Theorem uspgr2wlkeqi 16813

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 16796	. . . . 5 |- ( A e. (Walks ` G ) -> ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) )
2		wlkcprim 16796	. . . . 5 |- ( B e. (Walks ` G ) -> ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) )
3		wlkcl 16778	. . . . . 6 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> (♯ ` ( 1st ` A ) ) e. NN0 )
4		fveq2 5698	. . . . . . . . . . . 12 |- ( ( 2nd ` A ) = ( 2nd ` B ) -> (♯ ` ( 2nd ` A ) ) = (♯ ` ( 2nd ` B ) ) )
5	4	oveq1d 6103	. . . . . . . . . . 11 |- ( ( 2nd ` A ) = ( 2nd ` B ) -> ( (♯ ` ( 2nd ` A ) ) - 1 ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
6	5	eqcomd 2244	. . . . . . . . . 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 16787	. . . . . . . . . . 11 |- ( ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) -> (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
9		wlklenvm1 16787	. . . . . . . . . . 11 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> (♯ ` ( 1st ` A ) ) = ( (♯ ` ( 2nd ` A ) ) - 1 ) )
10	8, 9	eqeqan12rd 2255	. . . . . . . . . 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 1046	. 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 2239	. . . . . . 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 16812	. . . . . 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 1278	. . . . 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 1231	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   1stc1st 6376   2ndc2nd 6377   1c1 8186   - cmin 8504   NN0cn0 9574  ♯chash 11236  USPGraphcuspgr 16599  Walkscwlks 16763

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  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-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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-2o 6692  df-er 6811  df-map 6928  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-dec 9789  df-uz 9938  df-fz 10429  df-fzo 10567  df-ihash 11237  df-word 11327  df-ndx 13413  df-slot 13414  df-base 13416  df-edgf 16451  df-vtx 16460  df-iedg 16461  df-edg 16504  df-uhgrm 16515  df-upgren 16539  df-uspgren 16601  df-wlks 16764

This theorem is used by: (None)