 Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uspgr2wlkeqi Structured version   Visualization version   GIF version

Theorem uspgr2wlkeqi 40856
 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 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐴 = 𝐵)

Proof of Theorem uspgr2wlkeqi
StepHypRef Expression
1 1wlkcpr 40833 . . . . 5 (𝐴 ∈ (1Walks‘𝐺) ↔ (1st𝐴)(1Walks‘𝐺)(2nd𝐴))
2 1wlkcpr 40833 . . . . . 6 (𝐵 ∈ (1Walks‘𝐺) ↔ (1st𝐵)(1Walks‘𝐺)(2nd𝐵))
3 1wlkcl 40820 . . . . . . 7 ((1st𝐴)(1Walks‘𝐺)(2nd𝐴) → (#‘(1st𝐴)) ∈ ℕ0)
4 fveq2 6103 . . . . . . . . . . . . 13 ((2nd𝐴) = (2nd𝐵) → (#‘(2nd𝐴)) = (#‘(2nd𝐵)))
54oveq1d 6564 . . . . . . . . . . . 12 ((2nd𝐴) = (2nd𝐵) → ((#‘(2nd𝐴)) − 1) = ((#‘(2nd𝐵)) − 1))
65eqcomd 2616 . . . . . . . . . . 11 ((2nd𝐴) = (2nd𝐵) → ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1))
76adantl 481 . . . . . . . . . 10 ((((1st𝐴)(1Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(1Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1))
8 1wlklenvm1 40826 . . . . . . . . . . . 12 ((1st𝐵)(1Walks‘𝐺)(2nd𝐵) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))
9 1wlklenvm1 40826 . . . . . . . . . . . 12 ((1st𝐴)(1Walks‘𝐺)(2nd𝐴) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1))
108, 9eqeqan12rd 2628 . . . . . . . . . . 11 (((1st𝐴)(1Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(1Walks‘𝐺)(2nd𝐵)) → ((#‘(1st𝐵)) = (#‘(1st𝐴)) ↔ ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1)))
1110adantr 480 . . . . . . . . . 10 ((((1st𝐴)(1Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(1Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(1st𝐵)) = (#‘(1st𝐴)) ↔ ((#‘(2nd𝐵)) − 1) = ((#‘(2nd𝐴)) − 1)))
127, 11mpbird 246 . . . . . . . . 9 ((((1st𝐴)(1Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(1Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵)) → (#‘(1st𝐵)) = (#‘(1st𝐴)))
1312anim2i 591 . . . . . . . 8 (((#‘(1st𝐴)) ∈ ℕ0 ∧ (((1st𝐴)(1Walks‘𝐺)(2nd𝐴) ∧ (1st𝐵)(1Walks‘𝐺)(2nd𝐵)) ∧ (2nd𝐴) = (2nd𝐵))) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))
1413exp44 639 . . . . . . 7 ((#‘(1st𝐴)) ∈ ℕ0 → ((1st𝐴)(1Walks‘𝐺)(2nd𝐴) → ((1st𝐵)(1Walks‘𝐺)(2nd𝐵) → ((2nd𝐴) = (2nd𝐵) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))))))
153, 14mpcom 37 . . . . . 6 ((1st𝐴)(1Walks‘𝐺)(2nd𝐴) → ((1st𝐵)(1Walks‘𝐺)(2nd𝐵) → ((2nd𝐴) = (2nd𝐵) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))))
162, 15syl5bi 231 . . . . 5 ((1st𝐴)(1Walks‘𝐺)(2nd𝐴) → (𝐵 ∈ (1Walks‘𝐺) → ((2nd𝐴) = (2nd𝐵) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))))
171, 16sylbi 206 . . . 4 (𝐴 ∈ (1Walks‘𝐺) → (𝐵 ∈ (1Walks‘𝐺) → ((2nd𝐴) = (2nd𝐵) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))))
1817imp31 447 . . 3 (((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))
19183adant1 1072 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))
20 simpl 472 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) → 𝐺 ∈ USPGraph )
21 simpl 472 . . . . . . 7 (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → (#‘(1st𝐴)) ∈ ℕ0)
2220, 21anim12i 588 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) ∧ ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → (𝐺 ∈ USPGraph ∧ (#‘(1st𝐴)) ∈ ℕ0))
23 simpl 472 . . . . . . . 8 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → 𝐴 ∈ (1Walks‘𝐺))
2423adantl 481 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) → 𝐴 ∈ (1Walks‘𝐺))
25 eqidd 2611 . . . . . . 7 (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → (#‘(1st𝐴)) = (#‘(1st𝐴)))
2624, 25anim12i 588 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) ∧ ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝐴)) = (#‘(1st𝐴))))
27 simpr 476 . . . . . . . 8 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → 𝐵 ∈ (1Walks‘𝐺))
2827adantl 481 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) → 𝐵 ∈ (1Walks‘𝐺))
29 simpr 476 . . . . . . 7 (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → (#‘(1st𝐵)) = (#‘(1st𝐴)))
3028, 29anim12i 588 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) ∧ ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))))
31 uspgr2wlkeq2 40855 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (#‘(1st𝐴)) ∈ ℕ0) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝐴)) = (#‘(1st𝐴))) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3222, 26, 30, 31syl3anc 1318 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) ∧ ((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴)))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))
3332ex 449 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) → (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵)))
3433com23 84 . . 3 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) → ((2nd𝐴) = (2nd𝐵) → (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → 𝐴 = 𝐵)))
35343impia 1253 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → (((#‘(1st𝐴)) ∈ ℕ0 ∧ (#‘(1st𝐵)) = (#‘(1st𝐴))) → 𝐴 = 𝐵))
3619, 35mpd 15 1 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  1c1 9816   − cmin 10145  ℕ0cn0 11169  #chash 12979   USPGraph cuspgr 40378  1Walksc1wlks 40796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-1wlks 40800  df-wlks 40801 This theorem is referenced by:  1wlkpwwlkf1ouspgr  41076
 Copyright terms: Public domain W3C validator