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Theorem uspgr2wlkeq2 29538
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 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))

Proof of Theorem uspgr2wlkeq2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 simpr 483 . . . . . 6 ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁) → (♯‘(1st𝐵)) = 𝑁)
21eqcomd 2731 . . . . 5 ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁) → 𝑁 = (♯‘(1st𝐵)))
323ad2ant3 1132 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → 𝑁 = (♯‘(1st𝐵)))
43adantr 479 . . 3 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝑁 = (♯‘(1st𝐵)))
5 fveq1 6895 . . . . 5 ((2nd𝐴) = (2nd𝐵) → ((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))
65adantl 480 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → ((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))
76ralrimivw 3139 . . 3 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → ∀𝑖 ∈ (0...𝑁)((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))
8 simpl1l 1221 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐺 ∈ USPGraph)
9 simpl 481 . . . . . . 7 ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) → 𝐴 ∈ (Walks‘𝐺))
10 simpl 481 . . . . . . 7 ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁) → 𝐵 ∈ (Walks‘𝐺))
119, 10anim12i 611 . . . . . 6 (((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)))
12113adant1 1127 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)))
1312adantr 479 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)))
14 simpr 483 . . . . . . 7 ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) → (♯‘(1st𝐴)) = 𝑁)
1514eqcomd 2731 . . . . . 6 ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) → 𝑁 = (♯‘(1st𝐴)))
16153ad2ant2 1131 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → 𝑁 = (♯‘(1st𝐴)))
1716adantr 479 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝑁 = (♯‘(1st𝐴)))
18 uspgr2wlkeq 29537 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑖 ∈ (0...𝑁)((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))))
198, 13, 17, 18syl3anc 1368 . . 3 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑖 ∈ (0...𝑁)((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))))
204, 7, 19mpbir2and 711 . 2 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐴 = 𝐵)
2120ex 411 1 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  cfv 6549  (class class class)co 7419  1st c1st 7992  2nd c2nd 7993  0cc0 11145  0cn0 12510  ...cfz 13524  chash 14330  USPGraphcuspgr 29038  Walkscwlks 29487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9931  df-card 9969  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-nn 12251  df-2 12313  df-n0 12511  df-xnn0 12583  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-hash 14331  df-word 14506  df-edg 28938  df-uhgr 28948  df-upgr 28972  df-uspgr 29040  df-wlks 29490
This theorem is referenced by:  uspgr2wlkeqi  29539
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