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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 12515 | . . . . 5 ⊢ 0 ∈ ℤ | |
2 | 4z 12542 | . . . . 5 ⊢ 4 ∈ ℤ | |
3 | 2z 12540 | . . . . 5 ⊢ 2 ∈ ℤ | |
4 | 1, 2, 3 | 3pm3.2i 1340 | . . . 4 ⊢ (0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) |
5 | 0le2 12260 | . . . . 5 ⊢ 0 ≤ 2 | |
6 | 2re 12232 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 12242 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2lt4 12333 | . . . . . 6 ⊢ 2 < 4 | |
9 | 6, 7, 8 | ltleii 11283 | . . . . 5 ⊢ 2 ≤ 4 |
10 | 5, 9 | pm3.2i 472 | . . . 4 ⊢ (0 ≤ 2 ∧ 2 ≤ 4) |
11 | elfz4 13440 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ∈ ℤ) ∧ (0 ≤ 2 ∧ 2 ≤ 4)) → 2 ∈ (0...4)) | |
12 | 4, 10, 11 | mp2an 691 | . . 3 ⊢ 2 ∈ (0...4) |
13 | bcval2 14211 | . . 3 ⊢ (2 ∈ (0...4) → (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2)))) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ (4C2) = ((!‘4) / ((!‘(4 − 2)) · (!‘2))) |
15 | 3nn0 12436 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
16 | facp1 14184 | . . . . . 6 ⊢ (3 ∈ ℕ0 → (!‘(3 + 1)) = ((!‘3) · (3 + 1))) | |
17 | 15, 16 | ax-mp 5 | . . . . 5 ⊢ (!‘(3 + 1)) = ((!‘3) · (3 + 1)) |
18 | df-4 12223 | . . . . . 6 ⊢ 4 = (3 + 1) | |
19 | 18 | fveq2i 6846 | . . . . 5 ⊢ (!‘4) = (!‘(3 + 1)) |
20 | 18 | oveq2i 7369 | . . . . 5 ⊢ ((!‘3) · 4) = ((!‘3) · (3 + 1)) |
21 | 17, 19, 20 | 3eqtr4i 2771 | . . . 4 ⊢ (!‘4) = ((!‘3) · 4) |
22 | 4cn 12243 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
23 | 2cn 12233 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
24 | 2p2e4 12293 | . . . . . . . . 9 ⊢ (2 + 2) = 4 | |
25 | 22, 23, 23, 24 | subaddrii 11495 | . . . . . . . 8 ⊢ (4 − 2) = 2 |
26 | 25 | fveq2i 6846 | . . . . . . 7 ⊢ (!‘(4 − 2)) = (!‘2) |
27 | fac2 14185 | . . . . . . 7 ⊢ (!‘2) = 2 | |
28 | 26, 27 | eqtri 2761 | . . . . . 6 ⊢ (!‘(4 − 2)) = 2 |
29 | 28, 27 | oveq12i 7370 | . . . . 5 ⊢ ((!‘(4 − 2)) · (!‘2)) = (2 · 2) |
30 | 2t2e4 12322 | . . . . 5 ⊢ (2 · 2) = 4 | |
31 | 29, 30 | eqtri 2761 | . . . 4 ⊢ ((!‘(4 − 2)) · (!‘2)) = 4 |
32 | 21, 31 | oveq12i 7370 | . . 3 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = (((!‘3) · 4) / 4) |
33 | faccl 14189 | . . . . . . 7 ⊢ (3 ∈ ℕ0 → (!‘3) ∈ ℕ) | |
34 | 15, 33 | ax-mp 5 | . . . . . 6 ⊢ (!‘3) ∈ ℕ |
35 | 34 | nncni 12168 | . . . . 5 ⊢ (!‘3) ∈ ℂ |
36 | 4ne0 12266 | . . . . 5 ⊢ 4 ≠ 0 | |
37 | 35, 22, 36 | divcan4i 11907 | . . . 4 ⊢ (((!‘3) · 4) / 4) = (!‘3) |
38 | fac3 14186 | . . . 4 ⊢ (!‘3) = 6 | |
39 | 37, 38 | eqtri 2761 | . . 3 ⊢ (((!‘3) · 4) / 4) = 6 |
40 | 32, 39 | eqtri 2761 | . 2 ⊢ ((!‘4) / ((!‘(4 − 2)) · (!‘2))) = 6 |
41 | 14, 40 | eqtri 2761 | 1 ⊢ (4C2) = 6 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 + caddc 11059 · cmul 11061 ≤ cle 11195 − cmin 11390 / cdiv 11817 ℕcn 12158 2c2 12213 3c3 12214 4c4 12215 6c6 12217 ℕ0cn0 12418 ℤcz 12504 ...cfz 13430 !cfa 14179 Ccbc 14208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-seq 13913 df-fac 14180 df-bc 14209 |
This theorem is referenced by: bpoly4 15947 ex-bc 29438 5bc2eq10 40596 |
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