![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 4bc2eq6 | Structured version Visualization version 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 11802 | . . . . 5 ⊢ 0 ∈ ℤ | |
2 | 4z 11827 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 2z 11825 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | 1, 2, 3 | 3pm3.2i 1320 | . . . 4 ⊢ (0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) |
5 | 0le2 11547 | . . . . 5 ⊢ 0 ≤ 2 | |
6 | 2re 11512 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 11523 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2lt4 11620 | . . . . . 6 ⊢ 2 < 4 | |
9 | 6, 7, 8 | ltleii 10561 | . . . . 5 ⊢ 2 ≤ 4 |
10 | 5, 9 | pm3.2i 463 | . . . 4 ⊢ (0 ≤ 2 ∧ 2 ≤ 4) |
11 | elfz4 12715 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (0 ≤ 2 ∧ 2 ≤ 4)) → 2 ∈ (0...4)) | |
12 | 4, 10, 11 | mp2an 680 | . . 3 ⊢ 2 ∈ (0...4) |
13 | bcval2 13478 | . . 3 ⊢ (2 ∈ (0...4) → (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2)))) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2))) |
15 | 3nn0 11725 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
16 | facp1 13451 | . . . . . 6 ⊢ (3 ∈ ℕ0 → (!‘(3 + 1)) = ((!‘3) · (3 + 1))) | |
17 | 15, 16 | ax-mp 5 | . . . . 5 ⊢ (!‘(3 + 1)) = ((!‘3) · (3 + 1)) |
18 | df-4 11503 | . . . . . 6 ⊢ 4 = (3 + 1) | |
19 | 18 | fveq2i 6499 | . . . . 5 ⊢ (!‘4) = (!‘(3 + 1)) |
20 | 18 | oveq2i 6985 | . . . . 5 ⊢ ((!‘3) · 4) = ((!‘3) · (3 + 1)) |
21 | 17, 19, 20 | 3eqtr4i 2805 | . . . 4 ⊢ (!‘4) = ((!‘3) · 4) |
22 | 4cn 11524 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
23 | 2cn 11513 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
24 | 2p2e4 11580 | . . . . . . . . 9 ⊢ (2 + 2) = 4 | |
25 | 22, 23, 23, 24 | subaddrii 10774 | . . . . . . . 8 ⊢ (4 − 2) = 2 |
26 | 25 | fveq2i 6499 | . . . . . . 7 ⊢ (!‘(4 − 2)) = (!‘2) |
27 | fac2 13452 | . . . . . . 7 ⊢ (!‘2) = 2 | |
28 | 26, 27 | eqtri 2795 | . . . . . 6 ⊢ (!‘(4 − 2)) = 2 |
29 | 28, 27 | oveq12i 6986 | . . . . 5 ⊢ ((!‘(4 − 2)) · (!‘2)) = (2 · 2) |
30 | 2t2e4 11609 | . . . . 5 ⊢ (2 · 2) = 4 | |
31 | 29, 30 | eqtri 2795 | . . . 4 ⊢ ((!‘(4 − 2)) · (!‘2)) = 4 |
32 | 21, 31 | oveq12i 6986 | . . 3 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = (((!‘3) · 4) / 4) |
33 | faccl 13456 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → (!‘3) ∈ ℕ) | |
34 | 15, 33 | ax-mp 5 | . . . . . 6 ⊢ (!‘3) ∈ ℕ |
35 | 34 | nncni 11448 | . . . . 5 ⊢ (!‘3) ∈ ℂ |
36 | 4ne0 11553 | . . . . 5 ⊢ 4 ≠ 0 | |
37 | 35, 22, 36 | divcan4i 11186 | . . . 4 ⊢ (((!‘3) · 4) / 4) = (!‘3) |
38 | fac3 13453 | . . . 4 ⊢ (!‘3) = 6 | |
39 | 37, 38 | eqtri 2795 | . . 3 ⊢ (((!‘3) · 4) / 4) = 6 |
40 | 32, 39 | eqtri 2795 | . 2 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = 6 |
41 | 14, 40 | eqtri 2795 | 1 ⊢ (4C2) = 6 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 class class class wbr 4925 ‘cfv 6185 (class class class)co 6974 0cc0 10333 1c1 10334 + caddc 10336 · cmul 10338 ≤ cle 10473 − cmin 10668 / cdiv 11096 ℕcn 11437 2c2 11493 3c3 11494 4c4 11495 6c6 11497 ℕ0cn0 11705 ℤcz 11791 ...cfz 12706 !cfa 13446 Ccbc 13475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 df-seq 13183 df-fac 13447 df-bc 13476 |
This theorem is referenced by: bpoly4 15271 ex-bc 28024 |
Copyright terms: Public domain | W3C validator |