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Theorem uspgr2wlkeq2 26777
 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 Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
Assertion
Ref Expression
uspgr2wlkeq2 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))

Proof of Theorem uspgr2wlkeq2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 simpr 473 . . . . . 6 ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁) → (♯‘(1st𝐵)) = 𝑁)
21eqcomd 2819 . . . . 5 ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁) → 𝑁 = (♯‘(1st𝐵)))
323ad2ant3 1158 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → 𝑁 = (♯‘(1st𝐵)))
43adantr 468 . . 3 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝑁 = (♯‘(1st𝐵)))
5 fveq1 6410 . . . . 5 ((2nd𝐴) = (2nd𝐵) → ((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))
65adantl 469 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → ((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))
76ralrimivw 3162 . . 3 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → ∀𝑖 ∈ (0...𝑁)((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))
8 simpl1l 1286 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐺 ∈ USPGraph)
9 simpl 470 . . . . . . 7 ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) → 𝐴 ∈ (Walks‘𝐺))
10 simpl 470 . . . . . . 7 ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁) → 𝐵 ∈ (Walks‘𝐺))
119, 10anim12i 602 . . . . . 6 (((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)))
12113adant1 1153 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)))
1312adantr 468 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)))
14 simpr 473 . . . . . . 7 ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) → (♯‘(1st𝐴)) = 𝑁)
1514eqcomd 2819 . . . . . 6 ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) → 𝑁 = (♯‘(1st𝐴)))
16153ad2ant2 1157 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → 𝑁 = (♯‘(1st𝐴)))
1716adantr 468 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝑁 = (♯‘(1st𝐴)))
18 uspgr2wlkeq 26776 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑖 ∈ (0...𝑁)((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))))
198, 13, 17, 18syl3anc 1483 . . 3 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑖 ∈ (0...𝑁)((2nd𝐴)‘𝑖) = ((2nd𝐵)‘𝑖))))
204, 7, 19mpbir2and 695 . 2 ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐴 = 𝐵)
2120ex 399 1 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1100   = wceq 1637   ∈ wcel 2157  ∀wral 3103  ‘cfv 6104  (class class class)co 6877  1st c1st 7399  2nd c2nd 7400  0cc0 10224  ℕ0cn0 11562  ...cfz 12552  ♯chash 13340  USPGraphcuspgr 26264  Walkscwlks 26726 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 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301 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 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-cda 9278  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-nn 11309  df-2 11367  df-n0 11563  df-xnn0 11633  df-z 11647  df-uz 11908  df-fz 12553  df-fzo 12693  df-hash 13341  df-word 13513  df-edg 26160  df-uhgr 26173  df-upgr 26197  df-uspgr 26266  df-wlks 26729 This theorem is referenced by:  uspgr2wlkeqi  26778  wlknwwlksninjOLD  27022  wlkwwlkinjOLD  27030
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