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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 9266 | . . . . 5 ⊢ 0 ∈ ℤ | |
2 | 4z 9285 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 2z 9283 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | 1, 2, 3 | 3pm3.2i 1175 | . . . 4 ⊢ (0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) |
5 | 0le2 9011 | . . . . 5 ⊢ 0 ≤ 2 | |
6 | 2re 8991 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 8998 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2lt4 9094 | . . . . . 6 ⊢ 2 < 4 | |
9 | 6, 7, 8 | ltleii 8062 | . . . . 5 ⊢ 2 ≤ 4 |
10 | 5, 9 | pm3.2i 272 | . . . 4 ⊢ (0 ≤ 2 ∧ 2 ≤ 4) |
11 | elfz4 10020 | . . . 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 10732 | . . 3 ⊢ (2 ∈ (0...4) → (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2)))) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2))) |
15 | 3nn0 9196 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
16 | facp1 10712 | . . . . . 6 ⊢ (3 ∈ ℕ0 → (!‘(3 + 1)) = ((!‘3) · (3 + 1))) | |
17 | 15, 16 | ax-mp 5 | . . . . 5 ⊢ (!‘(3 + 1)) = ((!‘3) · (3 + 1)) |
18 | df-4 8982 | . . . . . 6 ⊢ 4 = (3 + 1) | |
19 | 18 | fveq2i 5520 | . . . . 5 ⊢ (!‘4) = (!‘(3 + 1)) |
20 | 18 | oveq2i 5888 | . . . . 5 ⊢ ((!‘3) · 4) = ((!‘3) · (3 + 1)) |
21 | 17, 19, 20 | 3eqtr4i 2208 | . . . 4 ⊢ (!‘4) = ((!‘3) · 4) |
22 | 4cn 8999 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
23 | 2cn 8992 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
24 | 2p2e4 9048 | . . . . . . . . 9 ⊢ (2 + 2) = 4 | |
25 | 22, 23, 23, 24 | subaddrii 8248 | . . . . . . . 8 ⊢ (4 − 2) = 2 |
26 | 25 | fveq2i 5520 | . . . . . . 7 ⊢ (!‘(4 − 2)) = (!‘2) |
27 | fac2 10713 | . . . . . . 7 ⊢ (!‘2) = 2 | |
28 | 26, 27 | eqtri 2198 | . . . . . 6 ⊢ (!‘(4 − 2)) = 2 |
29 | 28, 27 | oveq12i 5889 | . . . . 5 ⊢ ((!‘(4 − 2)) · (!‘2)) = (2 · 2) |
30 | 2t2e4 9075 | . . . . 5 ⊢ (2 · 2) = 4 | |
31 | 29, 30 | eqtri 2198 | . . . 4 ⊢ ((!‘(4 − 2)) · (!‘2)) = 4 |
32 | 21, 31 | oveq12i 5889 | . . 3 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = (((!‘3) · 4) / 4) |
33 | faccl 10717 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → (!‘3) ∈ ℕ) | |
34 | 15, 33 | ax-mp 5 | . . . . . 6 ⊢ (!‘3) ∈ ℕ |
35 | 34 | nncni 8931 | . . . . 5 ⊢ (!‘3) ∈ ℂ |
36 | 4ap0 9020 | . . . . 5 ⊢ 4 # 0 | |
37 | 35, 22, 36 | divcanap4i 8718 | . . . 4 ⊢ (((!‘3) · 4) / 4) = (!‘3) |
38 | fac3 10714 | . . . 4 ⊢ (!‘3) = 6 | |
39 | 37, 38 | eqtri 2198 | . . 3 ⊢ (((!‘3) · 4) / 4) = 6 |
40 | 32, 39 | eqtri 2198 | . 2 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = 6 |
41 | 14, 40 | eqtri 2198 | 1 ⊢ (4C2) = 6 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4005 ‘cfv 5218 (class class class)co 5877 0cc0 7813 1c1 7814 + caddc 7816 · cmul 7818 ≤ cle 7995 − cmin 8130 / cdiv 8631 ℕcn 8921 2c2 8972 3c3 8973 4c4 8974 6c6 8976 ℕ0cn0 9178 ℤcz 9255 ...cfz 10010 !cfa 10707 Ccbc 10729 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-fz 10011 df-seqfrec 10448 df-fac 10708 df-bc 10730 |
This theorem is referenced by: ex-bc 14566 |
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