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Theorem uspgr2wlkeqi 27440
 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 27421 . . . . 5 (𝐴 ∈ (Walks‘𝐺) ↔ (1st𝐴)(Walks‘𝐺)(2nd𝐴))
2 wlkcpr 27421 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) ↔ (1st𝐵)(Walks‘𝐺)(2nd𝐵))
3 wlkcl 27408 . . . . . . 7 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (♯‘(1st𝐴)) ∈ ℕ0)
4 fveq2 6649 . . . . . . . . . . . . 13 ((2nd𝐴) = (2nd𝐵) → (♯‘(2nd𝐴)) = (♯‘(2nd𝐵)))
54oveq1d 7154 . . . . . . . . . . . 12 ((2nd𝐴) = (2nd𝐵) → ((♯‘(2nd𝐴)) − 1) = ((♯‘(2nd𝐵)) − 1))
65eqcomd 2807 . . . . . . . . . . 11 ((2nd𝐴) = (2nd𝐵) → ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1))
76adantl 485 . . . . . . . . . 10 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1))
8 wlklenvm1 27414 . . . . . . . . . . . 12 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1))
9 wlklenvm1 27414 . . . . . . . . . . . 12 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1))
108, 9eqeqan12rd 2820 . . . . . . . . . . 11 (((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) → ((♯‘(1st𝐵)) = (♯‘(1st𝐴)) ↔ ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1)))
1110adantr 484 . . . . . . . . . 10 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(1st𝐵)) = (♯‘(1st𝐴)) ↔ ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1)))
127, 11mpbird 260 . . . . . . . . 9 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → (♯‘(1st𝐵)) = (♯‘(1st𝐴)))
1312anim2i 619 . . . . . . . 8 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵))) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
1413exp44 441 . . . . . . 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 245 . . . . 5 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (𝐵 ∈ (Walks‘𝐺) → ((2nd𝐴) = (2nd𝐵) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))))
171, 16sylbi 220 . . . 4 (𝐴 ∈ (Walks‘𝐺) → (𝐵 ∈ (Walks‘𝐺) → ((2nd𝐴) = (2nd𝐵) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))))
1817imp31 421 . . 3 (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
19183adant1 1127 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
20 simpl 486 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐺 ∈ USPGraph)
21 simpl 486 . . . . . . 7 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))) → (♯‘(1st𝐴)) ∈ ℕ0)
2220, 21anim12i 615 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → (𝐺 ∈ USPGraph ∧ (♯‘(1st𝐴)) ∈ ℕ0))
23 simpl 486 . . . . . . . 8 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐴 ∈ (Walks‘𝐺))
2423adantl 485 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐴 ∈ (Walks‘𝐺))
25 eqidd 2802 . . . . . . 7 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))) → (♯‘(1st𝐴)) = (♯‘(1st𝐴)))
2624, 25anim12i 615 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = (♯‘(1st𝐴))))
27 simpr 488 . . . . . . . 8 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐵 ∈ (Walks‘𝐺))
2827adantl 485 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐵 ∈ (Walks‘𝐺))
29 simpr 488 . . . . . . 7 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))) → (♯‘(1st𝐵)) = (♯‘(1st𝐴)))
3028, 29anim12i 615 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
31 uspgr2wlkeq2 27439 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (♯‘(1st𝐴)) ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = (♯‘(1st𝐴))) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3222, 26, 30, 31syl3anc 1368 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3332ex 416 . . . 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 1114 . 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  1c1 10531   − cmin 10863  ℕ0cn0 11889  ♯chash 13690  USPGraphcuspgr 26944  Walkscwlks 27389 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-hash 13691  df-word 13862  df-edg 26844  df-uhgr 26854  df-upgr 26878  df-uspgr 26946  df-wlks 27392 This theorem is referenced by:  wlkswwlksf1o  27668
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