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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 3963 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 3 | bccm1k.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ) | |
| 4 | 3 | nncnd 12282 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
| 5 | 1cnd 11256 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
| 6 | 4, 5 | subcld 11620 | . . . 4 ⊢ (𝜑 → (𝐾 − 1) ∈ ℂ) |
| 7 | 2, 6 | subcld 11620 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐾 − 1)) ∈ ℂ) |
| 8 | 3 | nnne0d 12316 | . . 3 ⊢ (𝜑 → 𝐾 ≠ 0) |
| 9 | 7, 4, 8 | divcld 12043 | . 2 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / 𝐾) ∈ ℂ) |
| 10 | nnm1nn0 12567 | . . . 4 ⊢ (𝐾 ∈ ℕ → (𝐾 − 1) ∈ ℕ0) | |
| 11 | 3, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ0) |
| 12 | 2, 11 | bcccl 44358 | . 2 ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) ∈ ℂ) |
| 13 | eldifsni 4790 | . . . . 5 ⊢ (𝐶 ∈ (ℂ ∖ {(𝐾 − 1)}) → 𝐶 ≠ (𝐾 − 1)) | |
| 14 | 1, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ (𝐾 − 1)) |
| 15 | 2, 6, 14 | subne0d 11629 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐾 − 1)) ≠ 0) |
| 16 | 7, 4, 15, 8 | divne0d 12059 | . 2 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / 𝐾) ≠ 0) |
| 17 | 2, 11 | bccp1k 44360 | . . . 4 ⊢ (𝜑 → (𝐶C𝑐((𝐾 − 1) + 1)) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1)))) |
| 18 | 4, 5 | npcand 11624 | . . . . 5 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾) |
| 19 | 18 | oveq2d 7447 | . . . 4 ⊢ (𝜑 → (𝐶C𝑐((𝐾 − 1) + 1)) = (𝐶C𝑐𝐾)) |
| 20 | 18 | oveq2d 7447 | . . . . 5 ⊢ (𝜑 → ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1)) = ((𝐶 − (𝐾 − 1)) / 𝐾)) |
| 21 | 20 | oveq2d 7447 | . . . 4 ⊢ (𝜑 → ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / ((𝐾 − 1) + 1))) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾))) |
| 22 | 17, 19, 21 | 3eqtr3d 2785 | . . 3 ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾))) |
| 23 | 12, 9 | mulcomd 11282 | . . 3 ⊢ (𝜑 → ((𝐶C𝑐(𝐾 − 1)) · ((𝐶 − (𝐾 − 1)) / 𝐾)) = (((𝐶 − (𝐾 − 1)) / 𝐾) · (𝐶C𝑐(𝐾 − 1)))) |
| 24 | 22, 23 | eqtr2d 2778 | . 2 ⊢ (𝜑 → (((𝐶 − (𝐾 − 1)) / 𝐾) · (𝐶C𝑐(𝐾 − 1))) = (𝐶C𝑐𝐾)) |
| 25 | 9, 12, 16, 24 | mvllmuld 12099 | 1 ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) = ((𝐶C𝑐𝐾) / ((𝐶 − (𝐾 − 1)) / 𝐾))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 / cdiv 11920 ℕcn 12266 ℕ0cn0 12526 C𝑐cbcc 44355 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-fac 14313 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-prod 15940 df-fallfac 16043 df-bcc 44356 |
| This theorem is referenced by: (None) |
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