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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bccbc | Structured version Visualization version GIF version | ||
| Description: The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.) | 
| Ref | Expression | 
|---|---|
| bccbc.c | ⊢ (𝜑 → 𝑁 ∈ ℕ0) | 
| bccbc.k | ⊢ (𝜑 → 𝐾 ∈ ℕ0) | 
| Ref | Expression | 
|---|---|
| bccbc | ⊢ (𝜑 → (𝑁C𝑐𝐾) = (𝑁C𝐾)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bccbc.c | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
| 2 | 1 | nn0cnd 12589 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ) | 
| 3 | bccbc.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
| 4 | 2, 3 | bccval 44357 | . . . 4 ⊢ (𝜑 → (𝑁C𝑐𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) | 
| 5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) | 
| 6 | bcfallfac 16080 | . . . 4 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) | |
| 7 | 6 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = ((𝑁 FallFac 𝐾) / (!‘𝐾))) | 
| 8 | 5, 7 | eqtr4d 2780 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = (𝑁C𝐾)) | 
| 9 | nn0split 13683 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) | |
| 10 | 1, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) | 
| 11 | 3, 10 | eleqtrd 2843 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) | 
| 12 | elun 4153 | . . . . . . 7 ⊢ (𝐾 ∈ ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))) ↔ (𝐾 ∈ (0...𝑁) ∨ 𝐾 ∈ (ℤ≥‘(𝑁 + 1)))) | |
| 13 | 11, 12 | sylib 218 | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ (0...𝑁) ∨ 𝐾 ∈ (ℤ≥‘(𝑁 + 1)))) | 
| 14 | 13 | orcanai 1005 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) | 
| 15 | eluzle 12891 | . . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝐾) | |
| 16 | 15 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) → (𝑁 + 1) ≤ 𝐾) | 
| 17 | 1 | nn0zd 12639 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 18 | 3 | nn0zd 12639 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ) | 
| 19 | zltp1le 12667 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 < 𝐾 ↔ (𝑁 + 1) ≤ 𝐾)) | |
| 20 | 17, 18, 19 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (𝑁 < 𝐾 ↔ (𝑁 + 1) ≤ 𝐾)) | 
| 21 | 20 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) → (𝑁 < 𝐾 ↔ (𝑁 + 1) ≤ 𝐾)) | 
| 22 | 16, 21 | mpbird 257 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑁 < 𝐾) | 
| 23 | 14, 22 | syldan 591 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → 𝑁 < 𝐾) | 
| 24 | 1 | nn0ge0d 12590 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝑁) | 
| 25 | 0zd 12625 | . . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℤ) | |
| 26 | elfzo 13701 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 ∈ (0..^𝐾) ↔ (0 ≤ 𝑁 ∧ 𝑁 < 𝐾))) | |
| 27 | 17, 25, 18, 26 | syl3anc 1373 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ (0..^𝐾) ↔ (0 ≤ 𝑁 ∧ 𝑁 < 𝐾))) | 
| 28 | 27 | biimpar 477 | . . . . . . 7 ⊢ ((𝜑 ∧ (0 ≤ 𝑁 ∧ 𝑁 < 𝐾)) → 𝑁 ∈ (0..^𝐾)) | 
| 29 | fzoval 13700 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (0..^𝐾) = (0...(𝐾 − 1))) | |
| 30 | 18, 29 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (0..^𝐾) = (0...(𝐾 − 1))) | 
| 31 | 30 | eleq2d 2827 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∈ (0..^𝐾) ↔ 𝑁 ∈ (0...(𝐾 − 1)))) | 
| 32 | 31 | biimpa 476 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (0..^𝐾)) → 𝑁 ∈ (0...(𝐾 − 1))) | 
| 33 | 2, 3 | bcc0 44359 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑁C𝑐𝐾) = 0 ↔ 𝑁 ∈ (0...(𝐾 − 1)))) | 
| 34 | 33 | biimpar 477 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (0...(𝐾 − 1))) → (𝑁C𝑐𝐾) = 0) | 
| 35 | 32, 34 | syldan 591 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (0..^𝐾)) → (𝑁C𝑐𝐾) = 0) | 
| 36 | 28, 35 | syldan 591 | . . . . . 6 ⊢ ((𝜑 ∧ (0 ≤ 𝑁 ∧ 𝑁 < 𝐾)) → (𝑁C𝑐𝐾) = 0) | 
| 37 | 24, 36 | sylanr1 682 | . . . . 5 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝑁 < 𝐾)) → (𝑁C𝑐𝐾) = 0) | 
| 38 | 37 | anabss5 668 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 < 𝐾) → (𝑁C𝑐𝐾) = 0) | 
| 39 | 23, 38 | syldan 591 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = 0) | 
| 40 | 1, 18 | jca 511 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ)) | 
| 41 | bcval3 14345 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0) | |
| 42 | 41 | 3expa 1119 | . . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0) | 
| 43 | 40, 42 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0) | 
| 44 | 39, 43 | eqtr4d 2780 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝑐𝐾) = (𝑁C𝐾)) | 
| 45 | 8, 44 | pm2.61dan 813 | 1 ⊢ (𝜑 → (𝑁C𝑐𝐾) = (𝑁C𝐾)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕ0cn0 12526 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 ..^cfzo 13694 !cfa 14312 Ccbc 14341 FallFac cfallfac 16040 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-bc 14342 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: binomcxplemnn0 44368 | 
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