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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 40537 | . 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
3 | cusgrexg 40663 | . . . 4 ⊢ (𝑉 ∈ Fin → ∃𝑒〈𝑉, 𝑒〉 ∈ ComplUSGraph) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → ∃𝑒〈𝑉, 𝑒〉 ∈ ComplUSGraph) |
5 | fusgrmaxsize.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
6 | fvex 6113 | . . . . . . . . 9 ⊢ (Vtx‘𝐺) ∈ V | |
7 | 1, 6 | eqeltri 2684 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
8 | vex 3176 | . . . . . . . 8 ⊢ 𝑒 ∈ V | |
9 | opvtxfv 25681 | . . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝑒 ∈ V) → (Vtx‘〈𝑉, 𝑒〉) = 𝑉) | |
10 | 7, 8, 9 | mp2an 704 | . . . . . . 7 ⊢ (Vtx‘〈𝑉, 𝑒〉) = 𝑉 |
11 | 10 | eqcomi 2619 | . . . . . 6 ⊢ 𝑉 = (Vtx‘〈𝑉, 𝑒〉) |
12 | eqid 2610 | . . . . . 6 ⊢ (Edg‘〈𝑉, 𝑒〉) = (Edg‘〈𝑉, 𝑒〉) | |
13 | 1, 5, 11, 12 | sizusglecusg 40679 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 〈𝑉, 𝑒〉 ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘(Edg‘〈𝑉, 𝑒〉))) |
14 | 13 | adantlr 747 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ 〈𝑉, 𝑒〉 ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘(Edg‘〈𝑉, 𝑒〉))) |
15 | 11, 12 | cusgrsize 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))) | |
17 | 16 | biimpd 218 | . . . . . . . 8 ⊢ ((#‘(Edg‘〈𝑉, 𝑒〉)) = ((#‘𝑉)C2) → ((#‘𝐸) ≤ (#‘(Edg‘〈𝑉, 𝑒〉)) → (#‘𝐸) ≤ ((#‘𝑉)C2))) |
18 | 15, 17 | syl 17 | . . . . . . 7 ⊢ ((〈𝑉, 𝑒〉 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → ((#‘𝐸) ≤ (#‘(Edg‘〈𝑉, 𝑒〉)) → (#‘𝐸) ≤ ((#‘𝑉)C2))) |
19 | 18 | expcom 450 | . . . . . 6 ⊢ (𝑉 ∈ Fin → (〈𝑉, 𝑒〉 ∈ ComplUSGraph → ((#‘𝐸) ≤ (#‘(Edg‘〈𝑉, 𝑒〉)) → (#‘𝐸) ≤ ((#‘𝑉)C2)))) |
20 | 19 | adantl 481 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (〈𝑉, 𝑒〉 ∈ ComplUSGraph → ((#‘𝐸) ≤ (#‘(Edg‘〈𝑉, 𝑒〉)) → (#‘𝐸) ≤ ((#‘𝑉)C2)))) |
21 | 20 | imp 444 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ 〈𝑉, 𝑒〉 ∈ ComplUSGraph) → ((#‘𝐸) ≤ (#‘(Edg‘〈𝑉, 𝑒〉)) → (#‘𝐸) ≤ ((#‘𝑉)C2))) |
22 | 14, 21 | mpd 15 | . . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ 〈𝑉, 𝑒〉 ∈ ComplUSGraph) → (#‘𝐸) ≤ ((#‘𝑉)C2)) |
23 | 4, 22 | exlimddv 1850 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) ≤ ((#‘𝑉)C2)) |
24 | 2, 23 | sylbi 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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