Theorem uspgr2wlkeqi 16379

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 16362	. . . . 5 |- ( A e. (Walks ` G ) -> ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) )
2		wlkcprim 16362	. . . . 5 |- ( B e. (Walks ` G ) -> ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) )
3		wlkcl 16344	. . . . . 6 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> (♯ ` ( 1st ` A ) ) e. NN0 )
4		fveq2 5672	. . . . . . . . . . . 12 |- ( ( 2nd ` A ) = ( 2nd ` B ) -> (♯ ` ( 2nd ` A ) ) = (♯ ` ( 2nd ` B ) ) )
5	4	oveq1d 6067	. . . . . . . . . . 11 |- ( ( 2nd ` A ) = ( 2nd ` B ) -> ( (♯ ` ( 2nd ` A ) ) - 1 ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
6	5	eqcomd 2240	. . . . . . . . . 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 16353	. . . . . . . . . . 11 |- ( ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) -> (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
9		wlklenvm1 16353	. . . . . . . . . . 11 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) )
10	8, 9	eqeqan12rd 2251	. . . . . . . . . 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 1042	. 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 2235	. . . . . . 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 16378	. . . . . 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 1274	. . . . 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 1227	. 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 1005   = wceq 1398   e. wcel 2205   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   1stc1st 6334   2ndc2nd 6335   1c1 8130   - cmin 8446   NN0cn0 9498  ♯chash 11142  USPGraphcuspgr 16165  Walkscwlks 16329

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-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-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  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-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-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-frec 6624  df-1o 6649  df-2o 6650  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-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-fz 10346  df-fzo 10481  df-ihash 11143  df-word 11229  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-upgren 16105  df-uspgren 16167  df-wlks 16330

This theorem is referenced by: (None)