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Theorem usg2spthonot 26415
 Description: A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
Assertion
Ref Expression
usg2spthonot ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))

Proof of Theorem usg2spthonot
StepHypRef Expression
1 ne0i 3880 . . . . 5 (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ≠ ∅)
2 2spontn0vne 26414 . . . . 5 ((𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ≠ ∅ → 𝐴𝐶)
31, 2syl 17 . . . 4 (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → 𝐴𝐶)
4 simpl 472 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝑉 USGrph 𝐸)
54adantl 481 . . . . . . . . 9 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → 𝑉 USGrph 𝐸)
6 3simpb 1052 . . . . . . . . . . 11 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐴𝑉𝐶𝑉))
76adantl 481 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴𝑉𝐶𝑉))
87adantl 481 . . . . . . . . 9 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (𝐴𝑉𝐶𝑉))
9 simpl 472 . . . . . . . . 9 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → 𝐴𝐶)
10 2pthwlkonot 26412 . . . . . . . . 9 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) = (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))
115, 8, 9, 10syl3anc 1318 . . . . . . . 8 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) = (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))
1211eleq2d 2673 . . . . . . 7 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)))
139adantr 480 . . . . . . . . 9 (((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) → 𝐴𝐶)
14 usg2wlkonot 26410 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
1514adantl 481 . . . . . . . . . 10 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
1615biimpa 500 . . . . . . . . 9 (((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
17 3anass 1035 . . . . . . . . 9 ((𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ (𝐴𝐶 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
1813, 16, 17sylanbrc 695 . . . . . . . 8 (((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) → (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
1918ex 449 . . . . . . 7 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) → (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
2012, 19sylbid 229 . . . . . 6 ((𝐴𝐶 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
2120ex 449 . . . . 5 (𝐴𝐶 → ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
2221com23 84 . . . 4 (𝐴𝐶 → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
233, 22mpcom 37 . . 3 (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
2423com12 32 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
2514bicomd 212 . . . . 5 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)))
2625anbi2d 736 . . . 4 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴𝐶 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (𝐴𝐶 ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))))
2717, 26syl5bb 271 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ (𝐴𝐶 ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))))
284adantr 480 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → 𝑉 USGrph 𝐸)
297adantr 480 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → (𝐴𝑉𝐶𝑉))
30 simpr 476 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → 𝐴𝐶)
3110eqcomd 2616 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) = (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))
3228, 29, 30, 31syl3anc 1318 . . . . . 6 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) = (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))
3332eleq2d 2673 . . . . 5 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)))
3433biimpd 218 . . . 4 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ 𝐴𝐶) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)))
3534expimpd 627 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴𝐶 ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)))
3627, 35sylbid 229 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)))
3724, 36impbid 201 1 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  {cpr 4127  ⟨cotp 4133   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   2WalksOnOt c2wlkonot 26382   2SPathOnOt c2pthonot 26384 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-ot 4134  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-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2wlkonot 26385  df-2spthonot 26387 This theorem is referenced by: (None)
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