Theorem eupth2lem3lem7fi 16698

Description: Lemma for eupth2lem3fi 16700: Combining trlsegvdegfi 16691, eupth2lem3lem3fi 16694, eupth2lem3lem4fi 16697 and eupth2lem3lem6fi 16695. (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 16336	. . . . . 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 16691	. . . 4 |- ( ph -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )
18	17	breq2d 4140	. . 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 16610	. . . . . . . 8 |- ( F (Trails ` G ) P -> F (Walks ` G ) P )
23	1	wlkp 16558	. . . . . . . 8 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
24	6, 22, 23	3syl 17	. . . . . . 7 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
25		elfzofz 10553	. . . . . . . 8 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. ( 0 ... (♯ ` F ) ) )
26	4, 25	syl 14	. . . . . . 7 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
27	24, 26	ffvelcdmd 5838	. . . . . 6 |- ( ph -> ( P ` N ) e. V )
28		fzofzp1 10628	. . . . . . . 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 5838	. . . . . 6 |- ( ph -> ( P ` ( N + 1 ) ) e. V )
31		fidceq 7165	. . . . . 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 3818	. . . . 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 16694	. . 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 16696	. . . . . 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 16697	. . . . 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 16695	. . . . . 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 7165	. . . . . . . 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 7165	. . . . . . . 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 ) ) } ) ) )

Syntax hints:   -. 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 3708   {cpr 3709   <.cop 3711   class class class wbr 4128   |` cres 4774   "cima 4775   Fun wfun 5369   -->wf 5371   ` cfv 5375  (class class class)co 6079   Fincfn 7016   0cc0 8173   1c1 8174   + caddc 8176   2c2 9338   ...cfz 10394  ..^cfzo 10532  ♯chash 11197   || cdvds 12537  Vtxcvtx 16236  iEdgciedg 16237  UPGraphcupgr 16315  UMGraphcumgr 16316  VtxDegcvtxdg 16510  Walkscwlks 16541  Trailsctrls 16604

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-map 6918  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-q 10003  df-rp 10038  df-xadd 10158  df-fz 10395  df-fzo 10533  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-word 11288  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-dvds 12538  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-edg 16282  df-uhgrm 16293  df-ushgrm 16294  df-upgren 16317  df-umgren 16318  df-uspgren 16379  df-subgr 16478  df-vtxdg 16511  df-wlks 16542  df-trls 16605

This theorem is referenced by:  eupth2lem3fi  16700