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| 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 12626 | . . . . 5 ⊢ 0 ∈ ℤ | |
| 2 | 4z 12653 | . . . . 5 ⊢ 4 ∈ ℤ | |
| 3 | 2z 12651 | . . . . 5 ⊢ 2 ∈ ℤ | |
| 4 | 1, 2, 3 | 3pm3.2i 1339 | . . . 4 ⊢ (0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) | 
| 5 | 0le2 12369 | . . . . 5 ⊢ 0 ≤ 2 | |
| 6 | 2re 12341 | . . . . . 6 ⊢ 2 ∈ ℝ | |
| 7 | 4re 12351 | . . . . . 6 ⊢ 4 ∈ ℝ | |
| 8 | 2lt4 12442 | . . . . . 6 ⊢ 2 < 4 | |
| 9 | 6, 7, 8 | ltleii 11385 | . . . . 5 ⊢ 2 ≤ 4 | 
| 10 | 5, 9 | pm3.2i 470 | . . . 4 ⊢ (0 ≤ 2 ∧ 2 ≤ 4) | 
| 11 | elfz4 13558 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (0 ≤ 2 ∧ 2 ≤ 4)) → 2 ∈ (0...4)) | |
| 12 | 4, 10, 11 | mp2an 692 | . . 3 ⊢ 2 ∈ (0...4) | 
| 13 | bcval2 14345 | . . 3 ⊢ (2 ∈ (0...4) → (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2)))) | |
| 14 | 12, 13 | ax-mp 5 | . 2 ⊢ (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2))) | 
| 15 | 3nn0 12546 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
| 16 | facp1 14318 | . . . . . 6 ⊢ (3 ∈ ℕ0 → (!‘(3 + 1)) = ((!‘3) · (3 + 1))) | |
| 17 | 15, 16 | ax-mp 5 | . . . . 5 ⊢ (!‘(3 + 1)) = ((!‘3) · (3 + 1)) | 
| 18 | df-4 12332 | . . . . . 6 ⊢ 4 = (3 + 1) | |
| 19 | 18 | fveq2i 6908 | . . . . 5 ⊢ (!‘4) = (!‘(3 + 1)) | 
| 20 | 18 | oveq2i 7443 | . . . . 5 ⊢ ((!‘3) · 4) = ((!‘3) · (3 + 1)) | 
| 21 | 17, 19, 20 | 3eqtr4i 2774 | . . . 4 ⊢ (!‘4) = ((!‘3) · 4) | 
| 22 | 4cn 12352 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
| 23 | 2cn 12342 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
| 24 | 2p2e4 12402 | . . . . . . . . 9 ⊢ (2 + 2) = 4 | |
| 25 | 22, 23, 23, 24 | subaddrii 11599 | . . . . . . . 8 ⊢ (4 − 2) = 2 | 
| 26 | 25 | fveq2i 6908 | . . . . . . 7 ⊢ (!‘(4 − 2)) = (!‘2) | 
| 27 | fac2 14319 | . . . . . . 7 ⊢ (!‘2) = 2 | |
| 28 | 26, 27 | eqtri 2764 | . . . . . 6 ⊢ (!‘(4 − 2)) = 2 | 
| 29 | 28, 27 | oveq12i 7444 | . . . . 5 ⊢ ((!‘(4 − 2)) · (!‘2)) = (2 · 2) | 
| 30 | 2t2e4 12431 | . . . . 5 ⊢ (2 · 2) = 4 | |
| 31 | 29, 30 | eqtri 2764 | . . . 4 ⊢ ((!‘(4 − 2)) · (!‘2)) = 4 | 
| 32 | 21, 31 | oveq12i 7444 | . . 3 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = (((!‘3) · 4) / 4) | 
| 33 | faccl 14323 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → (!‘3) ∈ ℕ) | |
| 34 | 15, 33 | ax-mp 5 | . . . . . 6 ⊢ (!‘3) ∈ ℕ | 
| 35 | 34 | nncni 12277 | . . . . 5 ⊢ (!‘3) ∈ ℂ | 
| 36 | 4ne0 12375 | . . . . 5 ⊢ 4 ≠ 0 | |
| 37 | 35, 22, 36 | divcan4i 12015 | . . . 4 ⊢ (((!‘3) · 4) / 4) = (!‘3) | 
| 38 | fac3 14320 | . . . 4 ⊢ (!‘3) = 6 | |
| 39 | 37, 38 | eqtri 2764 | . . 3 ⊢ (((!‘3) · 4) / 4) = 6 | 
| 40 | 32, 39 | eqtri 2764 | . 2 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = 6 | 
| 41 | 14, 40 | eqtri 2764 | 1 ⊢ (4C2) = 6 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 ≤ cle 11297 − cmin 11493 / cdiv 11921 ℕcn 12267 2c2 12322 3c3 12323 4c4 12324 6c6 12326 ℕ0cn0 12528 ℤcz 12615 ...cfz 13548 !cfa 14313 Ccbc 14342 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-seq 14044 df-fac 14314 df-bc 14343 | 
| This theorem is referenced by: bpoly4 16096 ex-bc 30472 5bc2eq10 42144 | 
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