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Theorem uspgr2wlkeqi 26772
 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 26752 . . . . 5 (𝐴 ∈ (Walks‘𝐺) ↔ (1st𝐴)(Walks‘𝐺)(2nd𝐴))
2 wlkcpr 26752 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) ↔ (1st𝐵)(Walks‘𝐺)(2nd𝐵))
3 wlkcl 26739 . . . . . . 7 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (♯‘(1st𝐴)) ∈ ℕ0)
4 fveq2 6408 . . . . . . . . . . . . 13 ((2nd𝐴) = (2nd𝐵) → (♯‘(2nd𝐴)) = (♯‘(2nd𝐵)))
54oveq1d 6889 . . . . . . . . . . . 12 ((2nd𝐴) = (2nd𝐵) → ((♯‘(2nd𝐴)) − 1) = ((♯‘(2nd𝐵)) − 1))
65eqcomd 2812 . . . . . . . . . . 11 ((2nd𝐴) = (2nd𝐵) → ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1))
76adantl 469 . . . . . . . . . 10 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1))
8 wlklenvm1 26745 . . . . . . . . . . . 12 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1))
9 wlklenvm1 26745 . . . . . . . . . . . 12 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1))
108, 9eqeqan12rd 2824 . . . . . . . . . . 11 (((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) → ((♯‘(1st𝐵)) = (♯‘(1st𝐴)) ↔ ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1)))
1110adantr 468 . . . . . . . . . 10 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(1st𝐵)) = (♯‘(1st𝐴)) ↔ ((♯‘(2nd𝐵)) − 1) = ((♯‘(2nd𝐴)) − 1)))
127, 11mpbird 248 . . . . . . . . 9 ((((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → (♯‘(1st𝐵)) = (♯‘(1st𝐴)))
1312anim2i 605 . . . . . . . 8 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (((1st𝐴)(Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵))) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
1413exp44 426 . . . . . . 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 233 . . . . 5 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (𝐵 ∈ (Walks‘𝐺) → ((2nd𝐴) = (2nd𝐵) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))))
171, 16sylbi 208 . . . 4 (𝐴 ∈ (Walks‘𝐺) → (𝐵 ∈ (Walks‘𝐺) → ((2nd𝐴) = (2nd𝐵) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))))
1817imp31 406 . . 3 (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
19183adant1 1153 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
20 simpl 470 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐺 ∈ USPGraph)
21 simpl 470 . . . . . . 7 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))) → (♯‘(1st𝐴)) ∈ ℕ0)
2220, 21anim12i 602 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → (𝐺 ∈ USPGraph ∧ (♯‘(1st𝐴)) ∈ ℕ0))
23 simpl 470 . . . . . . . 8 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐴 ∈ (Walks‘𝐺))
2423adantl 469 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐴 ∈ (Walks‘𝐺))
25 eqidd 2807 . . . . . . 7 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))) → (♯‘(1st𝐴)) = (♯‘(1st𝐴)))
2624, 25anim12i 602 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = (♯‘(1st𝐴))))
27 simpr 473 . . . . . . . 8 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐵 ∈ (Walks‘𝐺))
2827adantl 469 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐵 ∈ (Walks‘𝐺))
29 simpr 473 . . . . . . 7 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))) → (♯‘(1st𝐵)) = (♯‘(1st𝐴)))
3028, 29anim12i 602 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴))))
31 uspgr2wlkeq2 26771 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (♯‘(1st𝐴)) ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = (♯‘(1st𝐴))) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3222, 26, 30, 31syl3anc 1483 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1st𝐴)) ∈ ℕ0 ∧ (♯‘(1st𝐵)) = (♯‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3332ex 399 . . . 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 1138 . 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 197   ∧ wa 384   ∧ w3a 1100   = wceq 1637   ∈ wcel 2156   class class class wbr 4844  ‘cfv 6101  (class class class)co 6874  1st c1st 7396  2nd c2nd 7397  1c1 10222   − cmin 10551  ℕ0cn0 11559  ♯chash 13337  USPGraphcuspgr 26258  Walkscwlks 26720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ifp 1079  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-map 8094  df-pm 8095  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-card 9048  df-cda 9275  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-nn 11306  df-2 11364  df-n0 11560  df-xnn0 11630  df-z 11644  df-uz 11905  df-fz 12550  df-fzo 12690  df-hash 13338  df-word 13510  df-edg 26154  df-uhgr 26167  df-upgr 26191  df-uspgr 26260  df-wlks 26723 This theorem is referenced by:  wlkswwlksf1o  27006
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