Theorem eupth2lem3lem7fi 16486

Description: Lemma for eupth2lem3fi 16488: Combining trlsegvdegfi 16479, eupth2lem3lem3fi 16482, eupth2lem3lem4fi 16485 and eupth2lem3lem6fi 16483. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-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 ) ) ) )
eupth2lem3lem7fi.g	|- ( ph -> G e. UMGraph )
eupth2lem3lem7fi.v	|- ( ph -> V e. Fin )
eupth2lem3lem7fi.o	|- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
eupth2lem3lem7fi.e	|- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )

Assertion

Ref	Expression
eupth2lem3lem7fi	|- ( ph -> ( -. 2 || ( (VtxDeg ` Z ) ` 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 eupth2lem3lem7fi

Step	Hyp	Ref	Expression
1		trlsegvdeg.v	. . . . 5 |- V = (Vtx ` G )
2		trlsegvdeg.i	. . . . 5 |- I = (iEdg ` G )
3		trlsegvdeg.f	. . . . 5 |- ( ph -> Fun I )
4		trlsegvdeg.n	. . . . 5 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
5		trlsegvdeg.u	. . . . 5 |- ( ph -> U e. V )
6		trlsegvdeg.w	. . . . 5 |- ( ph -> F (Trails ` G ) P )
7		trlsegvdeg.vx	. . . . 5 |- ( ph -> (Vtx ` X ) = V )
8		trlsegvdeg.vy	. . . . 5 |- ( ph -> (Vtx ` Y ) = V )
9		trlsegvdeg.vz	. . . . 5 |- ( ph -> (Vtx ` Z ) = V )
10		trlsegvdeg.ix	. . . . 5 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
11		trlsegvdeg.iy	. . . . 5 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
12		trlsegvdeg.iz	. . . . 5 |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
13		eupth2lem3lem7fi.g	. . . . . 6 |- ( ph -> G e. UMGraph )
14		umgrupgr 16124	. . . . . 6 |- ( G e. UMGraph -> G e. UPGraph )
15	13, 14	syl 14	. . . . 5 |- ( ph -> G e. UPGraph )
16		eupth2lem3lem7fi.v	. . . . 5 |- ( ph -> V e. Fin )
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16	trlsegvdegfi 16479	. . . 4 |- ( ph -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )
18	17	breq2d 4123	. . 3 |- ( ph -> ( 2 || ( (VtxDeg ` Z ) ` U ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
19	18	notbid 673	. 2 |- ( ph -> ( -. 2 || ( (VtxDeg ` Z ) ` U ) <-> -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
20		eupth2lem3lem7fi.o	. . . 4 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
21		eupth2lem3lem7fi.e	. . . . 5 |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
22		trliswlk 16398	. . . . . . . 8 |- ( F (Trails ` G ) P -> F (Walks ` G ) P )
23	1	wlkp 16346	. . . . . . . 8 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
24	6, 22, 23	3syl 17	. . . . . . 7 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
25		elfzofz 10501	. . . . . . . 8 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. ( 0 ... (♯ ` F ) ) )
26	4, 25	syl 14	. . . . . . 7 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
27	24, 26	ffvelcdmd 5815	. . . . . 6 |- ( ph -> ( P ` N ) e. V )
28		fzofzp1 10576	. . . . . . . 8 |- ( N e. ( 0..^ (♯ ` F ) ) -> ( N + 1 ) e. ( 0 ... (♯ ` F ) ) )
29	4, 28	syl 14	. . . . . . 7 |- ( ph -> ( N + 1 ) e. ( 0 ... (♯ ` F ) ) )
30	24, 29	ffvelcdmd 5815	. . . . . 6 |- ( ph -> ( P ` ( N + 1 ) ) e. V )
31		fidceq 7126	. . . . . 6 |- ( ( V e. Fin /\ ( P ` N ) e. V /\ ( P ` ( N + 1 ) ) e. V ) -> DECID ( P ` N ) = ( P ` ( N + 1 ) ) )
32	16, 27, 30, 31	syl3anc 1274	. . . . 5 |- ( ph -> DECID ( P ` N ) = ( P ` ( N + 1 ) ) )
33		ifpprsnssdc 3801	. . . . 5 |- ( ( ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } /\ DECID ( 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 ) ) ) )
34	21, 32, 33	syl2anc 411	. . . 4 |- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 20, 34	eupth2lem3lem3fi 16482	. . 3 |- ( ( 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 ) ) } ) ) )
36	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 21	eupth2lem3lem5 16484	. . . . . 6 |- ( ph -> ( I ` ( F ` N ) ) e. ~P V )
37	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 20, 34, 36	eupth2lem3lem4fi 16485	. . . . 5 |- ( ( 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 ) ) } ) ) )
38	37	3expa 1230	. . . 4 |- ( ( ( 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 ) ) } ) ) )
39		neanior 2501	. . . . 5 |- ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) <-> -. ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) )
40	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 20, 21	eupth2lem3lem6fi 16483	. . . . . 6 |- ( ( 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 ) ) } ) ) )
41	40	3expa 1230	. . . . 5 |- ( ( ( 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 ) ) } ) ) )
42	39, 41	sylan2br 288	. . . 4 |- ( ( ( 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 ) ) } ) ) )
43		fidceq 7126	. . . . . . . 8 |- ( ( V e. Fin /\ U e. V /\ ( P ` N ) e. V ) -> DECID U = ( P ` N ) )
44	16, 5, 27, 43	syl3anc 1274	. . . . . . 7 |- ( ph -> DECID U = ( P ` N ) )
45	44	adantr 276	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> DECID U = ( P ` N ) )
46		fidceq 7126	. . . . . . . 8 |- ( ( V e. Fin /\ U e. V /\ ( P ` ( N + 1 ) ) e. V ) -> DECID U = ( P ` ( N + 1 ) ) )
47	16, 5, 30, 46	syl3anc 1274	. . . . . . 7 |- ( ph -> DECID U = ( P ` ( N + 1 ) ) )
48	47	adantr 276	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> DECID U = ( P ` ( N + 1 ) ) )
49		dcor 944	. . . . . 6 |- (DECID U = ( P ` N ) -> (DECID U = ( P ` ( N + 1 ) ) -> DECID ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) ) )
50	45, 48, 49	sylc 62	. . . . 5 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> DECID ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) )
51		exmiddc 844	. . . . 5 |- (DECID ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) -> ( ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) \/ -. ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) ) )
52	50, 51	syl 14	. . . 4 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) \/ -. ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) ) )
53	38, 42, 52	mpjaodan 806	. . 3 |- ( ( 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 ) ) } ) ) )
54		dcne 2425	. . . 4 |- (DECID ( P ` N ) = ( P ` ( N + 1 ) ) <-> ( ( P ` N ) = ( P ` ( N + 1 ) ) \/ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) )
55	32, 54	sylib 122	. . 3 |- ( ph -> ( ( P ` N ) = ( P ` ( N + 1 ) ) \/ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) )
56	35, 53, 55	mpjaodan 806	. 2 |- ( ph -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
57	19, 56	bitrd 188	1 |- ( ph -> ( -. 2 || ( (VtxDeg ` Z ) ` 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   = wceq 1398   e. wcel 2205   =/= wne 2414   {crab 2526   C_ wss 3213   (/)c0 3510   ifcif 3622   {csn 3691   {cpr 3692   <.cop 3694   class class class wbr 4111   |` cres 4753   "cima 4754   Fun wfun 5348   -->wf 5350   ` cfv 5354  (class class class)co 6052   Fincfn 6977   0cc0 8129   1c1 8130   + caddc 8132   2c2 9290   ...cfz 10345  ..^cfzo 10480  ♯chash 11142   || cdvds 12477  Vtxcvtx 16024  iEdgciedg 16025  UPGraphcupgr 16103  UMGraphcumgr 16104  VtxDegcvtxdg 16298  Walkscwlks 16329  Trailsctrls 16392

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-map 6886  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-z 9580  df-dec 9713  df-uz 9857  df-q 9955  df-rp 9990  df-xadd 10109  df-fz 10346  df-fzo 10481  df-fl 10634  df-mod 10689  df-seqfrec 10814  df-exp 10905  df-ihash 11143  df-word 11229  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-dvds 12478  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-edg 16070  df-uhgrm 16081  df-ushgrm 16082  df-upgren 16105  df-umgren 16106  df-uspgren 16167  df-subgr 16266  df-vtxdg 16299  df-wlks 16330  df-trls 16393

This theorem is referenced by:  eupth2lem3fi  16488