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Theorem fusgrmaxsize 40680
 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 40537 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3 cusgrexg 40663 . . . 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 6113 . . . . . . . . 9 (Vtx‘𝐺) ∈ V
71, 6eqeltri 2684 . . . . . . . 8 𝑉 ∈ V
8 vex 3176 . . . . . . . 8 𝑒 ∈ V
9 opvtxfv 25681 . . . . . . . 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 40679 . . . . 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 40670 . . . . . . . 8 ((⟨𝑉, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘(Edg‘⟨𝑉, 𝑒⟩)) = ((#‘𝑉)C2))
16 breq2 4587 . . . . . . . . 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 4583  ‘cfv 5804  (class class class)co 6549  Fincfn 7841   ≤ cle 9954  2c2 10947  Ccbc 12951  #chash 12979  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   FinUSGraph cfusgr 40535  ComplUSGraphccusgr 40553 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-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 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-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-seq 12664  df-fac 12923  df-bc 12952  df-hash 12980  df-vtx 25675  df-iedg 25676  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-nbgr 40554  df-uvtxa 40556  df-cplgr 40557  df-cusgr 40558 This theorem is referenced by: (None)
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