![]() |
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 3921 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
3 | bccm1k.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ) | |
4 | 3 | nncnd 12128 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
5 | 1cnd 11109 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
6 | 4, 5 | subcld 11471 | . . . 4 ⊢ (𝜑 → (𝐾 − 1) ∈ ℂ) |
7 | 2, 6 | subcld 11471 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐾 − 1)) ∈ ℂ) |
8 | 3 | nnne0d 12162 | . . 3 ⊢ (𝜑 → 𝐾 ≠ 0) |
9 | 7, 4, 8 | divcld 11890 | . 2 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / 𝐾) ∈ ℂ) |
10 | nnm1nn0 12413 | . . . 4 ⊢ (𝐾 ∈ ℕ → (𝐾 − 1) ∈ ℕ0) | |
11 | 3, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ0) |
12 | 2, 11 | bcccl 42524 | . 2 ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) ∈ ℂ) |
13 | eldifsni 4749 | . . . . 5 ⊢ (𝐶 ∈ (ℂ ∖ {(𝐾 − 1)}) → 𝐶 ≠ (𝐾 − 1)) | |
14 | 1, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ (𝐾 − 1)) |
15 | 2, 6, 14 | subne0d 11480 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐾 − 1)) ≠ 0) |
16 | 7, 4, 15, 8 | divne0d 11906 | . 2 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / 𝐾) ≠ 0) |
17 | 2, 11 | bccp1k 42526 | . . . 4 ⊢ (𝜑 → (𝐶C𝑐((𝐾 − 1) + 1)) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1)))) |
18 | 4, 5 | npcand 11475 | . . . . 5 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
19 | 18 | oveq2d 7368 | . . . 4 ⊢ (𝜑 → (𝐶C𝑐((𝐾 − 1) + 1)) = (𝐶C𝑐𝐾)) |
20 | 18 | oveq2d 7368 | . . . . 5 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1)) = ((𝐶 − (𝐾 − 1)) / 𝐾)) |
21 | 20 | oveq2d 7368 | . . . 4 ⊢ (𝜑 → ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1))) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾))) |
22 | 17, 19, 21 | 3eqtr3d 2786 | . . 3 ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾))) |
23 | 12, 9 | mulcomd 11135 | . . 3 ⊢ (𝜑 → ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾)) = (((𝐶 − (𝐾 − 1)) / 𝐾) · (𝐶C𝑐(𝐾 − 1)))) |
24 | 22, 23 | eqtr2d 2779 | . 2 ⊢ (𝜑 → (((𝐶 − (𝐾 − 1)) / 𝐾) · (𝐶C𝑐(𝐾 − 1))) = (𝐶C𝑐𝐾)) |
25 | 9, 12, 16, 24 | mvllmuld 11946 | 1 ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) = ((𝐶C𝑐𝐾) / ((𝐶 − (𝐾 − 1)) / 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∖ cdif 3906 {csn 4585 (class class class)co 7352 ℂcc 11008 1c1 11011 + caddc 11013 · cmul 11015 − cmin 11344 / cdiv 11771 ℕcn 12112 ℕ0cn0 12372 C𝑐cbcc 42521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-inf2 9536 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-sup 9337 df-oi 9405 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-n0 12373 df-z 12459 df-uz 12723 df-rp 12871 df-fz 13380 df-fzo 13523 df-seq 13862 df-exp 13923 df-fac 14128 df-hash 14185 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-clim 15330 df-prod 15749 df-fallfac 15850 df-bcc 42522 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |