![]() |
Mathbox for Steve Rodriguez |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bccm1k | Structured version Visualization version GIF version |
Description: Generalized binomial coefficient: 𝐶 choose (𝐾 − 1), when 𝐶 is not (𝐾 − 1). (Contributed by Steve Rodriguez, 22-Apr-2020.) |
Ref | Expression |
---|---|
bccm1k.c | ⊢ (𝜑 → 𝐶 ∈ (ℂ ∖ {(𝐾 − 1)})) |
bccm1k.k | ⊢ (𝜑 → 𝐾 ∈ ℕ) |
Ref | Expression |
---|---|
bccm1k | ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) = ((𝐶C𝑐𝐾) / ((𝐶 − (𝐾 − 1)) / 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bccm1k.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (ℂ ∖ {(𝐾 − 1)})) | |
2 | 1 | eldifad 3974 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
3 | bccm1k.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ) | |
4 | 3 | nncnd 12279 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
5 | 1cnd 11253 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
6 | 4, 5 | subcld 11617 | . . . 4 ⊢ (𝜑 → (𝐾 − 1) ∈ ℂ) |
7 | 2, 6 | subcld 11617 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐾 − 1)) ∈ ℂ) |
8 | 3 | nnne0d 12313 | . . 3 ⊢ (𝜑 → 𝐾 ≠ 0) |
9 | 7, 4, 8 | divcld 12040 | . 2 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / 𝐾) ∈ ℂ) |
10 | nnm1nn0 12564 | . . . 4 ⊢ (𝐾 ∈ ℕ → (𝐾 − 1) ∈ ℕ0) | |
11 | 3, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ0) |
12 | 2, 11 | bcccl 44334 | . 2 ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) ∈ ℂ) |
13 | eldifsni 4794 | . . . . 5 ⊢ (𝐶 ∈ (ℂ ∖ {(𝐾 − 1)}) → 𝐶 ≠ (𝐾 − 1)) | |
14 | 1, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ (𝐾 − 1)) |
15 | 2, 6, 14 | subne0d 11626 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐾 − 1)) ≠ 0) |
16 | 7, 4, 15, 8 | divne0d 12056 | . 2 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / 𝐾) ≠ 0) |
17 | 2, 11 | bccp1k 44336 | . . . 4 ⊢ (𝜑 → (𝐶C𝑐((𝐾 − 1) + 1)) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1)))) |
18 | 4, 5 | npcand 11621 | . . . . 5 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
19 | 18 | oveq2d 7446 | . . . 4 ⊢ (𝜑 → (𝐶C𝑐((𝐾 − 1) + 1)) = (𝐶C𝑐𝐾)) |
20 | 18 | oveq2d 7446 | . . . . 5 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1)) = ((𝐶 − (𝐾 − 1)) / 𝐾)) |
21 | 20 | oveq2d 7446 | . . . 4 ⊢ (𝜑 → ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1))) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾))) |
22 | 17, 19, 21 | 3eqtr3d 2782 | . . 3 ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾))) |
23 | 12, 9 | mulcomd 11279 | . . 3 ⊢ (𝜑 → ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾)) = (((𝐶 − (𝐾 − 1)) / 𝐾) · (𝐶C𝑐(𝐾 − 1)))) |
24 | 22, 23 | eqtr2d 2775 | . 2 ⊢ (𝜑 → (((𝐶 − (𝐾 − 1)) / 𝐾) · (𝐶C𝑐(𝐾 − 1))) = (𝐶C𝑐𝐾)) |
25 | 9, 12, 16, 24 | mvllmuld 12096 | 1 ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) = ((𝐶C𝑐𝐾) / ((𝐶 − (𝐾 − 1)) / 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 {csn 4630 (class class class)co 7430 ℂcc 11150 1c1 11153 + caddc 11155 · cmul 11157 − cmin 11489 / cdiv 11917 ℕcn 12263 ℕ0cn0 12523 C𝑐cbcc 44331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-fac 14309 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-prod 15936 df-fallfac 16039 df-bcc 44332 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |