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Theorem uspgr2wlkeqi 27132
 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 27113 . . . . 5 (𝐴 ∈ (Walks‘𝐺) ↔ (1st𝐴)(Walks‘𝐺)(2nd𝐴))
2 wlkcpr 27113 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) ↔ (1st𝐵)(Walks‘𝐺)(2nd𝐵))
3 wlkcl 27100 . . . . . . 7 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (♯‘(1st𝐴)) ∈ ℕ0)
4 fveq2 6499 . . . . . . . . . . . . 13 ((2nd𝐴) = (2nd𝐵) → (♯‘(2nd𝐴)) = (♯‘(2nd𝐵)))
54oveq1d 6991 . . . . . . . . . . . 12 ((2nd𝐴) = (2nd𝐵) → ((♯‘(2nd𝐴)) − 1) = ((♯‘(2nd𝐵)) − 1))
65eqcomd 2785 . . . . . . . . . . 11 ((2nd𝐴) = (2nd𝐵) → ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1))
76adantl 474 . . . . . . . . . 10 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1))
8 wlklenvm1 27106 . . . . . . . . . . . 12 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1))
9 wlklenvm1 27106 . . . . . . . . . . . 12 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1))
108, 9eqeqan12rd 2797 . . . . . . . . . . 11 (((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) → ((♯‘(1st𝐵)) = (♯‘(1st𝐴)) ↔ ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1)))
1110adantr 473 . . . . . . . . . 10 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(1st𝐵)) = (♯‘(1st𝐴)) ↔ ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1)))
127, 11mpbird 249 . . . . . . . . 9 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → (♯‘(1st𝐵)) = (♯‘(1st𝐴)))
1312anim2i 607 . . . . . . . 8 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵))) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
1413exp44 430 . . . . . . 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 234 . . . . 5 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (𝐵 ∈ (Walks‘𝐺) → ((2nd𝐴) = (2nd𝐵) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))))
171, 16sylbi 209 . . . 4 (𝐴 ∈ (Walks‘𝐺) → (𝐵 ∈ (Walks‘𝐺) → ((2nd𝐴) = (2nd𝐵) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))))
1817imp31 410 . . 3 (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
19183adant1 1110 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
20 simpl 475 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐺 ∈ USPGraph)
21 simpl 475 . . . . . . 7 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))) → (♯‘(1st𝐴)) ∈ ℕ0)
2220, 21anim12i 603 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → (𝐺 ∈ USPGraph ∧ (♯‘(1st𝐴)) ∈ ℕ0))
23 simpl 475 . . . . . . . 8 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐴 ∈ (Walks‘𝐺))
2423adantl 474 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐴 ∈ (Walks‘𝐺))
25 eqidd 2780 . . . . . . 7 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))) → (♯‘(1st𝐴)) = (♯‘(1st𝐴)))
2624, 25anim12i 603 . . . . . 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 474 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐵 ∈ (Walks‘𝐺))
29 simpr 477 . . . . . . 7 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))) → (♯‘(1st𝐵)) = (♯‘(1st𝐴)))
3028, 29anim12i 603 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
31 uspgr2wlkeq2 27131 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (♯‘(1st𝐴)) ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = (♯‘(1st𝐴))) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3222, 26, 30, 31syl3anc 1351 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3332ex 405 . . . 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 1097 . 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 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   class class class wbr 4929  ‘cfv 6188  (class class class)co 6976  1st c1st 7499  2nd c2nd 7500  1c1 10336   − cmin 10670  ℕ0cn0 11707  ♯chash 13505  USPGraphcuspgr 26636  Walkscwlks 27081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ifp 1044  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-2o 7906  df-oadd 7909  df-er 8089  df-map 8208  df-pm 8209  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-dju 9124  df-card 9162  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-2 11503  df-n0 11708  df-xnn0 11780  df-z 11794  df-uz 12059  df-fz 12709  df-fzo 12850  df-hash 13506  df-word 13673  df-edg 26536  df-uhgr 26546  df-upgr 26570  df-uspgr 26638  df-wlks 27084 This theorem is referenced by:  wlkswwlksf1o  27365
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