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Theorem fusgrmaxsize 26591
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 26430 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3 cusgrexg 26571 . . . 4 (𝑉 ∈ Fin → ∃𝑒𝑉, 𝑒⟩ ∈ ComplUSGraph)
43adantl 473 . . 3 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → ∃𝑒𝑉, 𝑒⟩ ∈ ComplUSGraph)
5 fusgrmaxsize.e . . . . . 6 𝐸 = (Edg‘𝐺)
6 fvex 6363 . . . . . . . . 9 (Vtx‘𝐺) ∈ V
71, 6eqeltri 2835 . . . . . . . 8 𝑉 ∈ V
8 vex 3343 . . . . . . . 8 𝑒 ∈ V
9 opvtxfv 26104 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝑒 ∈ V) → (Vtx‘⟨𝑉, 𝑒⟩) = 𝑉)
107, 8, 9mp2an 710 . . . . . . 7 (Vtx‘⟨𝑉, 𝑒⟩) = 𝑉
1110eqcomi 2769 . . . . . 6 𝑉 = (Vtx‘⟨𝑉, 𝑒⟩)
12 eqid 2760 . . . . . 6 (Edg‘⟨𝑉, 𝑒⟩) = (Edg‘⟨𝑉, 𝑒⟩)
131, 5, 11, 12sizusglecusg 26590 . . . . 5 ((𝐺 ∈ USGraph ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)))
1413adantlr 753 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)))
1511, 12cusgrsize 26581 . . . . . . . 8 ((⟨𝑉, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘(Edg‘⟨𝑉, 𝑒⟩)) = ((♯‘𝑉)C2))
16 breq2 4808 . . . . . . . . 9 ((♯‘(Edg‘⟨𝑉, 𝑒⟩)) = ((♯‘𝑉)C2) → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) ↔ (♯‘𝐸) ≤ ((♯‘𝑉)C2)))
1716biimpd 219 . . . . . . . 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 473 . . . . 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 2012 . 2 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))
242, 23sylbi 207 1 (𝐺 ∈ FinUSGraph → (♯‘𝐸) ≤ ((♯‘𝑉)C2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  cop 4327   class class class wbr 4804  cfv 6049  (class class class)co 6814  Fincfn 8123  cle 10287  2c2 11282  Ccbc 13303  chash 13331  Vtxcvtx 26094  Edgcedg 26159  USGraphcusgr 26264  FinUSGraphcfusgr 26428  ComplUSGraphccusgr 26536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-n0 11505  df-xnn0 11576  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-seq 13016  df-fac 13275  df-bc 13304  df-hash 13332  df-vtx 26096  df-iedg 26097  df-edg 26160  df-uhgr 26173  df-upgr 26197  df-umgr 26198  df-uspgr 26265  df-usgr 26266  df-fusgr 26429  df-nbgr 26445  df-uvtx 26507  df-cplgr 26537  df-cusgr 26538
This theorem is referenced by: (None)
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