Theorem uspgr2wlkeq2 16607

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 5694	. . . . 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 16606	. . . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   0cc0 8179   NN0cn0 9563   ...cfz 10411  ♯chash 11214  USPGraphcuspgr 16394  Walkscwlks 16558

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-uhgrm 16310  df-upgren 16334  df-uspgren 16396  df-wlks 16559

This theorem is used by:  uspgr2wlkeqi  16608