Theorem eupth2lem3lem7fi 16715

Description: Lemma for eupth2lem3fi 16717: Combining trlsegvdegfi 16708, eupth2lem3lem3fi 16711, eupth2lem3lem4fi 16714 and eupth2lem3lem6fi 16712. (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 16353	. . . . . 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 16708	. . . 4 |- ( ph -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )
18	17	breq2d 4142	. . 3 |- ( ph -> ( 2 || ( (VtxDeg ` Z ) ` U ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
19	18	notbid 677	. 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 16627	. . . . . . . 8 |- ( F (Trails ` G ) P -> F (Walks ` G ) P )
23	1	wlkp 16575	. . . . . . . 8 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
24	6, 22, 23	3syl 17	. . . . . . 7 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
25		elfzofz 10570	. . . . . . . 8 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. ( 0 ... (♯ ` F ) ) )
26	4, 25	syl 14	. . . . . . 7 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
27	24, 26	ffvelcdmd 5844	. . . . . 6 |- ( ph -> ( P ` N ) e. V )
28		fzofzp1 10645	. . . . . . . 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 5844	. . . . . 6 |- ( ph -> ( P ` ( N + 1 ) ) e. V )
31		fidceq 7171	. . . . . 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 1278	. . . . 5 |- ( ph -> DECID ( P ` N ) = ( P ` ( N + 1 ) ) )
33		ifpprsnssdc 3820	. . . . 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 415	. . . 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 16711	. . 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 16713	. . . . . 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 16714	. . . . 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 1234	. . . 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 2507	. . . . 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 16712	. . . . . 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 1234	. . . . 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 7171	. . . . . . . 8 |- ( ( V e. Fin /\ U e. V /\ ( P ` N ) e. V ) -> DECID U = ( P ` N ) )
44	16, 5, 27, 43	syl3anc 1278	. . . . . . 7 |- ( ph -> DECID U = ( P ` N ) )
45	44	adantr 276	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> DECID U = ( P ` N ) )
46		fidceq 7171	. . . . . . . 8 |- ( ( V e. Fin /\ U e. V /\ ( P ` ( N + 1 ) ) e. V ) -> DECID U = ( P ` ( N + 1 ) ) )
47	16, 5, 30, 46	syl3anc 1278	. . . . . . 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 948	. . . . . 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 848	. . . . 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 810	. . 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 2431	. . . 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 810	. 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 ) ) } ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846  if-wif 990   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   C_ wss 3220   (/)c0 3520   ifcif 3638   {csn 3709   {cpr 3710   <.cop 3712   class class class wbr 4130   |` cres 4776   "cima 4777   Fun wfun 5371   -->wf 5373   ` cfv 5377  (class class class)co 6085   Fincfn 7022   0cc0 8179   1c1 8180   + caddc 8182   2c2 9355   ...cfz 10411  ..^cfzo 10549  ♯chash 11214   || cdvds 12554  Vtxcvtx 16253  iEdgciedg 16254  UPGraphcupgr 16332  UMGraphcumgr 16333  VtxDegcvtxdg 16527  Walkscwlks 16558  Trailsctrls 16621

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-q 10020  df-rp 10055  df-xadd 10175  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-word 11305  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-uhgrm 16310  df-ushgrm 16311  df-upgren 16334  df-umgren 16335  df-uspgren 16396  df-subgr 16495  df-vtxdg 16528  df-wlks 16559  df-trls 16622

This theorem is used by:  eupth2lem3fi  16717