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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 9277 | . . . . 5 ⊢ 0 ∈ ℤ | |
2 | 4z 9296 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 2z 9294 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | 1, 2, 3 | 3pm3.2i 1176 | . . . 4 ⊢ (0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) |
5 | 0le2 9022 | . . . . 5 ⊢ 0 ≤ 2 | |
6 | 2re 9002 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 9009 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2lt4 9105 | . . . . . 6 ⊢ 2 < 4 | |
9 | 6, 7, 8 | ltleii 8073 | . . . . 5 ⊢ 2 ≤ 4 |
10 | 5, 9 | pm3.2i 272 | . . . 4 ⊢ (0 ≤ 2 ∧ 2 ≤ 4) |
11 | elfz4 10031 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (0 ≤ 2 ∧ 2 ≤ 4)) → 2 ∈ (0...4)) | |
12 | 4, 10, 11 | mp2an 426 | . . 3 ⊢ 2 ∈ (0...4) |
13 | bcval2 10743 | . . 3 ⊢ (2 ∈ (0...4) → (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2)))) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2))) |
15 | 3nn0 9207 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
16 | facp1 10723 | . . . . . 6 ⊢ (3 ∈ ℕ0 → (!‘(3 + 1)) = ((!‘3) · (3 + 1))) | |
17 | 15, 16 | ax-mp 5 | . . . . 5 ⊢ (!‘(3 + 1)) = ((!‘3) · (3 + 1)) |
18 | df-4 8993 | . . . . . 6 ⊢ 4 = (3 + 1) | |
19 | 18 | fveq2i 5530 | . . . . 5 ⊢ (!‘4) = (!‘(3 + 1)) |
20 | 18 | oveq2i 5899 | . . . . 5 ⊢ ((!‘3) · 4) = ((!‘3) · (3 + 1)) |
21 | 17, 19, 20 | 3eqtr4i 2218 | . . . 4 ⊢ (!‘4) = ((!‘3) · 4) |
22 | 4cn 9010 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
23 | 2cn 9003 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
24 | 2p2e4 9059 | . . . . . . . . 9 ⊢ (2 + 2) = 4 | |
25 | 22, 23, 23, 24 | subaddrii 8259 | . . . . . . . 8 ⊢ (4 − 2) = 2 |
26 | 25 | fveq2i 5530 | . . . . . . 7 ⊢ (!‘(4 − 2)) = (!‘2) |
27 | fac2 10724 | . . . . . . 7 ⊢ (!‘2) = 2 | |
28 | 26, 27 | eqtri 2208 | . . . . . 6 ⊢ (!‘(4 − 2)) = 2 |
29 | 28, 27 | oveq12i 5900 | . . . . 5 ⊢ ((!‘(4 − 2)) · (!‘2)) = (2 · 2) |
30 | 2t2e4 9086 | . . . . 5 ⊢ (2 · 2) = 4 | |
31 | 29, 30 | eqtri 2208 | . . . 4 ⊢ ((!‘(4 − 2)) · (!‘2)) = 4 |
32 | 21, 31 | oveq12i 5900 | . . 3 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = (((!‘3) · 4) / 4) |
33 | faccl 10728 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → (!‘3) ∈ ℕ) | |
34 | 15, 33 | ax-mp 5 | . . . . . 6 ⊢ (!‘3) ∈ ℕ |
35 | 34 | nncni 8942 | . . . . 5 ⊢ (!‘3) ∈ ℂ |
36 | 4ap0 9031 | . . . . 5 ⊢ 4 # 0 | |
37 | 35, 22, 36 | divcanap4i 8729 | . . . 4 ⊢ (((!‘3) · 4) / 4) = (!‘3) |
38 | fac3 10725 | . . . 4 ⊢ (!‘3) = 6 | |
39 | 37, 38 | eqtri 2208 | . . 3 ⊢ (((!‘3) · 4) / 4) = 6 |
40 | 32, 39 | eqtri 2208 | . 2 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = 6 |
41 | 14, 40 | eqtri 2208 | 1 ⊢ (4C2) = 6 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∧ w3a 979 = wceq 1363 ∈ wcel 2158 class class class wbr 4015 ‘cfv 5228 (class class class)co 5888 0cc0 7824 1c1 7825 + caddc 7827 · cmul 7829 ≤ cle 8006 − cmin 8141 / cdiv 8642 ℕcn 8932 2c2 8983 3c3 8984 4c4 8985 6c6 8987 ℕ0cn0 9189 ℤcz 9266 ...cfz 10021 !cfa 10718 Ccbc 10740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-fz 10022 df-seqfrec 10459 df-fac 10719 df-bc 10741 |
This theorem is referenced by: ex-bc 14708 |
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