MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  umgr2adedgwlkonALT Structured version   Visualization version   GIF version

Theorem umgr2adedgwlkonALT 27276
Description: Alternate proof for umgr2adedgwlkon 27275, using umgr2adedgwlk 27274, but with a much longer proof! In a multigraph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
umgr2adedgwlk.e 𝐸 = (Edg‘𝐺)
umgr2adedgwlk.i 𝐼 = (iEdg‘𝐺)
umgr2adedgwlk.f 𝐹 = ⟨“𝐽𝐾”⟩
umgr2adedgwlk.p 𝑃 = ⟨“𝐴𝐵𝐶”⟩
umgr2adedgwlk.g (𝜑𝐺 ∈ UMGraph)
umgr2adedgwlk.a (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
umgr2adedgwlk.j (𝜑 → (𝐼𝐽) = {𝐴, 𝐵})
umgr2adedgwlk.k (𝜑 → (𝐼𝐾) = {𝐵, 𝐶})
Assertion
Ref Expression
umgr2adedgwlkonALT (𝜑𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)

Proof of Theorem umgr2adedgwlkonALT
StepHypRef Expression
1 umgr2adedgwlk.e . . . 4 𝐸 = (Edg‘𝐺)
2 umgr2adedgwlk.i . . . 4 𝐼 = (iEdg‘𝐺)
3 umgr2adedgwlk.f . . . 4 𝐹 = ⟨“𝐽𝐾”⟩
4 umgr2adedgwlk.p . . . 4 𝑃 = ⟨“𝐴𝐵𝐶”⟩
5 umgr2adedgwlk.g . . . 4 (𝜑𝐺 ∈ UMGraph)
6 umgr2adedgwlk.a . . . 4 (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
7 umgr2adedgwlk.j . . . 4 (𝜑 → (𝐼𝐽) = {𝐴, 𝐵})
8 umgr2adedgwlk.k . . . 4 (𝜑 → (𝐼𝐾) = {𝐵, 𝐶})
91, 2, 3, 4, 5, 6, 7, 8umgr2adedgwlk 27274 . . 3 (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))))
10 simp1 1172 . . . 4 ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → 𝐹(Walks‘𝐺)𝑃)
11 id 22 . . . . . . 7 ((𝑃‘0) = 𝐴 → (𝑃‘0) = 𝐴)
1211eqcoms 2833 . . . . . 6 (𝐴 = (𝑃‘0) → (𝑃‘0) = 𝐴)
13123ad2ant1 1169 . . . . 5 ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → (𝑃‘0) = 𝐴)
14133ad2ant3 1171 . . . 4 ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘0) = 𝐴)
15 fveq2 6433 . . . . . . . . . . . 12 (2 = (♯‘𝐹) → (𝑃‘2) = (𝑃‘(♯‘𝐹)))
1615eqcoms 2833 . . . . . . . . . . 11 ((♯‘𝐹) = 2 → (𝑃‘2) = (𝑃‘(♯‘𝐹)))
1716eqeq1d 2827 . . . . . . . . . 10 ((♯‘𝐹) = 2 → ((𝑃‘2) = 𝐶 ↔ (𝑃‘(♯‘𝐹)) = 𝐶))
1817biimpcd 241 . . . . . . . . 9 ((𝑃‘2) = 𝐶 → ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = 𝐶))
1918eqcoms 2833 . . . . . . . 8 (𝐶 = (𝑃‘2) → ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = 𝐶))
20193ad2ant3 1171 . . . . . . 7 ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = 𝐶))
2120com12 32 . . . . . 6 ((♯‘𝐹) = 2 → ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → (𝑃‘(♯‘𝐹)) = 𝐶))
2221a1i 11 . . . . 5 (𝐹(Walks‘𝐺)𝑃 → ((♯‘𝐹) = 2 → ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → (𝑃‘(♯‘𝐹)) = 𝐶)))
23223imp 1143 . . . 4 ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘(♯‘𝐹)) = 𝐶)
2410, 14, 233jca 1164 . . 3 ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶))
259, 24syl 17 . 2 (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶))
26 3anass 1122 . . . . . 6 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (𝐺 ∈ UMGraph ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
275, 6, 26sylanbrc 580 . . . . 5 (𝜑 → (𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
281umgr2adedgwlklem 27273 . . . . 5 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴𝐵𝐵𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
29 3simpb 1186 . . . . . 6 ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
3029adantl 475 . . . . 5 (((𝐴𝐵𝐵𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
3127, 28, 303syl 18 . . . 4 (𝜑 → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
32 3anass 1122 . . . 4 ((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ↔ (𝐺 ∈ UMGraph ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
335, 31, 32sylanbrc 580 . . 3 (𝜑 → (𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
34 s2cli 14001 . . . . 5 ⟨“𝐽𝐾”⟩ ∈ Word V
353, 34eqeltri 2902 . . . 4 𝐹 ∈ Word V
36 s3cli 14002 . . . . 5 ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
374, 36eqeltri 2902 . . . 4 𝑃 ∈ Word V
3835, 37pm3.2i 464 . . 3 (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)
39 id 22 . . . . . 6 ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
40393adant1 1166 . . . . 5 ((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
4140anim1i 610 . . . 4 (((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)))
42 eqid 2825 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
4342iswlkon 26954 . . . 4 (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)) → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶)))
4441, 43syl 17 . . 3 (((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)) → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶)))
4533, 38, 44sylancl 582 . 2 (𝜑 → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶)))
4625, 45mpbird 249 1 (𝜑𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wne 2999  Vcvv 3414  {cpr 4399   class class class wbr 4873  cfv 6123  (class class class)co 6905  0cc0 10252  1c1 10253  2c2 11406  chash 13410  Word cword 13574  ⟨“cs2 13962  ⟨“cs3 13963  Vtxcvtx 26294  iEdgciedg 26295  Edgcedg 26345  UMGraphcumgr 26379  Walkscwlks 26894  WalksOncwlkson 26895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-n0 11619  df-z 11705  df-uz 11969  df-fz 12620  df-fzo 12761  df-hash 13411  df-word 13575  df-concat 13631  df-s1 13656  df-s2 13969  df-s3 13970  df-edg 26346  df-umgr 26381  df-wlks 26897  df-wlkson 26898
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator