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Theorem cusgraisrusgra 26465
 Description: A complete undirected simple graph with n vertices (at least one) is (n-1)-regular. (Contributed by Alexander van der Vekens, 10-Jul-2018.)
Assertion
Ref Expression
cusgraisrusgra ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ⟨𝑉, 𝐸⟩ RegUSGrph ((#‘𝑉) − 1))

Proof of Theorem cusgraisrusgra
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nbcusgra 25992 . . . 4 ((𝑉 ComplUSGrph 𝐸𝑣𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣}))
21ralrimiva 2949 . . 3 (𝑉 ComplUSGrph 𝐸 → ∀𝑣𝑉 (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣}))
3 cusisusgra 25987 . . . . . . . 8 (𝑉 ComplUSGrph 𝐸𝑉 USGrph 𝐸)
43adantr 480 . . . . . . 7 ((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 𝑉 USGrph 𝐸)
54adantr 480 . . . . . 6 (((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ ∀𝑣𝑉 (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣})) → 𝑉 USGrph 𝐸)
6 hashnncl 13018 . . . . . . . . 9 (𝑉 ∈ Fin → ((#‘𝑉) ∈ ℕ ↔ 𝑉 ≠ ∅))
7 nnm1nn0 11211 . . . . . . . . 9 ((#‘𝑉) ∈ ℕ → ((#‘𝑉) − 1) ∈ ℕ0)
86, 7syl6bir 243 . . . . . . . 8 (𝑉 ∈ Fin → (𝑉 ≠ ∅ → ((#‘𝑉) − 1) ∈ ℕ0))
98imp 444 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) − 1) ∈ ℕ0)
109ad2antlr 759 . . . . . 6 (((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ ∀𝑣𝑉 (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣})) → ((#‘𝑉) − 1) ∈ ℕ0)
114anim1i 590 . . . . . . . . . . . 12 (((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ 𝑣𝑉) → (𝑉 USGrph 𝐸𝑣𝑉))
1211adantr 480 . . . . . . . . . . 11 ((((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ 𝑣𝑉) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣})) → (𝑉 USGrph 𝐸𝑣𝑉))
13 hashnbgravdg 26440 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑣𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) = ((𝑉 VDeg 𝐸)‘𝑣))
1412, 13syl 17 . . . . . . . . . 10 ((((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ 𝑣𝑉) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣})) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) = ((𝑉 VDeg 𝐸)‘𝑣))
15 fveq2 6103 . . . . . . . . . . 11 ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣}) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) = (#‘(𝑉 ∖ {𝑣})))
16 simprl 790 . . . . . . . . . . . 12 ((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 𝑉 ∈ Fin)
17 hashdifsn 13063 . . . . . . . . . . . 12 ((𝑉 ∈ Fin ∧ 𝑣𝑉) → (#‘(𝑉 ∖ {𝑣})) = ((#‘𝑉) − 1))
1816, 17sylan 487 . . . . . . . . . . 11 (((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ 𝑣𝑉) → (#‘(𝑉 ∖ {𝑣})) = ((#‘𝑉) − 1))
1915, 18sylan9eqr 2666 . . . . . . . . . 10 ((((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ 𝑣𝑉) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣})) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) = ((#‘𝑉) − 1))
2014, 19eqtr3d 2646 . . . . . . . . 9 ((((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ 𝑣𝑉) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣})) → ((𝑉 VDeg 𝐸)‘𝑣) = ((#‘𝑉) − 1))
2120ex 449 . . . . . . . 8 (((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ 𝑣𝑉) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣}) → ((𝑉 VDeg 𝐸)‘𝑣) = ((#‘𝑉) − 1)))
2221ralimdva 2945 . . . . . . 7 ((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → (∀𝑣𝑉 (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣}) → ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = ((#‘𝑉) − 1)))
2322imp 444 . . . . . 6 (((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ ∀𝑣𝑉 (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣})) → ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = ((#‘𝑉) − 1))
24 usgrav 25867 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
253, 24syl 17 . . . . . . . . . 10 (𝑉 ComplUSGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2625adantr 480 . . . . . . . . 9 ((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
27 ovex 6577 . . . . . . . . . 10 ((#‘𝑉) − 1) ∈ V
2827a1i 11 . . . . . . . . 9 ((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → ((#‘𝑉) − 1) ∈ V)
29 df-3an 1033 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ ((#‘𝑉) − 1) ∈ V) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ ((#‘𝑉) − 1) ∈ V))
3026, 28, 29sylanbrc 695 . . . . . . . 8 ((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ ((#‘𝑉) − 1) ∈ V))
3130adantr 480 . . . . . . 7 (((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ ∀𝑣𝑉 (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣})) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ ((#‘𝑉) − 1) ∈ V))
32 isrusgra 26454 . . . . . . 7 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ ((#‘𝑉) − 1) ∈ V) → (⟨𝑉, 𝐸⟩ RegUSGrph ((#‘𝑉) − 1) ↔ (𝑉 USGrph 𝐸 ∧ ((#‘𝑉) − 1) ∈ ℕ0 ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = ((#‘𝑉) − 1))))
3331, 32syl 17 . . . . . 6 (((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ ∀𝑣𝑉 (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣})) → (⟨𝑉, 𝐸⟩ RegUSGrph ((#‘𝑉) − 1) ↔ (𝑉 USGrph 𝐸 ∧ ((#‘𝑉) − 1) ∈ ℕ0 ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = ((#‘𝑉) − 1))))
345, 10, 23, 33mpbir3and 1238 . . . . 5 (((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ ∀𝑣𝑉 (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣})) → ⟨𝑉, 𝐸⟩ RegUSGrph ((#‘𝑉) − 1))
3534expcom 450 . . . 4 (∀𝑣𝑉 (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣}) → ((𝑉 ComplUSGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → ⟨𝑉, 𝐸⟩ RegUSGrph ((#‘𝑉) − 1)))
3635expd 451 . . 3 (∀𝑣𝑉 (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = (𝑉 ∖ {𝑣}) → (𝑉 ComplUSGrph 𝐸 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ⟨𝑉, 𝐸⟩ RegUSGrph ((#‘𝑉) − 1))))
372, 36mpcom 37 . 2 (𝑉 ComplUSGrph 𝐸 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ⟨𝑉, 𝐸⟩ RegUSGrph ((#‘𝑉) − 1)))
38373impib 1254 1 ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ⟨𝑉, 𝐸⟩ RegUSGrph ((#‘𝑉) − 1))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  #chash 12979   USGrph cusg 25859   Neighbors cnbgra 25946   ComplUSGrph ccusgra 25947   VDeg cvdg 26420   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-hash 12980  df-usgra 25862  df-nbgra 25949  df-cusgra 25950  df-vdgr 26421  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  cusgraiffrusgra  26467
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