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Mirrors > Home > MPE Home > Th. List > Mathboxes > bcccl | Structured version Visualization version GIF version |
Description: Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
Ref | Expression |
---|---|
bccval.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
bccval.k | ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
Ref | Expression |
---|---|
bcccl | ⊢ (𝜑 → (𝐶C𝑐𝐾) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bccval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
2 | bccval.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
3 | 1, 2 | bccval 40690 | . 2 ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾))) |
4 | fallfaccl 15370 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (𝐶 FallFac 𝐾) ∈ ℂ) | |
5 | 1, 2, 4 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐶 FallFac 𝐾) ∈ ℂ) |
6 | faccl 13644 | . . . . 5 ⊢ (𝐾 ∈ ℕ0 → (!‘𝐾) ∈ ℕ) | |
7 | 2, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (!‘𝐾) ∈ ℕ) |
8 | 7 | nncnd 11654 | . . 3 ⊢ (𝜑 → (!‘𝐾) ∈ ℂ) |
9 | 7 | nnne0d 11688 | . . 3 ⊢ (𝜑 → (!‘𝐾) ≠ 0) |
10 | 5, 8, 9 | divcld 11416 | . 2 ⊢ (𝜑 → ((𝐶 FallFac 𝐾) / (!‘𝐾)) ∈ ℂ) |
11 | 3, 10 | eqeltrd 2913 | 1 ⊢ (𝜑 → (𝐶C𝑐𝐾) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 / cdiv 11297 ℕcn 11638 ℕ0cn0 11898 !cfa 13634 FallFac cfallfac 15358 C𝑐cbcc 40688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-fac 13635 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-prod 15260 df-fallfac 15361 df-bcc 40689 |
This theorem is referenced by: bccm1k 40694 binomcxplemnn0 40701 binomcxplemfrat 40703 binomcxplemradcnv 40704 binomcxplemcvg 40706 binomcxplemdvsum 40707 binomcxplemnotnn0 40708 |
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