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Theorem rusgraprop4 26471
 Description: The properties of a k-regular undirected simple graph expressed with trailing edges of walks (as words). (Contributed by Alexander van der Vekens, 2-Aug-2018.)
Assertion
Ref Expression
rusgraprop4 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ Word 𝑉(𝑝 ≠ ∅ → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾)))
Distinct variable groups:   𝑛,𝐸,𝑝   𝐾,𝑝   𝑛,𝑉,𝑝
Allowed substitution hint:   𝐾(𝑛)

Proof of Theorem rusgraprop4
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 rusgraprop3 26470 . 2 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾))
2 simp1 1054 . . 3 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾) → 𝑉 USGrph 𝐸)
3 simp2 1055 . . 3 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾) → 𝐾 ∈ ℕ0)
4 lswcl 13208 . . . . . . . . . . . 12 ((𝑝 ∈ Word 𝑉𝑝 ≠ ∅) → ( lastS ‘𝑝) ∈ 𝑉)
54adantll 746 . . . . . . . . . . 11 (((𝑉 USGrph 𝐸𝑝 ∈ Word 𝑉) ∧ 𝑝 ≠ ∅) → ( lastS ‘𝑝) ∈ 𝑉)
6 preq1 4212 . . . . . . . . . . . . . . . 16 (𝑣 = ( lastS ‘𝑝) → {𝑣, 𝑛} = {( lastS ‘𝑝), 𝑛})
76eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑣 = ( lastS ‘𝑝) → ({𝑣, 𝑛} ∈ ran 𝐸 ↔ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸))
87rabbidv 3164 . . . . . . . . . . . . . 14 (𝑣 = ( lastS ‘𝑝) → {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸} = {𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸})
98fveq2d 6107 . . . . . . . . . . . . 13 (𝑣 = ( lastS ‘𝑝) → (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}))
109eqeq1d 2612 . . . . . . . . . . . 12 (𝑣 = ( lastS ‘𝑝) → ((#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 ↔ (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾))
1110rspcv 3278 . . . . . . . . . . 11 (( lastS ‘𝑝) ∈ 𝑉 → (∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾))
125, 11syl 17 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑝 ∈ Word 𝑉) ∧ 𝑝 ≠ ∅) → (∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾))
1312ex 449 . . . . . . . . 9 ((𝑉 USGrph 𝐸𝑝 ∈ Word 𝑉) → (𝑝 ≠ ∅ → (∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾)))
1413com23 84 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑝 ∈ Word 𝑉) → (∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝑝 ≠ ∅ → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾)))
1514ex 449 . . . . . . 7 (𝑉 USGrph 𝐸 → (𝑝 ∈ Word 𝑉 → (∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝑝 ≠ ∅ → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾))))
1615com23 84 . . . . . 6 (𝑉 USGrph 𝐸 → (∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝑝 ∈ Word 𝑉 → (𝑝 ≠ ∅ → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾))))
1716a1d 25 . . . . 5 (𝑉 USGrph 𝐸 → (𝐾 ∈ ℕ0 → (∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝑝 ∈ Word 𝑉 → (𝑝 ≠ ∅ → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾)))))
18173imp1 1272 . . . 4 (((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾) ∧ 𝑝 ∈ Word 𝑉) → (𝑝 ≠ ∅ → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾))
1918ralrimiva 2949 . . 3 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾) → ∀𝑝 ∈ Word 𝑉(𝑝 ≠ ∅ → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾))
202, 3, 193jca 1235 . 2 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑣𝑉 (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾) → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ Word 𝑉(𝑝 ≠ ∅ → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾)))
211, 20syl 17 1 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ Word 𝑉(𝑝 ≠ ∅ → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  ∅c0 3874  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  ‘cfv 5804  ℕ0cn0 11169  #chash 12979  Word cword 13146   lastS clsw 13147   USGrph cusg 25859   RegUSGrph crusgra 26450 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-2o 7448  df-oadd 7451  df-er 7629  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-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-usgra 25862  df-nbgra 25949  df-vdgr 26421  df-rgra 26451  df-rusgra 26452 This theorem is referenced by: (None)
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