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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 12509 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
2 | 1 | nn0cnd 12530 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℂ) |
3 | 2z 12590 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | bccl 14278 | . . . . 5 ⊢ (((𝑁 − 1) ∈ ℕ0 ∧ 2 ∈ ℤ) → ((𝑁 − 1)C2) ∈ ℕ0) | |
5 | 1, 3, 4 | sylancl 586 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C2) ∈ ℕ0) |
6 | 5 | nn0cnd 12530 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C2) ∈ ℂ) |
7 | 2, 6 | addcomd 11412 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (((𝑁 − 1)C2) + (𝑁 − 1))) |
8 | bcn1 14269 | . . . . . 6 ⊢ ((𝑁 − 1) ∈ ℕ0 → ((𝑁 − 1)C1) = (𝑁 − 1)) | |
9 | 8 | eqcomd 2738 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ0 → (𝑁 − 1) = ((𝑁 − 1)C1)) |
10 | 1, 9 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) = ((𝑁 − 1)C1)) |
11 | 1e2m1 12335 | . . . . . 6 ⊢ 1 = (2 − 1) | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 = (2 − 1)) |
13 | 12 | oveq2d 7421 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C1) = ((𝑁 − 1)C(2 − 1))) |
14 | 10, 13 | eqtrd 2772 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) = ((𝑁 − 1)C(2 − 1))) |
15 | 14 | oveq2d 7421 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + (𝑁 − 1)) = (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1)))) |
16 | bcpasc 14277 | . . . 4 ⊢ (((𝑁 − 1) ∈ ℕ0 ∧ 2 ∈ ℤ) → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (((𝑁 − 1) + 1)C2)) | |
17 | 1, 3, 16 | sylancl 586 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (((𝑁 − 1) + 1)C2)) |
18 | nncn 12216 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
19 | 1cnd 11205 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) | |
20 | 18, 19 | npcand 11571 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁) |
21 | 20 | oveq1d 7420 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1) + 1)C2) = (𝑁C2)) |
22 | 17, 21 | eqtrd 2772 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (𝑁C2)) |
23 | 7, 15, 22 | 3eqtrd 2776 | 1 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (𝑁C2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 (class class class)co 7405 1c1 11107 + caddc 11109 − cmin 11440 ℕcn 12208 2c2 12263 ℕ0cn0 12468 ℤcz 12554 Ccbc 14258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-seq 13963 df-fac 14230 df-bc 14259 |
This theorem is referenced by: cusgrsize2inds 28699 |
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