|   | Mathbox for metakunt | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > 5bc2eq10 | Structured version Visualization version GIF version | ||
| Description: The value of 5 choose 2. (Contributed by metakunt, 8-Jun-2024.) | 
| Ref | Expression | 
|---|---|
| 5bc2eq10 | ⊢ (5C2) = ;10 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 4nn0 12547 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
| 2 | 2z 12651 | . . . . 5 ⊢ 2 ∈ ℤ | |
| 3 | bcpasc 14361 | . . . . 5 ⊢ ((4 ∈ ℕ0 ∧ 2 ∈ ℤ) → ((4C2) + (4C(2 − 1))) = ((4 + 1)C2)) | |
| 4 | 1, 2, 3 | mp2an 692 | . . . 4 ⊢ ((4C2) + (4C(2 − 1))) = ((4 + 1)C2) | 
| 5 | 4p1e5 12413 | . . . . 5 ⊢ (4 + 1) = 5 | |
| 6 | 5 | oveq1i 7442 | . . . 4 ⊢ ((4 + 1)C2) = (5C2) | 
| 7 | 4, 6 | eqtri 2764 | . . 3 ⊢ ((4C2) + (4C(2 − 1))) = (5C2) | 
| 8 | 7 | eqcomi 2745 | . 2 ⊢ (5C2) = ((4C2) + (4C(2 − 1))) | 
| 9 | 2m1e1 12393 | . . . . 5 ⊢ (2 − 1) = 1 | |
| 10 | 9 | oveq2i 7443 | . . . 4 ⊢ (4C(2 − 1)) = (4C1) | 
| 11 | 10 | oveq2i 7443 | . . 3 ⊢ ((4C2) + (4C(2 − 1))) = ((4C2) + (4C1)) | 
| 12 | 4bc2eq6 14369 | . . . 4 ⊢ (4C2) = 6 | |
| 13 | bcn1 14353 | . . . . 5 ⊢ (4 ∈ ℕ0 → (4C1) = 4) | |
| 14 | 1, 13 | ax-mp 5 | . . . 4 ⊢ (4C1) = 4 | 
| 15 | 12, 14 | oveq12i 7444 | . . 3 ⊢ ((4C2) + (4C1)) = (6 + 4) | 
| 16 | 11, 15 | eqtri 2764 | . 2 ⊢ ((4C2) + (4C(2 − 1))) = (6 + 4) | 
| 17 | 6p4e10 12807 | . 2 ⊢ (6 + 4) = ;10 | |
| 18 | 8, 16, 17 | 3eqtri 2768 | 1 ⊢ (5C2) = ;10 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 (class class class)co 7432 0cc0 11156 1c1 11157 + caddc 11159 − cmin 11493 2c2 12322 4c4 12324 5c5 12325 6c6 12326 ℕ0cn0 12528 ℤcz 12615 ;cdc 12735 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-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-fz 13549 df-seq 14044 df-fac 14314 df-bc 14343 | 
| This theorem is referenced by: 2ap1caineq 42147 | 
| Copyright terms: Public domain | W3C validator |