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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  0cn0 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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