) = ( P ` N ) )
101		simplr 529	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( P ` N ) =/= ( P ` ( N + 1 ) ) )
102		simpr 110	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( P ` N ) = U )
103	93, 100, 101, 102	eupth2lem2dc 16313	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
104	54, 86, 103	3bitrd 214	. . . . . 6 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
105	104	expcom 116	. . . . 5 |- ( ( P ` N ) = U -> ( ( 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 ) ) } ) ) ) )
106	105	eqcoms 2234	. . . 4 |- ( U = ( P ` N ) -> ( ( 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 ) ) } ) ) ) )
107	7	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( F ` N ) e. dom I )
108	13	simpld 112	. . . . . . . . . . . 12 |- ( ph -> ( P ` N ) e. V )
109	108	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` N ) e. V )
110	9	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> U e. V )
111		neeq2 2416	. . . . . . . . . . . . . 14 |- ( ( P ` ( N + 1 ) ) = U -> ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> ( P ` N ) =/= U ) )
112	111	biimpcd 159	. . . . . . . . . . . . 13 |- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( ( P ` ( N + 1 ) ) = U -> ( P ` N ) =/= U ) )
113	112	adantl 277	. . . . . . . . . . . 12 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` ( N + 1 ) ) = U -> ( P ` N ) =/= U ) )
114	113	imp 124	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` N ) =/= U )
115	20	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( I ` ( F ` N ) ) e. ~P V )
116	22	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
117		preq2 3749	. . . . . . . . . . . . . . . 16 |- ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , ( P ` ( N + 1 ) ) } = { ( P ` N ) , U } )
118	117	sseq1d 3256	. . . . . . . . . . . . . . 15 |- ( ( P ` ( N + 1 ) ) = U -> ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) <-> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) )
119	118	biimpcd 159	. . . . . . . . . . . . . 14 |- ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) -> ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) )
120	29, 119	biimtrdi 163	. . . . . . . . . . . . 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 ) ) ) -> ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) ) )
121	25, 120	mpd 13	. . . . . . . . . . . 12 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) )
122	121	imp 124	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) )
123	36	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> (Vtx ` Y ) = V )
124	38	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> V e. Fin )
125	49	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> Y e. UMGraph )
126	107, 109, 110, 114, 115, 116, 122, 123, 124, 125	1hegrvtxdg1rfi 16164	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( (VtxDeg ` Y ) ` U ) = 1 )
127	126	oveq2d 6034	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + 1 ) )
128	127	breq2d 4100	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
129	128	notbid 673	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
130	85	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) <-> -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
131		noel 3498	. . . . . . . . . . . 12 |- -. U e. (/)
132		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> ( P ` 0 ) = ( P ` ( N + 1 ) ) )
133	132	iftrued 3612	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) = (/) )
134	133	eleq2d 2301	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> ( U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> U e. (/) ) )
135	131, 134	mtbiri 681	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> -. U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) )
136	135	olcd 741	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> ( U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) \/ -. U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
137		df-dc 842	. . . . . . . . . 10 |- (DECID U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> ( U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) \/ -. U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
138	136, 137	sylibr 134	. . . . . . . . 9 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> DECID U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) )
139		fidceq 7056	. . . . . . . . . . . . 13 |- ( ( V e. Fin /\ U e. V /\ ( P ` 0 ) e. V ) -> DECID U = ( P ` 0 ) )
140	38, 9, 97, 139	syl3anc 1273	. . . . . . . . . . . 12 |- ( ph -> DECID U = ( P ` 0 ) )
141	140	ad3antrrr 492	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> DECID U = ( P ` 0 ) )
142		fidceq 7056	. . . . . . . . . . . . 13 |- ( ( V e. Fin /\ U e. V /\ ( P ` ( N + 1 ) ) e. V ) -> DECID U = ( P ` ( N + 1 ) ) )
143	38, 9, 14, 142	syl3anc 1273	. . . . . . . . . . . 12 |- ( ph -> DECID U = ( P ` ( N + 1 ) ) )
144	143	ad3antrrr 492	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> DECID U = ( P ` ( N + 1 ) ) )
145		dcor 943	. . . . . . . . . . 11 |- (DECID U = ( P ` 0 ) -> (DECID U = ( P ` ( N + 1 ) ) -> DECID ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) )
146	141, 144, 145	sylc 62	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> DECID ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) )
147		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> -. ( P ` 0 ) = ( P ` ( N + 1 ) ) )
148	147	iffalsed 3615	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) = { ( P ` 0 ) , ( P ` ( N + 1 ) ) } )
149	148	eleq2d 2301	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> ( U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> U e. { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) )
150		elprg 3689	. . . . . . . . . . . . . 14 |- ( U e. V -> ( U e. { ( P ` 0 ) , ( P ` ( N + 1 ) ) } <-> ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) )
151	9, 150	syl 14	. . . . . . . . . . . . 13 |- ( ph -> ( U e. { ( P ` 0 ) , ( P ` ( N + 1 ) ) } <-> ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) )
152	151	ad3antrrr 492	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> ( U e. { ( P ` 0 ) , ( P ` ( N + 1 ) ) } <-> ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) )
153	149, 152	bitrd 188	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> ( U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) )
154	153	dcbid 845	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> (DECID U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> DECID ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) )
155	146, 154	mpbird 167	. . . . . . . . 9 |- ( ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) /\ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) -> DECID U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) )
156	97	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` 0 ) e. V )
157	14	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` ( N + 1 ) ) e. V )
158		fidceq 7056	. . . . . . . . . . 11 |- ( ( V e. Fin /\ ( P ` 0 ) e. V /\ ( P ` ( N + 1 ) ) e. V ) -> DECID ( P ` 0 ) = ( P ` ( N + 1 ) ) )
159	124, 156, 157, 158	syl3anc 1273	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> DECID ( P ` 0 ) = ( P ` ( N + 1 ) ) )
160		exmiddc 843	. . . . . . . . . 10 |- (DECID ( P ` 0 ) = ( P ` ( N + 1 ) ) -> ( ( P ` 0 ) = ( P ` ( N + 1 ) ) \/ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) )
161	159, 160	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( ( P ` 0 ) = ( P ` ( N + 1 ) ) \/ -. ( P ` 0 ) = ( P ` ( N + 1 ) ) ) )
162	138, 155, 161	mpjaodan 805	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> DECID U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) )
163		simplr 529	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` N ) =/= ( P ` ( N + 1 ) ) )
164	163	necomd 2488	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` ( N + 1 ) ) =/= ( P ` N ) )
165		simpr 110	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` ( N + 1 ) ) = U )
166	157, 159, 164, 165	eupth2lem2dc 16313	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
167	166	a1d 22	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> (DECID U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) ) )
168	167	con1biddc 883	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> (DECID U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( 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 ) ) } ) ) ) )
169	162, 168	mpd 13	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
170	129, 130, 169	3bitrd 214	. . . . . 6 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
171	170	expcom 116	. . . . 5 |- ( ( P ` ( N + 1 ) ) = U -> ( ( 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 ) ) } ) ) ) )
172	171	eqcoms 2234	. . . 4 |- ( U = ( P ` ( N + 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 ) ) } ) ) ) )
173	106, 172	jaoi 723	. . 3 |- ( ( U = ( P ` N ) \/ U = ( P ` ( N + 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 ) ) } ) ) ) )
174	173	com12 30	. 2 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) )
175	174	3impia 1226	1 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U = ( P ` N ) \/ U = ( 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   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841  if-wif 985   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   {crab 2514   _Vcvv 2802   C_ wss 3200   (/)c0 3494   ifcif 3605   ~Pcpw 3652   {csn 3669   {cpr 3670   <.cop 3672   class class class wbr 4088   X. cxp 4723   dom cdm 4725   |` cres 4727   "cima 4728   Fun wfun 5320   Fn wfn 5321   -->wf 5322   -1-1->wf1 5323   ` cfv 5326  (class class class)co 6018   1stc1st 6301   Fincfn 6909   0cc0 8032   1c1 8033   + caddc 8035   < clt 8214   NNcn 9143   2c2 9194   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243  ..^cfzo 10377  ♯chash 11038   || cdvds 12350   Basecbs 13084  Vtxcvtx 15866  iEdgciedg 15867  UPGraphcupgr 15945  UMGraphcumgr 15946  VtxDegcvtxdg 16140  Walkscwlks 16171  Trailsctrls 16234

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-frec 6557  df-1o 6582  df-2o 6583  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-upgren 15947  df-umgren 15948  df-subgr 16108  df-vtxdg 16141  df-wlks 16172  df-trls 16235

This theorem is referenced by:  eupth2lem3lem7fi  16328