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 Description: In an undirected simple graph, two adjacent edges form a simple path of length 2. (Contributed by Alexander van der Vekens, 1-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
usgra2adedgspth ((𝑉 USGrph 𝐸𝐴𝐶) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐹(𝑉 SPaths 𝐸)𝑃))

StepHypRef Expression
1 usgrav 25867 . . . 4 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
21ad2antrr 758 . . 3 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3 usgraedgrnv 25906 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐴𝑉𝐵𝑉))
43adantrr 749 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴𝑉𝐵𝑉))
54simpld 474 . . . . 5 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐴𝑉)
64simprd 478 . . . . 5 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐵𝑉)
7 usgraedgrnv 25906 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐵𝑉𝐶𝑉))
87adantrl 748 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐵𝑉𝐶𝑉))
98simprd 478 . . . . 5 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐶𝑉)
105, 6, 93jca 1235 . . . 4 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴𝑉𝐵𝑉𝐶𝑉))
1110adantlr 747 . . 3 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴𝑉𝐵𝑉𝐶𝑉))
12 usgraedgrn 25910 . . . . 5 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴𝐵)
1312ad2ant2r 779 . . . 4 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐴𝐵)
14 simplr 788 . . . 4 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐴𝐶)
15 usgraedgrn 25910 . . . . 5 ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐵𝐶)
1615ad2ant2rl 781 . . . 4 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐵𝐶)
1713, 14, 163jca 1235 . . 3 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴𝐵𝐴𝐶𝐵𝐶))
18 usgra2adedgspthlem2 26140 . . 3 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}) ∧ (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))
19 fvex 6113 . . . . . 6 (𝐸‘{𝐴, 𝐵}) ∈ V
20 fvex 6113 . . . . . 6 (𝐸‘{𝐵, 𝐶}) ∈ V
2119, 20pm3.2i 470 . . . . 5 ((𝐸‘{𝐴, 𝐵}) ∈ V ∧ (𝐸‘{𝐵, 𝐶}) ∈ V)
22 usgra2adedgspth.f . . . . 5 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩}
23 usgra2adedgspth.p . . . . 5 𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}
2421, 22, 23constr2spth 26130 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}) ∧ (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}) → 𝐹(𝑉 SPaths 𝐸)𝑃))
2524imp 444 . . 3 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ ((𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}) ∧ (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶})) → 𝐹(𝑉 SPaths 𝐸)𝑃)
262, 11, 17, 18, 25syl31anc 1321 . 2 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐹(𝑉 SPaths 𝐸)𝑃)
2726ex 449 1 ((𝑉 USGrph 𝐸𝐴𝐶) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐹(𝑉 SPaths 𝐸)𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  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   USGrph cusg 25859   SPaths cspath 26029 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-trail 26037  df-spth 26039 This theorem is referenced by: (None)
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