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Mirrors > Home > MPE Home > Th. List > 4bc3eq4 | Structured version Visualization version GIF version |
Description: The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.) |
Ref | Expression |
---|---|
4bc3eq4 | ⊢ (4C3) = 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4cn 11300 | . . . 4 ⊢ 4 ∈ ℂ | |
2 | ax-1cn 10196 | . . . 4 ⊢ 1 ∈ ℂ | |
3 | 3cn 11297 | . . . 4 ⊢ 3 ∈ ℂ | |
4 | 3p1e4 11355 | . . . . 5 ⊢ (3 + 1) = 4 | |
5 | 3, 2, 4 | addcomli 10430 | . . . 4 ⊢ (1 + 3) = 4 |
6 | 1, 2, 3, 5 | subaddrii 10572 | . . 3 ⊢ (4 − 1) = 3 |
7 | 6 | oveq2i 6804 | . 2 ⊢ (4C(4 − 1)) = (4C3) |
8 | 4nn0 11513 | . . 3 ⊢ 4 ∈ ℕ0 | |
9 | bcnm1 13318 | . . 3 ⊢ (4 ∈ ℕ0 → (4C(4 − 1)) = 4) | |
10 | 8, 9 | ax-mp 5 | . 2 ⊢ (4C(4 − 1)) = 4 |
11 | 7, 10 | eqtr3i 2795 | 1 ⊢ (4C3) = 4 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ∈ wcel 2145 (class class class)co 6793 1c1 10139 − cmin 10468 3c3 11273 4c4 11274 ℕ0cn0 11494 Ccbc 13293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-seq 13009 df-fac 13265 df-bc 13294 |
This theorem is referenced by: bpoly4 14996 ex-bc 27651 |
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