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Theorem uspgr2wlkeqi 26407
 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 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐴 = 𝐵)

Proof of Theorem uspgr2wlkeqi
StepHypRef Expression
1 wlkcpr 26388 . . . . 5 (𝐴 ∈ (Walks‘𝐺) ↔ (1st𝐴)(Walks‘𝐺)(2nd𝐴))
2 wlkcpr 26388 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) ↔ (1st𝐵)(Walks‘𝐺)(2nd𝐵))
3 wlkcl 26375 . . . . . . 7 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (#‘(1st𝐴)) ∈ ℕ0)
4 fveq2 6150 . . . . . . . . . . . . 13 ((2nd𝐴) = (2nd𝐵) → (#‘(2nd𝐴)) = (#‘(2nd𝐵)))
54oveq1d 6620 . . . . . . . . . . . 12 ((2nd𝐴) = (2nd𝐵) → ((#‘(2nd𝐴)) − 1) = ((#‘(2nd𝐵)) − 1))
65eqcomd 2632 . . . . . . . . . . 11 ((2nd𝐴) = (2nd𝐵) → ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1))
76adantl 482 . . . . . . . . . 10 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1))
8 wlklenvm1 26381 . . . . . . . . . . . 12 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))
9 wlklenvm1 26381 . . . . . . . . . . . 12 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1))
108, 9eqeqan12rd 2644 . . . . . . . . . . 11 (((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) → ((#‘(1st𝐵)) = (#‘(1st𝐴)) ↔ ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1)))
1110adantr 481 . . . . . . . . . 10 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(1st𝐵)) = (#‘(1st𝐴)) ↔ ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1)))
127, 11mpbird 247 . . . . . . . . 9 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → (#‘(1st𝐵)) = (#‘(1st𝐴)))
1312anim2i 592 . . . . . . . 8 (((#‘(1st𝐴)) ∈ ℕ0 ∧ (((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵))) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))
1413exp44 640 . . . . . . 7 ((#‘(1st𝐴)) ∈ ℕ0 → ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → ((1st𝐵)(Walks‘𝐺)(2nd𝐵) → ((2nd𝐴) = (2nd𝐵) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))))))
153, 14mpcom 38 . . . . . 6 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → ((1st𝐵)(Walks‘𝐺)(2nd𝐵) → ((2nd𝐴) = (2nd𝐵) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))))
162, 15syl5bi 232 . . . . 5 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (𝐵 ∈ (Walks‘𝐺) → ((2nd𝐴) = (2nd𝐵) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))))
171, 16sylbi 207 . . . 4 (𝐴 ∈ (Walks‘𝐺) → (𝐵 ∈ (Walks‘𝐺) → ((2nd𝐴) = (2nd𝐵) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))))
1817imp31 448 . . 3 (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))
19183adant1 1077 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))
20 simpl 473 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐺 ∈ USPGraph )
21 simpl 473 . . . . . . 7 (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → (#‘(1st𝐴)) ∈ ℕ0)
2220, 21anim12i 589 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → (𝐺 ∈ USPGraph ∧ (#‘(1st𝐴)) ∈ ℕ0))
23 simpl 473 . . . . . . . 8 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐴 ∈ (Walks‘𝐺))
2423adantl 482 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐴 ∈ (Walks‘𝐺))
25 eqidd 2627 . . . . . . 7 (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → (#‘(1st𝐴)) = (#‘(1st𝐴)))
2624, 25anim12i 589 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → (𝐴 ∈ (Walks‘𝐺) ∧ (#‘(1st𝐴)) = (#‘(1st𝐴))))
27 simpr 477 . . . . . . . 8 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐵 ∈ (Walks‘𝐺))
2827adantl 482 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐵 ∈ (Walks‘𝐺))
29 simpr 477 . . . . . . 7 (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → (#‘(1st𝐵)) = (#‘(1st𝐴)))
3028, 29anim12i 589 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → (𝐵 ∈ (Walks‘𝐺) ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))
31 uspgr2wlkeq2 26406 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (#‘(1st𝐴)) ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (#‘(1st𝐴)) = (#‘(1st𝐴))) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3222, 26, 30, 31syl3anc 1323 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3332ex 450 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵)))
3433com23 86 . . 3 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → ((2nd𝐴) = (2nd𝐵) → (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → 𝐴 = 𝐵)))
35343impia 1258 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → 𝐴 = 𝐵))
3619, 35mpd 15 1 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1992   class class class wbr 4618  ‘cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  1c1 9882   − cmin 10211  ℕ0cn0 11237  #chash 13054   USPGraph cuspgr 25931  Walkscwlks 26356 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-edg 25835  df-uhgr 25844  df-upgr 25868  df-uspgr 25933  df-wlks 26359 This theorem is referenced by:  wlkpwwlkf1ouspgr  26628
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