Theorem uspgr2wlkeqi 16488

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 16471	. . . . 5 |- ( A e. (Walks ` G ) -> ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) )
2		wlkcprim 16471	. . . . 5 |- ( B e. (Walks ` G ) -> ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) )
3		wlkcl 16453	. . . . . 6 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> (♯ ` ( 1st ` A ) ) e. NN0 )
4		fveq2 5675	. . . . . . . . . . . 12 |- ( ( 2nd ` A ) = ( 2nd ` B ) -> (♯ ` ( 2nd ` A ) ) = (♯ ` ( 2nd ` B ) ) )
5	4	oveq1d 6073	. . . . . . . . . . 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 16462	. . . . . . . . . . 11 |- ( ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) -> (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
9		wlklenvm1 16462	. . . . . . . . . . 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 16487	. . . . . 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 4114   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   1c1 8144   - cmin 8460   NN0cn0 9513  ♯chash 11163  USPGraphcuspgr 16274  Walkscwlks 16438

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 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-uhgrm 16190  df-upgren 16214  df-uspgren 16276  df-wlks 16439

This theorem is referenced by: (None)