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Mirrors > Home > MPE Home > Th. List > fusgrmaxsize | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fusgrmaxsize.v | ⊢ 𝑉 = (Vtx‘𝐺) |
fusgrmaxsize.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
fusgrmaxsize | ⊢ (𝐺 ∈ FinUSGraph → (♯‘𝐸) ≤ ((♯‘𝑉)C2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fusgrmaxsize.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | isfusgr 27102 | . 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
3 | cusgrexg 27228 | . . . 4 ⊢ (𝑉 ∈ Fin → ∃𝑒〈𝑉, 𝑒〉 ∈ ComplUSGraph) | |
4 | 3 | adantl 484 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → ∃𝑒〈𝑉, 𝑒〉 ∈ ComplUSGraph) |
5 | fusgrmaxsize.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
6 | 1 | fvexi 6686 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
7 | vex 3499 | . . . . . . . 8 ⊢ 𝑒 ∈ V | |
8 | 6, 7 | opvtxfvi 26796 | . . . . . . 7 ⊢ (Vtx‘〈𝑉, 𝑒〉) = 𝑉 |
9 | 8 | eqcomi 2832 | . . . . . 6 ⊢ 𝑉 = (Vtx‘〈𝑉, 𝑒〉) |
10 | eqid 2823 | . . . . . 6 ⊢ (Edg‘〈𝑉, 𝑒〉) = (Edg‘〈𝑉, 𝑒〉) | |
11 | 1, 5, 9, 10 | sizusglecusg 27247 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 〈𝑉, 𝑒〉 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘(Edg‘〈𝑉, 𝑒〉))) |
12 | 11 | adantlr 713 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ 〈𝑉, 𝑒〉 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘(Edg‘〈𝑉, 𝑒〉))) |
13 | 9, 10 | cusgrsize 27238 | . . . . . . . 8 ⊢ ((〈𝑉, 𝑒〉 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘(Edg‘〈𝑉, 𝑒〉)) = ((♯‘𝑉)C2)) |
14 | breq2 5072 | . . . . . . . . 9 ⊢ ((♯‘(Edg‘〈𝑉, 𝑒〉)) = ((♯‘𝑉)C2) → ((♯‘𝐸) ≤ (♯‘(Edg‘〈𝑉, 𝑒〉)) ↔ (♯‘𝐸) ≤ ((♯‘𝑉)C2))) | |
15 | 14 | biimpd 231 | . . . . . . . 8 ⊢ ((♯‘(Edg‘〈𝑉, 𝑒〉)) = ((♯‘𝑉)C2) → ((♯‘𝐸) ≤ (♯‘(Edg‘〈𝑉, 𝑒〉)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))) |
16 | 13, 15 | syl 17 | . . . . . . 7 ⊢ ((〈𝑉, 𝑒〉 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → ((♯‘𝐸) ≤ (♯‘(Edg‘〈𝑉, 𝑒〉)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))) |
17 | 16 | expcom 416 | . . . . . 6 ⊢ (𝑉 ∈ Fin → (〈𝑉, 𝑒〉 ∈ ComplUSGraph → ((♯‘𝐸) ≤ (♯‘(Edg‘〈𝑉, 𝑒〉)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2)))) |
18 | 17 | adantl 484 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (〈𝑉, 𝑒〉 ∈ ComplUSGraph → ((♯‘𝐸) ≤ (♯‘(Edg‘〈𝑉, 𝑒〉)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2)))) |
19 | 18 | imp 409 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ 〈𝑉, 𝑒〉 ∈ ComplUSGraph) → ((♯‘𝐸) ≤ (♯‘(Edg‘〈𝑉, 𝑒〉)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))) |
20 | 12, 19 | mpd 15 | . . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ 〈𝑉, 𝑒〉 ∈ ComplUSGraph) → (♯‘𝐸) ≤ ((♯‘𝑉)C2)) |
21 | 4, 20 | exlimddv 1936 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) ≤ ((♯‘𝑉)C2)) |
22 | 2, 21 | sylbi 219 | 1 ⊢ (𝐺 ∈ FinUSGraph → (♯‘𝐸) ≤ ((♯‘𝑉)C2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ≤ cle 10678 2c2 11695 Ccbc 13665 ♯chash 13693 Vtxcvtx 26783 Edgcedg 26834 USGraphcusgr 26936 FinUSGraphcfusgr 27100 ComplUSGraphccusgr 27194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-seq 13373 df-fac 13637 df-bc 13666 df-hash 13694 df-vtx 26785 df-iedg 26786 df-edg 26835 df-uhgr 26845 df-upgr 26869 df-umgr 26870 df-uspgr 26937 df-usgr 26938 df-fusgr 27101 df-nbgr 27117 df-uvtx 27170 df-cplgr 27195 df-cusgr 27196 |
This theorem is referenced by: (None) |
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