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Mirrors > Home > MPE Home > Th. List > cusgrasize | Structured version Visualization version GIF version |
Description: The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2). (Contributed by Alexander van der Vekens, 11-Jan-2018.) |
Ref | Expression |
---|---|
cusgrasize | ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cusgra 25950 | . . 3 ⊢ ComplUSGrph = {〈𝑣, 𝑒〉 ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝑒)} | |
2 | 1 | relopabi 5167 | . 2 ⊢ Rel ComplUSGrph |
3 | vex 3176 | . . 3 ⊢ 𝑒 ∈ V | |
4 | 3 | resex 5363 | . 2 ⊢ (𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}) ∈ V |
5 | fveq2 6103 | . . 3 ⊢ (𝑒 = 𝐸 → (#‘𝑒) = (#‘𝐸)) | |
6 | fveq2 6103 | . . . 4 ⊢ (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉)) | |
7 | 6 | oveq1d 6564 | . . 3 ⊢ (𝑣 = 𝑉 → ((#‘𝑣)C2) = ((#‘𝑉)C2)) |
8 | 5, 7 | eqeqan12rd 2628 | . 2 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((#‘𝑒) = ((#‘𝑣)C2) ↔ (#‘𝐸) = ((#‘𝑉)C2))) |
9 | fveq2 6103 | . . 3 ⊢ (𝑒 = 𝑓 → (#‘𝑒) = (#‘𝑓)) | |
10 | fveq2 6103 | . . . 4 ⊢ (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤)) | |
11 | 10 | oveq1d 6564 | . . 3 ⊢ (𝑣 = 𝑤 → ((#‘𝑣)C2) = ((#‘𝑤)C2)) |
12 | 9, 11 | eqeqan12rd 2628 | . 2 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((#‘𝑒) = ((#‘𝑣)C2) ↔ (#‘𝑓) = ((#‘𝑤)C2))) |
13 | eqid 2610 | . . 3 ⊢ (𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}) = (𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}) | |
14 | 13 | cusgrares 26001 | . 2 ⊢ ((𝑣 ComplUSGrph 𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛}) ComplUSGrph (𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)})) |
15 | fveq2 6103 | . . . 4 ⊢ (𝑓 = (𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}) → (#‘𝑓) = (#‘(𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}))) | |
16 | 15 | adantl 481 | . . 3 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = (𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)})) → (#‘𝑓) = (#‘(𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}))) |
17 | fveq2 6103 | . . . . 5 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛}))) | |
18 | 17 | adantr 480 | . . . 4 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = (𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)})) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛}))) |
19 | 18 | oveq1d 6564 | . . 3 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = (𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)})) → ((#‘𝑤)C2) = ((#‘(𝑣 ∖ {𝑛}))C2)) |
20 | 16, 19 | eqeq12d 2625 | . 2 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = (𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)})) → ((#‘𝑓) = ((#‘𝑤)C2) ↔ (#‘(𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)})) = ((#‘(𝑣 ∖ {𝑛}))C2))) |
21 | cusgrasizeindb0 25999 | . 2 ⊢ ((𝑣 ComplUSGrph 𝑒 ∧ (#‘𝑣) = 0) → (#‘𝑒) = ((#‘𝑣)C2)) | |
22 | 13 | cusgrasize2inds 26005 | . . 3 ⊢ ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ComplUSGrph 𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) → ((#‘(𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)})) = ((#‘(𝑣 ∖ {𝑛}))C2) → (#‘𝑒) = ((#‘𝑣)C2)))) |
23 | 22 | imp31 447 | . 2 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣 ComplUSGrph 𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (#‘(𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)})) = ((#‘(𝑣 ∖ {𝑛}))C2)) → (#‘𝑒) = ((#‘𝑣)C2)) |
24 | 2, 4, 8, 12, 14, 20, 21, 23 | brfi1ind 13136 | 1 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 ∀wral 2896 {crab 2900 ∖ cdif 3537 {csn 4125 {cpr 4127 class class class wbr 4583 dom cdm 5038 ran crn 5039 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 1c1 9816 + caddc 9818 2c2 10947 ℕ0cn0 11169 Ccbc 12951 #chash 12979 USGrph cusg 25859 ComplUSGrph ccusgra 25947 |
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-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-usgra 25862 df-nbgra 25949 df-cusgra 25950 |
This theorem is referenced by: usgramaxsize 26015 |
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