Theorem uspgr2wlkeqi 16522

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 16505	. . . . 5 |- ( A e. (Walks ` G ) -> ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) )
2		wlkcprim 16505	. . . . 5 |- ( B e. (Walks ` G ) -> ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) )
3		wlkcl 16487	. . . . . 6 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> (♯ ` ( 1st ` A ) ) e. NN0 )
4		fveq2 5690	. . . . . . . . . . . 12 |- ( ( 2nd ` A ) = ( 2nd ` B ) -> (♯ ` ( 2nd ` A ) ) = (♯ ` ( 2nd ` B ) ) )
5	4	oveq1d 6090	. . . . . . . . . . 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 16496	. . . . . . . . . . 11 |- ( ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) -> (♯ ` ( 1st ` B ) ) = ( (♯ ` ( 2nd ` B ) ) - 1 ) )
9		wlklenvm1 16496	. . . . . . . . . . 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 16521	. . . . . 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 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   1c1 8170   - cmin 8487   NN0cn0 9542  ♯chash 11192  USPGraphcuspgr 16308  Walkscwlks 16472

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-2o 6678  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-z 9624  df-dec 9757  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-uhgrm 16224  df-upgren 16248  df-uspgren 16310  df-wlks 16473

This theorem is referenced by: (None)