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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 8822 | . . . . 5 ⊢ 0 ∈ ℤ | |
2 | 4z 8841 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 2z 8839 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | 1, 2, 3 | 3pm3.2i 1122 | . . . 4 ⊢ (0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) |
5 | 0le2 8573 | . . . . 5 ⊢ 0 ≤ 2 | |
6 | 2re 8553 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 8560 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2lt4 8650 | . . . . . 6 ⊢ 2 < 4 | |
9 | 6, 7, 8 | ltleii 7648 | . . . . 5 ⊢ 2 ≤ 4 |
10 | 5, 9 | pm3.2i 267 | . . . 4 ⊢ (0 ≤ 2 ∧ 2 ≤ 4) |
11 | elfz4 9494 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (0 ≤ 2 ∧ 2 ≤ 4)) → 2 ∈ (0...4)) | |
12 | 4, 10, 11 | mp2an 418 | . . 3 ⊢ 2 ∈ (0...4) |
13 | bcval2 10219 | . . 3 ⊢ (2 ∈ (0...4) → (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2)))) | |
14 | 12, 13 | ax-mp 7 | . 2 ⊢ (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2))) |
15 | 3nn0 8752 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
16 | facp1 10199 | . . . . . 6 ⊢ (3 ∈ ℕ0 → (!‘(3 + 1)) = ((!‘3) · (3 + 1))) | |
17 | 15, 16 | ax-mp 7 | . . . . 5 ⊢ (!‘(3 + 1)) = ((!‘3) · (3 + 1)) |
18 | df-4 8544 | . . . . . 6 ⊢ 4 = (3 + 1) | |
19 | 18 | fveq2i 5321 | . . . . 5 ⊢ (!‘4) = (!‘(3 + 1)) |
20 | 18 | oveq2i 5677 | . . . . 5 ⊢ ((!‘3) · 4) = ((!‘3) · (3 + 1)) |
21 | 17, 19, 20 | 3eqtr4i 2119 | . . . 4 ⊢ (!‘4) = ((!‘3) · 4) |
22 | 4cn 8561 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
23 | 2cn 8554 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
24 | 2p2e4 8604 | . . . . . . . . 9 ⊢ (2 + 2) = 4 | |
25 | 22, 23, 23, 24 | subaddrii 7832 | . . . . . . . 8 ⊢ (4 − 2) = 2 |
26 | 25 | fveq2i 5321 | . . . . . . 7 ⊢ (!‘(4 − 2)) = (!‘2) |
27 | fac2 10200 | . . . . . . 7 ⊢ (!‘2) = 2 | |
28 | 26, 27 | eqtri 2109 | . . . . . 6 ⊢ (!‘(4 − 2)) = 2 |
29 | 28, 27 | oveq12i 5678 | . . . . 5 ⊢ ((!‘(4 − 2)) · (!‘2)) = (2 · 2) |
30 | 2t2e4 8631 | . . . . 5 ⊢ (2 · 2) = 4 | |
31 | 29, 30 | eqtri 2109 | . . . 4 ⊢ ((!‘(4 − 2)) · (!‘2)) = 4 |
32 | 21, 31 | oveq12i 5678 | . . 3 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = (((!‘3) · 4) / 4) |
33 | faccl 10204 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → (!‘3) ∈ ℕ) | |
34 | 15, 33 | ax-mp 7 | . . . . . 6 ⊢ (!‘3) ∈ ℕ |
35 | 34 | nncni 8493 | . . . . 5 ⊢ (!‘3) ∈ ℂ |
36 | 4ap0 8582 | . . . . 5 ⊢ 4 # 0 | |
37 | 35, 22, 36 | divcanap4i 8287 | . . . 4 ⊢ (((!‘3) · 4) / 4) = (!‘3) |
38 | fac3 10201 | . . . 4 ⊢ (!‘3) = 6 | |
39 | 37, 38 | eqtri 2109 | . . 3 ⊢ (((!‘3) · 4) / 4) = 6 |
40 | 32, 39 | eqtri 2109 | . 2 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = 6 |
41 | 14, 40 | eqtri 2109 | 1 ⊢ (4C2) = 6 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 class class class wbr 3851 ‘cfv 5028 (class class class)co 5666 0cc0 7411 1c1 7412 + caddc 7414 · cmul 7416 ≤ cle 7584 − cmin 7714 / cdiv 8200 ℕcn 8483 2c2 8534 3c3 8535 4c4 8536 6c6 8538 ℕ0cn0 8734 ℤcz 8811 ...cfz 9485 !cfa 10194 Ccbc 10216 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 ax-pre-mulext 7524 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-frec 6170 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 df-div 8201 df-inn 8484 df-2 8542 df-3 8543 df-4 8544 df-5 8545 df-6 8546 df-n0 8735 df-z 8812 df-uz 9081 df-q 9166 df-fz 9486 df-iseq 9914 df-fac 10195 df-bc 10217 |
This theorem is referenced by: ex-bc 11929 |
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