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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 3804 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
3 | bccm1k.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ) | |
4 | 3 | nncnd 11396 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
5 | 1cnd 10373 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
6 | 4, 5 | subcld 10736 | . . . 4 ⊢ (𝜑 → (𝐾 − 1) ∈ ℂ) |
7 | 2, 6 | subcld 10736 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐾 − 1)) ∈ ℂ) |
8 | 3 | nnne0d 11429 | . . 3 ⊢ (𝜑 → 𝐾 ≠ 0) |
9 | 7, 4, 8 | divcld 11153 | . 2 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / 𝐾) ∈ ℂ) |
10 | nnm1nn0 11689 | . . . 4 ⊢ (𝐾 ∈ ℕ → (𝐾 − 1) ∈ ℕ0) | |
11 | 3, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ0) |
12 | 2, 11 | bcccl 39504 | . 2 ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) ∈ ℂ) |
13 | eldifsni 4553 | . . . . 5 ⊢ (𝐶 ∈ (ℂ ∖ {(𝐾 − 1)}) → 𝐶 ≠ (𝐾 − 1)) | |
14 | 1, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ (𝐾 − 1)) |
15 | 2, 6, 14 | subne0d 10745 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐾 − 1)) ≠ 0) |
16 | 7, 4, 15, 8 | divne0d 11169 | . 2 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / 𝐾) ≠ 0) |
17 | 2, 11 | bccp1k 39506 | . . . 4 ⊢ (𝜑 → (𝐶C𝑐((𝐾 − 1) + 1)) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1)))) |
18 | 4, 5 | npcand 10740 | . . . . 5 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
19 | 18 | oveq2d 6940 | . . . 4 ⊢ (𝜑 → (𝐶C𝑐((𝐾 − 1) + 1)) = (𝐶C𝑐𝐾)) |
20 | 18 | oveq2d 6940 | . . . . 5 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1)) = ((𝐶 − (𝐾 − 1)) / 𝐾)) |
21 | 20 | oveq2d 6940 | . . . 4 ⊢ (𝜑 → ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1))) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾))) |
22 | 17, 19, 21 | 3eqtr3d 2822 | . . 3 ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾))) |
23 | 12, 9 | mulcomd 10400 | . . 3 ⊢ (𝜑 → ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾)) = (((𝐶 − (𝐾 − 1)) / 𝐾) · (𝐶C𝑐(𝐾 − 1)))) |
24 | 22, 23 | eqtr2d 2815 | . 2 ⊢ (𝜑 → (((𝐶 − (𝐾 − 1)) / 𝐾) · (𝐶C𝑐(𝐾 − 1))) = (𝐶C𝑐𝐾)) |
25 | 9, 12, 16, 24 | mvllmuld 11209 | 1 ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) = ((𝐶C𝑐𝐾) / ((𝐶 − (𝐾 − 1)) / 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∖ cdif 3789 {csn 4398 (class class class)co 6924 ℂcc 10272 1c1 10275 + caddc 10277 · cmul 10279 − cmin 10608 / cdiv 11034 ℕcn 11378 ℕ0cn0 11646 C𝑐cbcc 39501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-n0 11647 df-z 11733 df-uz 11997 df-rp 12142 df-fz 12648 df-fzo 12789 df-seq 13124 df-exp 13183 df-fac 13383 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-clim 14631 df-prod 15043 df-fallfac 15144 df-bcc 39502 |
This theorem is referenced by: (None) |
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