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Theorem uspgr2wlkeq2 27445
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 488 . . . . . 6 ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁) → (♯‘(1st𝐵)) = 𝑁)
21eqcomd 2830 . . . . 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 484 . . 3 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝑁 = (♯‘(1st𝐵)))
5 fveq1 6662 . . . . 5 ((2nd𝐴) = (2nd𝐵) → ((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))
65adantl 485 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → ((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))
76ralrimivw 3178 . . 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 486 . . . . . . 7 ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) → 𝐴 ∈ (Walks‘𝐺))
10 simpl 486 . . . . . . 7 ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁) → 𝐵 ∈ (Walks‘𝐺))
119, 10anim12i 615 . . . . . 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 484 . . . 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 488 . . . . . . 7 ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) → (♯‘(1st𝐴)) = 𝑁)
1514eqcomd 2830 . . . . . 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 484 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝑁 = (♯‘(1st𝐴)))
18 uspgr2wlkeq 27444 . . . 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 712 . 2 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐴 = 𝐵)
2120ex 416 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  cfv 6345  (class class class)co 7151  1st c1st 7684  2nd c2nd 7685  0cc0 10537  0cn0 11896  ...cfz 12896  chash 13697  USPGraphcuspgr 26950  Walkscwlks 27395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-2o 8101  df-oadd 8104  df-er 8287  df-map 8406  df-pm 8407  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-dju 9329  df-card 9367  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11637  df-2 11699  df-n0 11897  df-xnn0 11967  df-z 11981  df-uz 12243  df-fz 12897  df-fzo 13040  df-hash 13698  df-word 13869  df-edg 26850  df-uhgr 26860  df-upgr 26884  df-uspgr 26952  df-wlks 27398
This theorem is referenced by:  uspgr2wlkeqi  27446
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