Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11941 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
2 | 1 | nn0cnd 11960 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℂ) |
3 | 2z 12017 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | bccl 13685 | . . . . 5 ⊢ (((𝑁 − 1) ∈ ℕ0 ∧ 2 ∈ ℤ) → ((𝑁 − 1)C2) ∈ ℕ0) | |
5 | 1, 3, 4 | sylancl 588 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C2) ∈ ℕ0) |
6 | 5 | nn0cnd 11960 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C2) ∈ ℂ) |
7 | 2, 6 | addcomd 10845 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (((𝑁 − 1)C2) + (𝑁 − 1))) |
8 | bcn1 13676 | . . . . . 6 ⊢ ((𝑁 − 1) ∈ ℕ0 → ((𝑁 − 1)C1) = (𝑁 − 1)) | |
9 | 8 | eqcomd 2830 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ0 → (𝑁 − 1) = ((𝑁 − 1)C1)) |
10 | 1, 9 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) = ((𝑁 − 1)C1)) |
11 | 1e2m1 11767 | . . . . . 6 ⊢ 1 = (2 − 1) | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 = (2 − 1)) |
13 | 12 | oveq2d 7175 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C1) = ((𝑁 − 1)C(2 − 1))) |
14 | 10, 13 | eqtrd 2859 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) = ((𝑁 − 1)C(2 − 1))) |
15 | 14 | oveq2d 7175 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + (𝑁 − 1)) = (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1)))) |
16 | bcpasc 13684 | . . . 4 ⊢ (((𝑁 − 1) ∈ ℕ0 ∧ 2 ∈ ℤ) → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (((𝑁 − 1) + 1)C2)) | |
17 | 1, 3, 16 | sylancl 588 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (((𝑁 − 1) + 1)C2)) |
18 | nncn 11649 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
19 | 1cnd 10639 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) | |
20 | 18, 19 | npcand 11004 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁) |
21 | 20 | oveq1d 7174 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1) + 1)C2) = (𝑁C2)) |
22 | 17, 21 | eqtrd 2859 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (𝑁C2)) |
23 | 7, 15, 22 | 3eqtrd 2863 | 1 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (𝑁C2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 (class class class)co 7159 1c1 10541 + caddc 10543 − cmin 10873 ℕcn 11641 2c2 11695 ℕ0cn0 11900 ℤcz 11984 Ccbc 13665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-seq 13373 df-fac 13637 df-bc 13666 |
This theorem is referenced by: cusgrsize2inds 27238 |
Copyright terms: Public domain | W3C validator |