Theorem eupth2lem3lem4fi 16455

Description: Lemma for eupth2lem3fi 16458. 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 16370	. . . . . . . . . . . . . 14 |- ( F (Trails ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )
4		f1f 5572	. . . . . . . . . . . . . 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 5812	. . . . . . . . . . . 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 16442	. . . . . . . . . . . . 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 2425	. . . . . . . . . . . . . 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 2413	. . . . . . . . . . . . . . . 16 |- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> -. ( P ` N ) = ( P ` ( N + 1 ) ) )
27		ifpfal 999	. . . . . . . . . . . . . . . 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 3767	. . . . . . . . . . . . . . . 16 |- ( ( P ` N ) = U -> { ( P ` N ) , ( P ` ( N + 1 ) ) } = { U , ( P ` ( N + 1 ) ) } )
31	30	sseq1d 3266	. . . . . . . . . . . . . . 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 2312	. . . . . . . . . . . . . 14 |- ( ph -> U e. (Vtx ` Y ) )
41		df-vtx 15996	. . . . . . . . . . . . . . 15 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
42	41	mptrcl 5759	. . . . . . . . . . . . . 14 |- ( U e. (Vtx ` Y ) -> Y e. _V )
43	40, 42	syl 14	. . . . . . . . . . . . 13 |- ( ph -> Y e. _V )
44	12	funfnd 5382	. . . . . . . . . . . . . . 15 |- ( ph -> I Fn dom I )
45		fnressn 5869	. . . . . . . . . . . . . . 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 2268	. . . . . . . . . . . . 13 |- ( ph -> (iEdg ` Y ) = ( I |` { ( F ` N ) } ) )
48		eupth2lem3lem4fi.g	. . . . . . . . . . . . 13 |- ( ph -> G e. UMGraph )
49	11, 2, 43, 36, 47, 48	umgrspan 16262	. . . . . . . . . . . 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 16291	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( (VtxDeg ` Y ) ` U ) = 1 )
52	51	oveq2d 6065	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + 1 ) )
53	52	breq2d 4120	. . . . . . . 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 16094	. . . . . . . . . . . . . . . 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 16450	. . . . . . . . . . . . . 14 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )
62	61	nn0zd 9694	. . . . . . . . . . . . 13 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. ZZ )
63		2nn 9395	. . . . . . . . . . . . . 14 |- 2 e. NN
64	63	a1i 9	. . . . . . . . . . . . 13 |- ( ph -> 2 e. NN )
65		1lt2 9403	. . . . . . . . . . . . . 14 |- 1 < 2
66	65	a1i 9	. . . . . . . . . . . . 13 |- ( ph -> 1 < 2 )
67		ndvdsp1 12611	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 9599	. . . . . . . . . . . . . 14 |- 1 e. ZZ
71		n2dvds1 12591	. . . . . . . . . . . . . 14 |- -. 2 || 1
72		opoe 12574	. . . . . . . . . . . . . 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 5669	. . . . . . . . . . . . . 14 |- ( x = U -> ( (VtxDeg ` X ) ` x ) = ( (VtxDeg ` X ) ` U ) )
78	77	breq2d 4120	. . . . . . . . . . . . 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 2973	. . . . . . . . . . 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 2302	. . . . . . . . . 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 16368	. . . . . . . . . . 11 |- ( F (Trails ` G ) P -> F (Walks ` G ) P )
88	11	wlkp 16316	. . . . . . . . . . 11 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
89	1, 87, 88	3syl 17	. . . . . . . . . 10 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
90		elfzofz 10493	. . . . . . . . . . 11 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. ( 0 ... (♯ ` F ) ) )
91	6, 90	syl 14	. . . . . . . . . 10 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
92	89, 91	ffvelcdmd 5812	. . . . . . . . 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 10359	. . . . . . . . . . . 12 |- ( N e. ( 0 ... (♯ ` F ) ) -> (♯ ` F ) e. ( ZZ>= ` 0 ) )
95		eluzfz1 10361	. . . . . . . . . . . 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 5812	. . . . . . . . . 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 7123	. . . . . . . . 9 |- ( ( V e. Fin /\ ( P ` 0 ) e. V /\ ( P ` N ) e. V ) -> DECID ( P ` 0 ) = ( P ` N ) )
100	39, 98, 93, 99	syl3anc 1274	. . . . . . . 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 16441	. . . . . . 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 2235	. . . 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 2426	. . . . . . . . . . . . . 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 3768	. . . . . . . . . . . . . . . 16 |- ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , ( P ` ( N + 1 ) ) } = { ( P ` N ) , U } )
118	117	sseq1d 3266	. . . . . . . . . . . . . . 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 16292	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( (VtxDeg ` Y ) ` U ) = 1 )
127	126	oveq2d 6065	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + 1 ) )
128	127	breq2d 4120	. . . . . . . 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 3511	. . . . . . . . . . . 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 3628	. . . . . . . . . . . . 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 2302	. . . . . . . . . . . 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 682	. . . . . . . . . . 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 742	. . . . . . . . . 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 843	. . . . . . . . . 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 7123	. . . . . . . . . . . . 13 |- ( ( V e. Fin /\ U e. V /\ ( P ` 0 ) e. V ) -> DECID U = ( P ` 0 ) )
140	38, 9, 97, 139	syl3anc 1274	. . . . . . . . . . . 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 7123	. . . . . . . . . . . . 13 |- ( ( V e. Fin /\ U e. V /\ ( P ` ( N + 1 ) ) e. V ) -> DECID U = ( P ` ( N + 1 ) ) )
143	38, 9, 14, 142	syl3anc 1274	. . . . . . . . . . . 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 944	. . . . . . . . . . 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 3631	. . . . . . . . . . . . 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 2302	. . . . . . . . . . . 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 3708	. . . . . . . . . . . . . 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 846	. . . . . . . . . 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 7123	. . . . . . . . . . 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 1274	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> DECID ( P ` 0 ) = ( P ` ( N + 1 ) ) )
160		exmiddc 844	. . . . . . . . . 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 806	. . . . . . . 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 2498	. . . . . . . . . . 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 16441	. . . . . . . . . 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 884	. . . . . . . 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 2235	. . . 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 724	. . 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 1227	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 716  DECID wdc 842  if-wif 986   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   {crab 2524   _Vcvv 2812   C_ wss 3210   (/)c0 3507   ifcif 3619   ~Pcpw 3668   {csn 3688   {cpr 3689   <.cop 3691   class class class wbr 4108   X. cxp 4746   dom cdm 4748   |` cres 4750   "cima 4751   Fun wfun 5345   Fn wfn 5346   -->wf 5347   -1-1->wf1 5348   ` cfv 5351  (class class class)co 6049   1stc1st 6331   Fincfn 6974   0cc0 8123   1c1 8124   + caddc 8126   < clt 8304   NNcn 9233   2c2 9284   ZZcz 9573   ZZ>=cuz 9849   ...cfz 10338  ..^cfzo 10472  ♯chash 11133   || cdvds 12466   Basecbs 13201  Vtxcvtx 15994  iEdgciedg 15995  UPGraphcupgr 16073  UMGraphcumgr 16074  VtxDegcvtxdg 16268  Walkscwlks 16299  Trailsctrls 16362

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-frec 6621  df-1o 6646  df-2o 6647  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-upgren 16075  df-umgren 16076  df-subgr 16236  df-vtxdg 16269  df-wlks 16300  df-trls 16363

This theorem is referenced by:  eupth2lem3lem7fi  16456