Theorem eupth2lem3lem7fi 16328

Description: Lemma for eupth2lem3fi 16330: Combining trlsegvdegfi 16321, eupth2lem3lem3fi 16324, eupth2lem3lem4fi 16327 and eupth2lem3lem6fi 16325. (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 15966	. . . . . 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 16321	. . . 4 |- ( ph -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )
18	17	breq2d 4100	. . 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 16240	. . . . . . . 8 |- ( F (Trails ` G ) P -> F (Walks ` G ) P )
23	1	wlkp 16188	. . . . . . . 8 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
24	6, 22, 23	3syl 17	. . . . . . 7 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
25		elfzofz 10398	. . . . . . . 8 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. ( 0 ... (♯ ` F ) ) )
26	4, 25	syl 14	. . . . . . 7 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
27	24, 26	ffvelcdmd 5783	. . . . . 6 |- ( ph -> ( P ` N ) e. V )
28		fzofzp1 10473	. . . . . . . 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 5783	. . . . . 6 |- ( ph -> ( P ` ( N + 1 ) ) e. V )
31		fidceq 7056	. . . . . 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 1273	. . . . 5 |- ( ph -> DECID ( P ` N ) = ( P ` ( N + 1 ) ) )
33		ifpprsnssdc 3779	. . . . 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 16324	. . 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 16326	. . . . . 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 16327	. . . . 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 1229	. . . 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 2489	. . . . 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 16325	. . . . . 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 1229	. . . . 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 7056	. . . . . . . 8 |- ( ( V e. Fin /\ U e. V /\ ( P ` N ) e. V ) -> DECID U = ( P ` N ) )
44	16, 5, 27, 43	syl3anc 1273	. . . . . . 7 |- ( ph -> DECID U = ( P ` N ) )
45	44	adantr 276	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> DECID U = ( P ` N ) )
46		fidceq 7056	. . . . . . . 8 |- ( ( V e. Fin /\ U e. V /\ ( P ` ( N + 1 ) ) e. V ) -> DECID U = ( P ` ( N + 1 ) ) )
47	16, 5, 30, 46	syl3anc 1273	. . . . . . 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 943	. . . . . 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 843	. . . . 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 805	. . 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 2413	. . . 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 805	. 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 715  DECID wdc 841  if-wif 985   = wceq 1397   e. wcel 2202   =/= wne 2402   {crab 2514   C_ wss 3200   (/)c0 3494   ifcif 3605   {csn 3669   {cpr 3670   <.cop 3672   class class class wbr 4088   |` cres 4727   "cima 4728   Fun wfun 5320   -->wf 5322   ` cfv 5326  (class class class)co 6018   Fincfn 6909   0cc0 8032   1c1 8033   + caddc 8035   2c2 9194   ...cfz 10243  ..^cfzo 10377  ♯chash 11038   || cdvds 12350  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-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  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-ushgrm 15924  df-upgren 15947  df-umgren 15948  df-uspgren 16009  df-subgr 16108  df-vtxdg 16141  df-wlks 16172  df-trls 16235

This theorem is referenced by:  eupth2lem3fi  16330