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Theorem uspgr2wlkeqi 26600
 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 26580 . . . . 5 (𝐴 ∈ (Walks‘𝐺) ↔ (1st𝐴)(Walks‘𝐺)(2nd𝐴))
2 wlkcpr 26580 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) ↔ (1st𝐵)(Walks‘𝐺)(2nd𝐵))
3 wlkcl 26567 . . . . . . 7 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (#‘(1st𝐴)) ∈ ℕ0)
4 fveq2 6229 . . . . . . . . . . . . 13 ((2nd𝐴) = (2nd𝐵) → (#‘(2nd𝐴)) = (#‘(2nd𝐵)))
54oveq1d 6705 . . . . . . . . . . . 12 ((2nd𝐴) = (2nd𝐵) → ((#‘(2nd𝐴)) − 1) = ((#‘(2nd𝐵)) − 1))
65eqcomd 2657 . . . . . . . . . . 11 ((2nd𝐴) = (2nd𝐵) → ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1))
76adantl 481 . . . . . . . . . 10 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1))
8 wlklenvm1 26573 . . . . . . . . . . . 12 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))
9 wlklenvm1 26573 . . . . . . . . . . . 12 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1))
108, 9eqeqan12rd 2669 . . . . . . . . . . 11 (((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) → ((#‘(1st𝐵)) = (#‘(1st𝐴)) ↔ ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1)))
1110adantr 480 . . . . . . . . . 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 447 . . 3 (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))
19183adant1 1099 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))
20 simpl 472 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐺 ∈ USPGraph)
21 simpl 472 . . . . . . 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 472 . . . . . . . 8 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐴 ∈ (Walks‘𝐺))
2423adantl 481 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐴 ∈ (Walks‘𝐺))
25 eqidd 2652 . . . . . . 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 476 . . . . . . . 8 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐵 ∈ (Walks‘𝐺))
2827adantl 481 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐵 ∈ (Walks‘𝐺))
29 simpr 476 . . . . . . 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 26599 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (#‘(1st𝐴)) ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (#‘(1st𝐴)) = (#‘(1st𝐴))) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3222, 26, 30, 31syl3anc 1366 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3332ex 449 . . . 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 1280 . 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 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  1c1 9975   − cmin 10304  ℕ0cn0 11330  #chash 13157  USPGraphcuspgr 26088  Walkscwlks 26548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-uspgr 26090  df-wlks 26551 This theorem is referenced by:  wlkpwwlkf1ouspgr  26833
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