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Mirrors > Home > ILE Home > Th. List > bcn2m1 | 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 9248 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
2 | 1 | nn0cnd 9262 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℂ) |
3 | 2z 9312 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | bccl 10782 | . . . . 5 ⊢ (((𝑁 − 1) ∈ ℕ0 ∧ 2 ∈ ℤ) → ((𝑁 − 1)C2) ∈ ℕ0) | |
5 | 1, 3, 4 | sylancl 413 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C2) ∈ ℕ0) |
6 | 5 | nn0cnd 9262 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C2) ∈ ℂ) |
7 | 2, 6 | addcomd 8139 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (((𝑁 − 1)C2) + (𝑁 − 1))) |
8 | bcn1 10773 | . . . . . 6 ⊢ ((𝑁 − 1) ∈ ℕ0 → ((𝑁 − 1)C1) = (𝑁 − 1)) | |
9 | 8 | eqcomd 2195 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ0 → (𝑁 − 1) = ((𝑁 − 1)C1)) |
10 | 1, 9 | syl 14 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) = ((𝑁 − 1)C1)) |
11 | 1e2m1 9069 | . . . . . 6 ⊢ 1 = (2 − 1) | |
12 | 11 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 = (2 − 1)) |
13 | 12 | oveq2d 5913 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1)C1) = ((𝑁 − 1)C(2 − 1))) |
14 | 10, 13 | eqtrd 2222 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) = ((𝑁 − 1)C(2 − 1))) |
15 | 14 | oveq2d 5913 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + (𝑁 − 1)) = (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1)))) |
16 | bcpasc 10781 | . . . 4 ⊢ (((𝑁 − 1) ∈ ℕ0 ∧ 2 ∈ ℤ) → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (((𝑁 − 1) + 1)C2)) | |
17 | 1, 3, 16 | sylancl 413 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (((𝑁 − 1) + 1)C2)) |
18 | nncn 8958 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
19 | 1cnd 8004 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) | |
20 | 18, 19 | npcand 8303 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁) |
21 | 20 | oveq1d 5912 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1) + 1)C2) = (𝑁C2)) |
22 | 17, 21 | eqtrd 2222 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 − 1)C2) + ((𝑁 − 1)C(2 − 1))) = (𝑁C2)) |
23 | 7, 15, 22 | 3eqtrd 2226 | 1 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (𝑁C2)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 (class class class)co 5897 1c1 7843 + caddc 7845 − cmin 8159 ℕcn 8950 2c2 9001 ℕ0cn0 9207 ℤcz 9284 Ccbc 10762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-n0 9208 df-z 9285 df-uz 9560 df-q 9652 df-rp 9686 df-fz 10041 df-seqfrec 10479 df-fac 10741 df-bc 10763 |
This theorem is referenced by: (None) |
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