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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 12607 | . . . . 5 ⊢ 0 ∈ ℤ | |
2 | 4z 12634 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 2z 12632 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | 1, 2, 3 | 3pm3.2i 1336 | . . . 4 ⊢ (0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) |
5 | 0le2 12352 | . . . . 5 ⊢ 0 ≤ 2 | |
6 | 2re 12324 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 12334 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2lt4 12425 | . . . . . 6 ⊢ 2 < 4 | |
9 | 6, 7, 8 | ltleii 11375 | . . . . 5 ⊢ 2 ≤ 4 |
10 | 5, 9 | pm3.2i 469 | . . . 4 ⊢ (0 ≤ 2 ∧ 2 ≤ 4) |
11 | elfz4 13534 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (0 ≤ 2 ∧ 2 ≤ 4)) → 2 ∈ (0...4)) | |
12 | 4, 10, 11 | mp2an 690 | . . 3 ⊢ 2 ∈ (0...4) |
13 | bcval2 14304 | . . 3 ⊢ (2 ∈ (0...4) → (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2)))) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2))) |
15 | 3nn0 12528 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
16 | facp1 14277 | . . . . . 6 ⊢ (3 ∈ ℕ0 → (!‘(3 + 1)) = ((!‘3) · (3 + 1))) | |
17 | 15, 16 | ax-mp 5 | . . . . 5 ⊢ (!‘(3 + 1)) = ((!‘3) · (3 + 1)) |
18 | df-4 12315 | . . . . . 6 ⊢ 4 = (3 + 1) | |
19 | 18 | fveq2i 6905 | . . . . 5 ⊢ (!‘4) = (!‘(3 + 1)) |
20 | 18 | oveq2i 7437 | . . . . 5 ⊢ ((!‘3) · 4) = ((!‘3) · (3 + 1)) |
21 | 17, 19, 20 | 3eqtr4i 2766 | . . . 4 ⊢ (!‘4) = ((!‘3) · 4) |
22 | 4cn 12335 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
23 | 2cn 12325 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
24 | 2p2e4 12385 | . . . . . . . . 9 ⊢ (2 + 2) = 4 | |
25 | 22, 23, 23, 24 | subaddrii 11587 | . . . . . . . 8 ⊢ (4 − 2) = 2 |
26 | 25 | fveq2i 6905 | . . . . . . 7 ⊢ (!‘(4 − 2)) = (!‘2) |
27 | fac2 14278 | . . . . . . 7 ⊢ (!‘2) = 2 | |
28 | 26, 27 | eqtri 2756 | . . . . . 6 ⊢ (!‘(4 − 2)) = 2 |
29 | 28, 27 | oveq12i 7438 | . . . . 5 ⊢ ((!‘(4 − 2)) · (!‘2)) = (2 · 2) |
30 | 2t2e4 12414 | . . . . 5 ⊢ (2 · 2) = 4 | |
31 | 29, 30 | eqtri 2756 | . . . 4 ⊢ ((!‘(4 − 2)) · (!‘2)) = 4 |
32 | 21, 31 | oveq12i 7438 | . . 3 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = (((!‘3) · 4) / 4) |
33 | faccl 14282 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → (!‘3) ∈ ℕ) | |
34 | 15, 33 | ax-mp 5 | . . . . . 6 ⊢ (!‘3) ∈ ℕ |
35 | 34 | nncni 12260 | . . . . 5 ⊢ (!‘3) ∈ ℂ |
36 | 4ne0 12358 | . . . . 5 ⊢ 4 ≠ 0 | |
37 | 35, 22, 36 | divcan4i 11999 | . . . 4 ⊢ (((!‘3) · 4) / 4) = (!‘3) |
38 | fac3 14279 | . . . 4 ⊢ (!‘3) = 6 | |
39 | 37, 38 | eqtri 2756 | . . 3 ⊢ (((!‘3) · 4) / 4) = 6 |
40 | 32, 39 | eqtri 2756 | . 2 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = 6 |
41 | 14, 40 | eqtri 2756 | 1 ⊢ (4C2) = 6 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 ≤ cle 11287 − cmin 11482 / cdiv 11909 ℕcn 12250 2c2 12305 3c3 12306 4c4 12307 6c6 12309 ℕ0cn0 12510 ℤcz 12596 ...cfz 13524 !cfa 14272 Ccbc 14301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-seq 14007 df-fac 14273 df-bc 14302 |
This theorem is referenced by: bpoly4 16043 ex-bc 30282 5bc2eq10 41646 |
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