Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgra2adedgwlkon Structured version   Visualization version   GIF version

 Description: In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
Hypotheses
Ref Expression
usgra2adedgspth.f 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩}
usgra2adedgspth.p 𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}
Assertion
Ref Expression
usgra2adedgwlkon (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃))

StepHypRef Expression
1 usgrav 25867 . . . . . . 7 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
21adantr 480 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3 usgraedgrnv 25906 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐴𝑉𝐵𝑉))
43ancomd 466 . . . . . . . . 9 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐵𝑉𝐴𝑉))
54adantrr 749 . . . . . . . 8 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐵𝑉𝐴𝑉))
65simprd 478 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐴𝑉)
73adantrr 749 . . . . . . . 8 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴𝑉𝐵𝑉))
87simprd 478 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐵𝑉)
9 usgraedgrnv 25906 . . . . . . . . 9 ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐵𝑉𝐶𝑉))
109adantrl 748 . . . . . . . 8 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐵𝑉𝐶𝑉))
1110simprd 478 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐶𝑉)
126, 8, 113jca 1235 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴𝑉𝐵𝑉𝐶𝑉))
132, 12jca 553 . . . . 5 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))
14 usgra2adedgspthlem1 26139 . . . . 5 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))
15 fvex 6113 . . . . . . 7 (𝐸‘{𝐴, 𝐵}) ∈ V
16 fvex 6113 . . . . . . 7 (𝐸‘{𝐵, 𝐶}) ∈ V
1715, 16pm3.2i 470 . . . . . 6 ((𝐸‘{𝐴, 𝐵}) ∈ V ∧ (𝐸‘{𝐵, 𝐶}) ∈ V)
18 usgra2adedgspth.f . . . . . 6 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩}
19 usgra2adedgspth.p . . . . . 6 𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}
2017, 18, 19constr2wlk 26128 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (((𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}) → 𝐹(𝑉 Walks 𝐸)𝑃))
2113, 14, 20sylc 63 . . . 4 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐹(𝑉 Walks 𝐸)𝑃)
223ex 449 . . . . . . 7 (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝐴𝑉𝐵𝑉)))
239ex 449 . . . . . . 7 (𝑉 USGrph 𝐸 → ({𝐵, 𝐶} ∈ ran 𝐸 → (𝐵𝑉𝐶𝑉)))
2422, 23anim12d 584 . . . . . 6 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))))
25192wlklemA 26084 . . . . . . . 8 (𝐴𝑉 → (𝑃‘0) = 𝐴)
2625adantr 480 . . . . . . 7 ((𝐴𝑉𝐵𝑉) → (𝑃‘0) = 𝐴)
2717, 182trllemA 26080 . . . . . . . . . 10 (#‘𝐹) = 2
2827fveq2i 6106 . . . . . . . . 9 (𝑃‘(#‘𝐹)) = (𝑃‘2)
29192wlklemC 26086 . . . . . . . . 9 (𝐶𝑉 → (𝑃‘2) = 𝐶)
3028, 29syl5eq 2656 . . . . . . . 8 (𝐶𝑉 → (𝑃‘(#‘𝐹)) = 𝐶)
3130adantl 481 . . . . . . 7 ((𝐵𝑉𝐶𝑉) → (𝑃‘(#‘𝐹)) = 𝐶)
3226, 31anim12i 588 . . . . . 6 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶))
3324, 32syl6 34 . . . . 5 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶)))
3433imp 444 . . . 4 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶))
35 3anass 1035 . . . 4 ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶) ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶)))
3621, 34, 35sylanbrc 695 . . 3 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶))
37 prex 4836 . . . . . . 7 {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩} ∈ V
3818, 37eqeltri 2684 . . . . . 6 𝐹 ∈ V
39 tpex 6855 . . . . . . 7 {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩} ∈ V
4019, 39eqeltri 2684 . . . . . 6 𝑃 ∈ V
4138, 40pm3.2i 470 . . . . 5 (𝐹 ∈ V ∧ 𝑃 ∈ V)
4241a1i 11 . . . 4 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
433simpld 474 . . . . 5 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴𝑉)
449simprd 478 . . . . 5 ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐶𝑉)
4543, 44anim12dan 878 . . . 4 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴𝑉𝐶𝑉))
46 iswlkon 26062 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐶𝑉)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶)))
472, 42, 45, 46syl3anc 1318 . . 3 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶)))
4836, 47mpbird 246 . 2 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃)
4948ex 449 1 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   USGrph cusg 25859   Walks cwalk 26026   WalkOn cwlkon 26030 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-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-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-usgra 25862  df-wlk 26036  df-wlkon 26042 This theorem is referenced by:  usg2wlkon  26146
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