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Theorem fusgrmaxsize 40672
Description: The maximum size of a finite simple graph with 𝑛 vertices is (((𝑛 − 1)∗𝑛) / 2). See statement in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 14-Nov-2020.)
Hypotheses
Ref Expression
fusgrmaxsize.v 𝑉 = (Vtx‘𝐺)
fusgrmaxsize.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
fusgrmaxsize (𝐺 ∈ FinUSGraph → (#‘𝐸) ≤ ((#‘𝑉)C2))

Proof of Theorem fusgrmaxsize
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 fusgrmaxsize.v . . 3 𝑉 = (Vtx‘𝐺)
21isfusgr 40529 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3 cusgrexg 40655 . . . 4 (𝑉 ∈ Fin → ∃𝑒𝑉, 𝑒⟩ ∈ ComplUSGraph)
43adantl 481 . . 3 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → ∃𝑒𝑉, 𝑒⟩ ∈ ComplUSGraph)
5 fusgrmaxsize.e . . . . . 6 𝐸 = (Edg‘𝐺)
6 fvex 6098 . . . . . . . . 9 (Vtx‘𝐺) ∈ V
71, 6eqeltri 2684 . . . . . . . 8 𝑉 ∈ V
8 vex 3176 . . . . . . . 8 𝑒 ∈ V
9 opvtxfv 40229 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝑒 ∈ V) → (Vtx‘⟨𝑉, 𝑒⟩) = 𝑉)
107, 8, 9mp2an 704 . . . . . . 7 (Vtx‘⟨𝑉, 𝑒⟩) = 𝑉
1110eqcomi 2619 . . . . . 6 𝑉 = (Vtx‘⟨𝑉, 𝑒⟩)
12 eqid 2610 . . . . . 6 (Edg‘⟨𝑉, 𝑒⟩) = (Edg‘⟨𝑉, 𝑒⟩)
131, 5, 11, 12sizusglecusg 40671 . . . . 5 ((𝐺 ∈ USGraph ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘(Edg‘⟨𝑉, 𝑒⟩)))
1413adantlr 747 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘(Edg‘⟨𝑉, 𝑒⟩)))
1511, 12cusgrsize 40662 . . . . . . . 8 ((⟨𝑉, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘(Edg‘⟨𝑉, 𝑒⟩)) = ((#‘𝑉)C2))
16 breq2 4582 . . . . . . . . 9 ((#‘(Edg‘⟨𝑉, 𝑒⟩)) = ((#‘𝑉)C2) → ((#‘𝐸) ≤ (#‘(Edg‘⟨𝑉, 𝑒⟩)) ↔ (#‘𝐸) ≤ ((#‘𝑉)C2)))
1716biimpd 218 . . . . . . . 8 ((#‘(Edg‘⟨𝑉, 𝑒⟩)) = ((#‘𝑉)C2) → ((#‘𝐸) ≤ (#‘(Edg‘⟨𝑉, 𝑒⟩)) → (#‘𝐸) ≤ ((#‘𝑉)C2)))
1815, 17syl 17 . . . . . . 7 ((⟨𝑉, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → ((#‘𝐸) ≤ (#‘(Edg‘⟨𝑉, 𝑒⟩)) → (#‘𝐸) ≤ ((#‘𝑉)C2)))
1918expcom 450 . . . . . 6 (𝑉 ∈ Fin → (⟨𝑉, 𝑒⟩ ∈ ComplUSGraph → ((#‘𝐸) ≤ (#‘(Edg‘⟨𝑉, 𝑒⟩)) → (#‘𝐸) ≤ ((#‘𝑉)C2))))
2019adantl 481 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (⟨𝑉, 𝑒⟩ ∈ ComplUSGraph → ((#‘𝐸) ≤ (#‘(Edg‘⟨𝑉, 𝑒⟩)) → (#‘𝐸) ≤ ((#‘𝑉)C2))))
2120imp 444 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → ((#‘𝐸) ≤ (#‘(Edg‘⟨𝑉, 𝑒⟩)) → (#‘𝐸) ≤ ((#‘𝑉)C2)))
2214, 21mpd 15 . . 3 (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (#‘𝐸) ≤ ((#‘𝑉)C2))
234, 22exlimddv 1850 . 2 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) ≤ ((#‘𝑉)C2))
242, 23sylbi 206 1 (𝐺 ∈ FinUSGraph → (#‘𝐸) ≤ ((#‘𝑉)C2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wex 1695  wcel 1977  Vcvv 3173  cop 4131   class class class wbr 4578  cfv 5790  (class class class)co 6527  Fincfn 7819  cle 9932  2c2 10920  Ccbc 12909  #chash 12937  Vtxcvtx 40221  Edgcedga 40343   USGraph cusgr 40371   FinUSGraph cfusgr 40527  ComplUSGraphccusgr 40545
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 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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 4368  df-int 4406  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-2o 7426  df-oadd 7429  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-card 8626  df-cda 8851  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-n0 11143  df-z 11214  df-uz 11523  df-rp 11668  df-fz 12156  df-seq 12622  df-fac 12881  df-bc 12910  df-hash 12938  df-vtx 40223  df-iedg 40224  df-uhgr 40272  df-upgr 40300  df-umgr 40301  df-edga 40344  df-uspgr 40372  df-usgr 40373  df-fusgr 40528  df-nbgr 40546  df-uvtxa 40548  df-cplgr 40549  df-cusgr 40550
This theorem is referenced by: (None)
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