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Theorem elwwlks2ons3 29805
Description: For each walk of length 2 between two vertices, there is a third vertex in the middle of the walk. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
Hypothesis
Ref Expression
wwlks2onv.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
elwwlks2ons3 (𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟚“𝐎𝑏𝐶”⟩ ∧ ⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)))
Distinct variable groups:   𝐎,𝑏   𝐶,𝑏   𝐺,𝑏   𝑉,𝑏   𝑊,𝑏

Proof of Theorem elwwlks2ons3
StepHypRef Expression
1 id 22 . . . 4 (𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) → 𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶))
2 wwlks2onv.v . . . . 5 𝑉 = (Vtx‘𝐺)
32elwwlks2ons3im 29804 . . . 4 (𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) → (𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
4 anass 467 . . . 4 (((𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) ↔ (𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ∧ (𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
51, 3, 4sylanbrc 581 . . 3 (𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) → ((𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉))
6 simpr 483 . . . 4 (((𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊‘1) ∈ 𝑉)
7 s3eq2 14848 . . . . . 6 (𝑏 = (𝑊‘1) → ⟚“𝐎𝑏𝐶”⟩ = ⟚“𝐎(𝑊‘1)𝐶”⟩)
8 eqeq2 2737 . . . . . . 7 (⟚“𝐎𝑏𝐶”⟩ = ⟚“𝐎(𝑊‘1)𝐶”⟩ → (𝑊 = ⟚“𝐎𝑏𝐶”⟩ ↔ 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩))
9 eleq1 2813 . . . . . . 7 (⟚“𝐎𝑏𝐶”⟩ = ⟚“𝐎(𝑊‘1)𝐶”⟩ → (⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ↔ ⟚“𝐎(𝑊‘1)𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)))
108, 9anbi12d 630 . . . . . 6 (⟚“𝐎𝑏𝐶”⟩ = ⟚“𝐎(𝑊‘1)𝐶”⟩ → ((𝑊 = ⟚“𝐎𝑏𝐶”⟩ ∧ ⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩ ∧ ⟚“𝐎(𝑊‘1)𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶))))
117, 10syl 17 . . . . 5 (𝑏 = (𝑊‘1) → ((𝑊 = ⟚“𝐎𝑏𝐶”⟩ ∧ ⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩ ∧ ⟚“𝐎(𝑊‘1)𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶))))
1211adantl 480 . . . 4 ((((𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) ∧ 𝑏 = (𝑊‘1)) → ((𝑊 = ⟚“𝐎𝑏𝐶”⟩ ∧ ⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩ ∧ ⟚“𝐎(𝑊‘1)𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶))))
13 simpr 483 . . . . . 6 ((𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩) → 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩)
14 eleq1 2813 . . . . . . 7 (𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩ → (𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ↔ ⟚“𝐎(𝑊‘1)𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)))
1514biimpac 477 . . . . . 6 ((𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩) → ⟚“𝐎(𝑊‘1)𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶))
1613, 15jca 510 . . . . 5 ((𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩) → (𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩ ∧ ⟚“𝐎(𝑊‘1)𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)))
1716adantr 479 . . . 4 (((𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩ ∧ ⟚“𝐎(𝑊‘1)𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)))
186, 12, 17rspcedvd 3605 . . 3 (((𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟚“𝐎(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟚“𝐎𝑏𝐶”⟩ ∧ ⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)))
195, 18syl 17 . 2 (𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟚“𝐎𝑏𝐶”⟩ ∧ ⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)))
20 eleq1 2813 . . . . 5 (⟚“𝐎𝑏𝐶”⟩ = 𝑊 → (⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)))
2120eqcoms 2733 . . . 4 (𝑊 = ⟚“𝐎𝑏𝐶”⟩ → (⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)))
2221biimpa 475 . . 3 ((𝑊 = ⟚“𝐎𝑏𝐶”⟩ ∧ ⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶))
2322rexlimivw 3141 . 2 (∃𝑏 ∈ 𝑉 (𝑊 = ⟚“𝐎𝑏𝐶”⟩ ∧ ⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶))
2419, 23impbii 208 1 (𝑊 ∈ (𝐎(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟚“𝐎𝑏𝐶”⟩ ∧ ⟚“𝐎𝑏𝐶”⟩ ∈ (𝐎(2 WWalksNOn 𝐺)𝐶)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  âˆƒwrex 3060  â€˜cfv 6543  (class class class)co 7413  1c1 11134  2c2 12292  âŸšâ€œcs3 14820  Vtxcvtx 28848   WWalksNOn cwwlksnon 29677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-n0 12498  df-z 12584  df-uz 12848  df-fz 13512  df-fzo 13655  df-hash 14317  df-word 14492  df-concat 14548  df-s1 14573  df-s2 14826  df-s3 14827  df-wwlks 29680  df-wwlksn 29681  df-wwlksnon 29682
This theorem is referenced by:  elwwlks2on  29809  frgr2wwlk1  30178
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