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Mirrors > Home > ILE Home > Th. List > 4bc2eq6 | GIF version |
Description: The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.) |
Ref | Expression |
---|---|
4bc2eq6 | ⊢ (4C2) = 6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 9058 | . . . . 5 ⊢ 0 ∈ ℤ | |
2 | 4z 9077 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 2z 9075 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | 1, 2, 3 | 3pm3.2i 1159 | . . . 4 ⊢ (0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) |
5 | 0le2 8803 | . . . . 5 ⊢ 0 ≤ 2 | |
6 | 2re 8783 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 8790 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2lt4 8886 | . . . . . 6 ⊢ 2 < 4 | |
9 | 6, 7, 8 | ltleii 7859 | . . . . 5 ⊢ 2 ≤ 4 |
10 | 5, 9 | pm3.2i 270 | . . . 4 ⊢ (0 ≤ 2 ∧ 2 ≤ 4) |
11 | elfz4 9792 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (0 ≤ 2 ∧ 2 ≤ 4)) → 2 ∈ (0...4)) | |
12 | 4, 10, 11 | mp2an 422 | . . 3 ⊢ 2 ∈ (0...4) |
13 | bcval2 10489 | . . 3 ⊢ (2 ∈ (0...4) → (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2)))) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2))) |
15 | 3nn0 8988 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
16 | facp1 10469 | . . . . . 6 ⊢ (3 ∈ ℕ0 → (!‘(3 + 1)) = ((!‘3) · (3 + 1))) | |
17 | 15, 16 | ax-mp 5 | . . . . 5 ⊢ (!‘(3 + 1)) = ((!‘3) · (3 + 1)) |
18 | df-4 8774 | . . . . . 6 ⊢ 4 = (3 + 1) | |
19 | 18 | fveq2i 5417 | . . . . 5 ⊢ (!‘4) = (!‘(3 + 1)) |
20 | 18 | oveq2i 5778 | . . . . 5 ⊢ ((!‘3) · 4) = ((!‘3) · (3 + 1)) |
21 | 17, 19, 20 | 3eqtr4i 2168 | . . . 4 ⊢ (!‘4) = ((!‘3) · 4) |
22 | 4cn 8791 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
23 | 2cn 8784 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
24 | 2p2e4 8840 | . . . . . . . . 9 ⊢ (2 + 2) = 4 | |
25 | 22, 23, 23, 24 | subaddrii 8044 | . . . . . . . 8 ⊢ (4 − 2) = 2 |
26 | 25 | fveq2i 5417 | . . . . . . 7 ⊢ (!‘(4 − 2)) = (!‘2) |
27 | fac2 10470 | . . . . . . 7 ⊢ (!‘2) = 2 | |
28 | 26, 27 | eqtri 2158 | . . . . . 6 ⊢ (!‘(4 − 2)) = 2 |
29 | 28, 27 | oveq12i 5779 | . . . . 5 ⊢ ((!‘(4 − 2)) · (!‘2)) = (2 · 2) |
30 | 2t2e4 8867 | . . . . 5 ⊢ (2 · 2) = 4 | |
31 | 29, 30 | eqtri 2158 | . . . 4 ⊢ ((!‘(4 − 2)) · (!‘2)) = 4 |
32 | 21, 31 | oveq12i 5779 | . . 3 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = (((!‘3) · 4) / 4) |
33 | faccl 10474 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → (!‘3) ∈ ℕ) | |
34 | 15, 33 | ax-mp 5 | . . . . . 6 ⊢ (!‘3) ∈ ℕ |
35 | 34 | nncni 8723 | . . . . 5 ⊢ (!‘3) ∈ ℂ |
36 | 4ap0 8812 | . . . . 5 ⊢ 4 # 0 | |
37 | 35, 22, 36 | divcanap4i 8512 | . . . 4 ⊢ (((!‘3) · 4) / 4) = (!‘3) |
38 | fac3 10471 | . . . 4 ⊢ (!‘3) = 6 | |
39 | 37, 38 | eqtri 2158 | . . 3 ⊢ (((!‘3) · 4) / 4) = 6 |
40 | 32, 39 | eqtri 2158 | . 2 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = 6 |
41 | 14, 40 | eqtri 2158 | 1 ⊢ (4C2) = 6 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 0cc0 7613 1c1 7614 + caddc 7616 · cmul 7618 ≤ cle 7794 − cmin 7926 / cdiv 8425 ℕcn 8713 2c2 8764 3c3 8765 4c4 8766 6c6 8768 ℕ0cn0 8970 ℤcz 9047 ...cfz 9783 !cfa 10464 Ccbc 10486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-fz 9784 df-seqfrec 10212 df-fac 10465 df-bc 10487 |
This theorem is referenced by: ex-bc 12930 |
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