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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 9089 | . . . . 5 ⊢ 0 ∈ ℤ | |
2 | 4z 9108 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 2z 9106 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | 1, 2, 3 | 3pm3.2i 1160 | . . . 4 ⊢ (0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) |
5 | 0le2 8834 | . . . . 5 ⊢ 0 ≤ 2 | |
6 | 2re 8814 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 8821 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2lt4 8917 | . . . . . 6 ⊢ 2 < 4 | |
9 | 6, 7, 8 | ltleii 7890 | . . . . 5 ⊢ 2 ≤ 4 |
10 | 5, 9 | pm3.2i 270 | . . . 4 ⊢ (0 ≤ 2 ∧ 2 ≤ 4) |
11 | elfz4 9830 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (0 ≤ 2 ∧ 2 ≤ 4)) → 2 ∈ (0...4)) | |
12 | 4, 10, 11 | mp2an 423 | . . 3 ⊢ 2 ∈ (0...4) |
13 | bcval2 10528 | . . 3 ⊢ (2 ∈ (0...4) → (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2)))) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2))) |
15 | 3nn0 9019 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
16 | facp1 10508 | . . . . . 6 ⊢ (3 ∈ ℕ0 → (!‘(3 + 1)) = ((!‘3) · (3 + 1))) | |
17 | 15, 16 | ax-mp 5 | . . . . 5 ⊢ (!‘(3 + 1)) = ((!‘3) · (3 + 1)) |
18 | df-4 8805 | . . . . . 6 ⊢ 4 = (3 + 1) | |
19 | 18 | fveq2i 5432 | . . . . 5 ⊢ (!‘4) = (!‘(3 + 1)) |
20 | 18 | oveq2i 5793 | . . . . 5 ⊢ ((!‘3) · 4) = ((!‘3) · (3 + 1)) |
21 | 17, 19, 20 | 3eqtr4i 2171 | . . . 4 ⊢ (!‘4) = ((!‘3) · 4) |
22 | 4cn 8822 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
23 | 2cn 8815 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
24 | 2p2e4 8871 | . . . . . . . . 9 ⊢ (2 + 2) = 4 | |
25 | 22, 23, 23, 24 | subaddrii 8075 | . . . . . . . 8 ⊢ (4 − 2) = 2 |
26 | 25 | fveq2i 5432 | . . . . . . 7 ⊢ (!‘(4 − 2)) = (!‘2) |
27 | fac2 10509 | . . . . . . 7 ⊢ (!‘2) = 2 | |
28 | 26, 27 | eqtri 2161 | . . . . . 6 ⊢ (!‘(4 − 2)) = 2 |
29 | 28, 27 | oveq12i 5794 | . . . . 5 ⊢ ((!‘(4 − 2)) · (!‘2)) = (2 · 2) |
30 | 2t2e4 8898 | . . . . 5 ⊢ (2 · 2) = 4 | |
31 | 29, 30 | eqtri 2161 | . . . 4 ⊢ ((!‘(4 − 2)) · (!‘2)) = 4 |
32 | 21, 31 | oveq12i 5794 | . . 3 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = (((!‘3) · 4) / 4) |
33 | faccl 10513 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → (!‘3) ∈ ℕ) | |
34 | 15, 33 | ax-mp 5 | . . . . . 6 ⊢ (!‘3) ∈ ℕ |
35 | 34 | nncni 8754 | . . . . 5 ⊢ (!‘3) ∈ ℂ |
36 | 4ap0 8843 | . . . . 5 ⊢ 4 # 0 | |
37 | 35, 22, 36 | divcanap4i 8543 | . . . 4 ⊢ (((!‘3) · 4) / 4) = (!‘3) |
38 | fac3 10510 | . . . 4 ⊢ (!‘3) = 6 | |
39 | 37, 38 | eqtri 2161 | . . 3 ⊢ (((!‘3) · 4) / 4) = 6 |
40 | 32, 39 | eqtri 2161 | . 2 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = 6 |
41 | 14, 40 | eqtri 2161 | 1 ⊢ (4C2) = 6 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 0cc0 7644 1c1 7645 + caddc 7647 · cmul 7649 ≤ cle 7825 − cmin 7957 / cdiv 8456 ℕcn 8744 2c2 8795 3c3 8796 4c4 8797 6c6 8799 ℕ0cn0 9001 ℤcz 9078 ...cfz 9821 !cfa 10503 Ccbc 10525 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-fz 9822 df-seqfrec 10250 df-fac 10504 df-bc 10526 |
This theorem is referenced by: ex-bc 13112 |
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