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Mirrors > Home > MPE Home > Th. List > bcn2m1 | Structured version Visualization version GIF version |
Description: Compute the binomial coefficient "𝑁 choose 2 " from "(𝑁 − 1) choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018.) |
Ref | Expression |
---|---|
bcn2m1 | ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (𝑁C2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnm1nn0 11750 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
2 | 1 | nn0cnd 11769 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℂ) |
3 | 2z 11827 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | bccl 13497 | . . . . 5 ⊢ (((𝑁 − 1) ∈ ℕ0 ∧ 2 ∈ ℤ) → ((𝑁 − 1)C2) ∈ ℕ0) | |
5 | 1, 3, 4 | sylancl 577 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C2) ∈ ℕ0) |
6 | 5 | nn0cnd 11769 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C2) ∈ ℂ) |
7 | 2, 6 | addcomd 10642 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (((𝑁 − 1)C2) + (𝑁 − 1))) |
8 | bcn1 13488 | . . . . . 6 ⊢ ((𝑁 − 1) ∈ ℕ0 → ((𝑁 − 1)C1) = (𝑁 − 1)) | |
9 | 8 | eqcomd 2785 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ0 → (𝑁 − 1) = ((𝑁 − 1)C1)) |
10 | 1, 9 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) = ((𝑁 − 1)C1)) |
11 | 1e2m1 11574 | . . . . . 6 ⊢ 1 = (2 − 1) | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 = (2 − 1)) |
13 | 12 | oveq2d 6992 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C1) = ((𝑁 − 1)C(2 − 1))) |
14 | 10, 13 | eqtrd 2815 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) = ((𝑁 − 1)C(2 − 1))) |
15 | 14 | oveq2d 6992 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + (𝑁 − 1)) = (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1)))) |
16 | bcpasc 13496 | . . . 4 ⊢ (((𝑁 − 1) ∈ ℕ0 ∧ 2 ∈ ℤ) → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (((𝑁 − 1) + 1)C2)) | |
17 | 1, 3, 16 | sylancl 577 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (((𝑁 − 1) + 1)C2)) |
18 | nncn 11448 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
19 | 1cnd 10434 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) | |
20 | 18, 19 | npcand 10802 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁) |
21 | 20 | oveq1d 6991 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1) + 1)C2) = (𝑁C2)) |
22 | 17, 21 | eqtrd 2815 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (𝑁C2)) |
23 | 7, 15, 22 | 3eqtrd 2819 | 1 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (𝑁C2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 (class class class)co 6976 1c1 10336 + caddc 10338 − cmin 10670 ℕcn 11439 2c2 11495 ℕ0cn0 11707 ℤcz 11793 Ccbc 13477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-fz 12709 df-seq 13185 df-fac 13449 df-bc 13478 |
This theorem is referenced by: cusgrsize2inds 26938 |
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