Theorem uspgr2wlkeq2 16521

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 Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)

Assertion

Ref	Expression
uspgr2wlkeq2	|- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> A = B ) )

Proof of Theorem uspgr2wlkeq2

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 |- ( ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) -> (♯ ` ( 1st ` B ) ) = N )
2	1	eqcomd 2244	. . . . 5 |- ( ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) -> N = (♯ ` ( 1st ` B ) ) )
3	2	3ad2ant3 1051	. . . 4 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) -> N = (♯ ` ( 1st ` B ) ) )
4	3	adantr 276	. . 3 |- ( ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> N = (♯ ` ( 1st ` B ) ) )
5		fveq1 5689	. . . . 5 |- ( ( 2nd ` A ) = ( 2nd ` B ) -> ( ( 2nd ` A ) ` i ) = ( ( 2nd ` B ) ` i ) )
6	5	adantl 277	. . . 4 |- ( ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> ( ( 2nd ` A ) ` i ) = ( ( 2nd ` B ) ` i ) )
7	6	ralrimivw 2624	. . 3 |- ( ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> A. i e. ( 0 ... N ) ( ( 2nd ` A ) ` i ) = ( ( 2nd ` B ) ` i ) )
8		simpl1l 1079	. . . 4 |- ( ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> G e. USPGraph )
9		simpl 109	. . . . . . 7 |- ( ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) -> A e. (Walks ` G ) )
10		simpl 109	. . . . . . 7 |- ( ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) -> B e. (Walks ` G ) )
11	9, 10	anim12i 338	. . . . . 6 |- ( ( ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) -> ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) )
12	11	3adant1 1046	. . . . 5 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) -> ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) )
13	12	adantr 276	. . . 4 |- ( ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) )
14		simpr 110	. . . . . . 7 |- ( ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) -> (♯ ` ( 1st ` A ) ) = N )
15	14	eqcomd 2244	. . . . . 6 |- ( ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) -> N = (♯ ` ( 1st ` A ) ) )
16	15	3ad2ant2 1050	. . . . 5 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) -> N = (♯ ` ( 1st ` A ) ) )
17	16	adantr 276	. . . 4 |- ( ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> N = (♯ ` ( 1st ` A ) ) )
18		uspgr2wlkeq 16520	. . . 4 |- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. i e. ( 0 ... N ) ( ( 2nd ` A ) ` i ) = ( ( 2nd ` B ) ` i ) ) ) )
19	8, 13, 17, 18	syl3anc 1278	. . 3 |- ( ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> ( A = B <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. i e. ( 0 ... N ) ( ( 2nd ` A ) ` i ) = ( ( 2nd ` B ) ` i ) ) ) )
20	4, 7, 19	mpbir2and 957	. 2 |- ( ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> A = B )
21	20	ex 115	1 |- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   0cc0 8169   NN0cn0 9542   ...cfz 10390  ♯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:  uspgr2wlkeqi  16522