Theorem uspgr2wlkeqi 16078

Description: Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 6-May-2021.)

Assertion

Ref	Expression
uspgr2wlkeqi	|- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> A = B )

Proof of Theorem uspgr2wlkeqi

Step	Hyp	Ref	Expression
1		wlkcprim 16061	. . . . 5 |- ( A e. (Walks ` G ) -> ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) )
2		wlkcprim 16061	. . . . 5 |- ( B e. (Walks ` G ) -> ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) )
3		wlkcl 16044	. . . . . 6 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> (♯ ` ( 1st ` A ) ) e. NN0 )
4		fveq2 5627	. . . . . . . . . . . 12 |- ( ( 2nd ` A ) = ( 2nd ` B ) -> (♯ ` ( 2nd ` A ) ) = (♯ ` ( 2nd ` B ) ) )
5	4	oveq1d 6016	. . . . . . . . . . 11 |- ( ( 2nd ` A ) = ( 2nd ` B ) -> ( (♯ ` ( 2nd ` A ) ) - 1 ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
6	5	eqcomd 2235	. . . . . . . . . 10 |- ( ( 2nd ` A ) = ( 2nd ` B ) -> ( (♯ ` ( 2nd ` B ) ) - 1 ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) )
7	6	adantl 277	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) /\ ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> ( (♯ ` ( 2nd ` B ) ) - 1 ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) )
8		wlklenvm1 16052	. . . . . . . . . . 11 |- ( ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) -> (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
9		wlklenvm1 16052	. . . . . . . . . . 11 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) )
10	8, 9	eqeqan12rd 2246	. . . . . . . . . 10 |- ( ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) /\ ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) ) -> ( (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) <-> ( (♯ ` ( 2nd ` B ) ) - 1 ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) ) )
11	10	adantr 276	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) /\ ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> ( (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) <-> ( (♯ ` ( 2nd ` B ) ) - 1 ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) ) )
12	7, 11	mpbird 167	. . . . . . . 8 |- ( ( ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) /\ ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) )
13	12	anim2i 342	. . . . . . 7 |- ( ( (♯ ` ( 1st ` A ) ) e. NN0 /\ ( ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) /\ ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) -> ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) )
14	13	exp44 373	. . . . . 6 |- ( (♯ ` ( 1st ` A ) ) e. NN0 -> ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> ( ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) ) ) ) )
15	3, 14	mpcom 36	. . . . 5 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> ( ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) ) ) )
16	1, 2, 15	syl2im 38	. . . 4 |- ( A e. (Walks ` G ) -> ( B e. (Walks ` G ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) ) ) )
17	16	imp31 256	. . 3 |- ( ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) )
18	17	3adant1 1039	. 2 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) )
19		simpl 109	. . . . . . 7 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) ) -> G e. USPGraph )
20		simpl 109	. . . . . . 7 |- ( ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) -> (♯ ` ( 1st ` A ) ) e. NN0 )
21	19, 20	anim12i 338	. . . . . 6 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) ) /\ ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) ) -> ( G e. USPGraph /\ (♯ ` ( 1st ` A ) ) e. NN0 ) )
22		simpl 109	. . . . . . . 8 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> A e. (Walks ` G ) )
23	22	adantl 277	. . . . . . 7 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) ) -> A e. (Walks ` G ) )
24		eqidd 2230	. . . . . . 7 |- ( ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) -> (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` A ) ) )
25	23, 24	anim12i 338	. . . . . 6 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) ) /\ ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) ) -> ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` A ) ) ) )
26		simpr 110	. . . . . . . 8 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> B e. (Walks ` G ) )
27	26	adantl 277	. . . . . . 7 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) ) -> B e. (Walks ` G ) )
28		simpr 110	. . . . . . 7 |- ( ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) -> (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) )
29	27, 28	anim12i 338	. . . . . 6 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) ) /\ ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) ) -> ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) )
30		uspgr2wlkeq2 16077	. . . . . 6 |- ( ( ( G e. USPGraph /\ (♯ ` ( 1st ` A ) ) e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = (♯ ` ( 1st ` A ) ) ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> A = B ) )
31	21, 25, 29, 30	syl3anc 1271	. . . . 5 |- ( ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) ) /\ ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> A = B ) )
32	31	ex 115	. . . 4 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) ) -> ( ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> A = B ) ) )
33	32	com23 78	. . 3 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> ( ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) -> A = B ) ) )
34	33	3impia 1224	. 2 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> ( ( (♯ ` ( 1st ` A ) ) e. NN0 /\ (♯ ` ( 1st ` B ) ) = (♯ ` ( 1st ` A ) ) ) -> A = B ) )
35	18, 34	mpd 13	1 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   1stc1st 6284   2ndc2nd 6285   1c1 8000   - cmin 8317   NN0cn0 9369  ♯chash 10997  USPGraphcuspgr 15951  Walkscwlks 16030

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-2o 6563  df-er 6680  df-map 6797  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-z 9447  df-dec 9579  df-uz 9723  df-fz 10205  df-fzo 10339  df-ihash 10998  df-word 11072  df-ndx 13035  df-slot 13036  df-base 13038  df-edgf 15806  df-vtx 15815  df-iedg 15816  df-edg 15859  df-uhgrm 15869  df-upgren 15893  df-uspgren 15953  df-wlks 16031

This theorem is referenced by: (None)